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Multiscale analysis of slow-fast neuronal learning models with noise

Mathieu Galtier12* and Gilles Wainrib3

Author Affiliations

1 NeuroMathComp Project Team, INRIA/ENS Paris, 23 avenue d’Italie, Paris, 75013, France

2 School of Engineering and Science, Jacobs University Bremen gGmbH, College Ring 1, P.O. Box 750 561, Bremen, 28725, Germany

3 Laboratoire Analyse Géométrie et Applications, Université Paris 13, 99 avenue Jean-Baptiste Clément, Villetaneuse, Paris, France

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The Journal of Mathematical Neuroscience 2012, 2:13  doi:10.1186/2190-8567-2-13

The electronic version of this article is the complete one and can be found online at: http://www.mathematical-neuroscience.com/content/2/1/13


Received:19 April 2012
Accepted:26 October 2012
Published:22 November 2012

© 2012 M. Galtier, G. Wainrib; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper deals with the application of temporal averaging methods to recurrent networks of noisy neurons undergoing a slow and unsupervised modification of their connectivity matrix called learning. Three time-scales arise for these models: (i) the fast neuronal dynamics, (ii) the intermediate external input to the system, and (iii) the slow learning mechanisms. Based on this time-scale separation, we apply an extension of the mathematical theory of stochastic averaging with periodic forcing in order to derive a reduced deterministic model for the connectivity dynamics. We focus on a class of models where the activity is linear to understand the specificity of several learning rules (Hebbian, trace or anti-symmetric learning). In a weakly connected regime, we study the equilibrium connectivity which gathers the entire ‘knowledge’ of the network about the inputs. We develop an asymptotic method to approximate this equilibrium. We show that the symmetric part of the connectivity post-learning encodes the correlation structure of the inputs, whereas the anti-symmetric part corresponds to the cross correlation between the inputs and their time derivative. Moreover, the time-scales ratio appears as an important parameter revealing temporal correlations.

Keywords:
slow-fast systems; stochastic differential equations; inhomogeneous Markov process; averaging; model reduction; recurrent networks; unsupervised learning; Hebbian learning; STDP

1 Introduction

Complex systems are made of a large number of interacting elements leading to non-trivial behaviors. They arise in various areas of research such as biology, social sciences, physics or communication networks. In particular in neuroscience, the nervous system is composed of billions of interconnected neurons interacting with their environment. Two specific features of this class of complex systems are that (i) external inputs and (ii) internal sources of random fluctuations influence their dynamics. Their theoretical understanding is a great challenge and involves high-dimensional non-linear mathematical models integrating non-autonomous and stochastic perturbations.

Modeling these systems gives rise to many different scales both in space and in time. In particular, learning processes in the brain involve three time-scales: from neuronal activity (fast), external stimulation (intermediate) to synaptic plasticity (slow). Here, fast time-scale corresponds to a few milliseconds and slow time-scale to minutes/hour, and intermediate time-scale generally ranges between fast and slow scales, although some stimuli may be faster than neuronal activity time-scale (e.g., submilliseconds auditory signals [1]). The separation of these time-scales is an important and useful property in their study. Indeed, multiscale methods appear particularly relevant to handle and simplify such complex systems.

First, stochastic averaging principle [2,3] is a powerful tool to analyze the impact of noise on slow-fast dynamical systems. This method relies on approximating the fast dynamics by its quasi-stationary measure and averaging the slow evolution with respect to this measure. In the asymptotic regime of perfect time-scale separation, this leads to a slow reduced system whose analysis enables a better understanding of the original stochastic model.

Second, periodic averaging theory [4], which has been originally developed for celestial mechanics, is particularly relevant to study the effect of fast deterministic and periodic perturbations (external input) on dynamical systems. This method also leads to a reduced model where the external perturbation is time-averaged.

It seems appropriate to gather these two methods to address our case of a noisy and input-driven slow-fast dynamical system. This combined approach provides a novel way to understand the interactions between the three time-scales relevant in our models. More precisely, we will consider the following class of multiscale stochastic differential equations (SDEs), with <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1">View MathML</a> two small parameters

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M2">View MathML</a>

(1)

where <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M3">View MathML</a> represents the fast activity of the individual elements, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M4">View MathML</a> represents the connectivity weights that vary slowly due to plasticity, and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M5">View MathML</a> represents the value of the external input at time t. Random perturbations are included in the form of a diffusion term, and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M6">View MathML</a> is a standard Brownian motion.

We are interested in the double limit <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M7">View MathML</a> and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M8">View MathML</a> to describe the evolution of the slow variable w in the asymptotic regime where both the variable v and the external input are much faster than w. This asymptotic regime corresponds to the study of a neuronal network in which both the external input u and the neuronal activity v operate on a faster time-scale than the slow plasticity-driven evolution of synaptic weights W. To account for the possible difference of time-scales between v and the input, we introduce the time-scale ratio <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M9">View MathML</a>. In the interesting case where <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M10">View MathML</a>, one needs to understand the long-time behavior of the rescaled periodically forced SDE for any <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M11">View MathML</a> fixed

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M12">View MathML</a>

Recently, in an important contribution [5], a precise understanding of the long-time behavior of such processes has been obtained using methods from partial differential equations. In particular, conditions ensuring the existence of a periodic family of probability measures to which the law of v converges as time grows have been identified, together with a sharp estimation of the speed of mixing. These results are at the heart of the extension of the classical stochastic averaging principle [2] to the case of periodically forced slow-fast SDEs [6]. As a result, we obtain a reduced equation describing the slow evolution of variable w in the form of an ordinary differential equation,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M13">View MathML</a>

where <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M14">View MathML</a> is constructed as an average of G with respect to a specific probability measure, as explained in Section 2.

This paper first introduces the appropriate mathematical framework and then focuses on applying these multiscale methods to learning neural networks.

The individual elements of these networks are neurons or populations of neurons. A common assumption at the basis of mathematical neuroscience [7] is to model their behavior by a stochastic differential equation which is made of four different contributions: (i) an intrinsic dynamics term, (ii) a communication term, (iii) a term for the external input, and (iv) a stochastic term for the intrinsic variability. Assuming that their activity is represented by the fast variable <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M15">View MathML</a>, the first equation of system (1) is a generic representation of a neural network (function F corresponds to the first three terms contributing to the dynamics). In the literature, the level of non-linearity of the function F ranges from a linear (or almost-linear) system to spiking neuron dynamics [8], yet the structure of the system is universal.

These neurons are interconnected through a connectivity matrix which represents the strength of the synapses connecting the real neurons together. The slow modification of the connectivity between the neurons is commonly thought to be the essence of learning. Unsupervised learning rules update the connectivity exclusively based on the value of the activity variable. Therefore, this mechanism is represented by the slow equation above, where <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M16">View MathML</a> is the connectivity matrix and G is the learning rule. Probably the most famous of these rules is the Hebbian learning rule introduced in [9]. It says that if both neurons A and B are active at the same time, then the synapses from A to B and B to A should be strengthened proportionally to the product of the activity of A and B. There are many different variations of this correlation-based principle which can be found in [10,11]. Another recent, unsupervised, biologically motivated learning rule is the spike-timing-dependent plasticity (STDP) reviewed in [12]. It is similar to Hebbian learning except that it focuses on causation instead of correlation and that it occurs on a faster time-scale. Both of these types of rule correspond to G being quadratic in v.

Previous literature about dynamic learning networks is thick, yet we take a significantly different approach to understand the problem. An historical focus was the understanding of feedforward deterministic networks [13-15]. Another approach consisted in precomputing the connectivity of a recurrent network according to the principles underlying the Hebbian rule [16]. Actually, most of current research in the field is focused on STDP and is based on the precise times of the spikes, making them explicit in computations [17-20]. Our approach is different from the others regarding at least one of the following points: (i) we consider recurrent networks, (ii) we study the evolution of the coupled system activity/connectivity, and (iii) we consider bounded dynamical systems for the activity without asking them to be spiking. Besides, our approach is a rigorous mathematical analysis in a field where most results rely heavily on heuristic arguments and numerical simulations. To our knowledge, this is the first time such models expressed in a slow-fast SDE formalism are analyzed using temporal averaging principles.

The purpose of this application is to understand what the network learns from the exposition to time-dependent inputs. In other words, we are interested in the evolution of the connectivity variable, which evolves on a slow time-scale, under the influence of the external input and some noise added on the fast variable. More precisely, we intend to explicitly compute the equilibrium connectivities of such systems. This final matrix corresponds to the knowledge the network has extracted from the inputs. Although the derivation of the results is mathematically tough for untrained readers, we have tried to extract widely understandable conclusions from our mathematical results and we believe this paper brings novel elements to the debate about the role and mechanisms of learning in large scale networks.

Although the averaging method is a generic principle, we have made significant assumptions to keep the analysis of the averaged system mathematically tractable. In particular, we will assume that the activity evolves according to a linear stochastic differential equation. This is not very realistic when modeling individual neurons, but it seems more reasonable to model populations of neurons; see Chapter 11 of [7].

The paper is organized as follows. Section 2 is devoted to introducing the temporal averaging theory. Theorem 2.2 is the main result of this section. It provides the technical tool to tackle learning neural networks. Section 3 corresponds to application of the mathematical tools developed in the previous section onto the models of learning neural networks. A generic model is described and three different particular models of increasing complexity are analyzed. First, Hebbian learning, then trace-learning, and finally STDP learning are analyzed for linear activities. Finally, Section 4 is a discussion of the consequences of the previous results from the viewpoint of their biological interpretation.

2 Averaging principles: theory

In this section, we present multiscale theoretical results concerning stochastic averaging of periodically forced SDEs (Section 2.3). These results combine ideas from singular perturbations, classical periodic averaging and stochastic averaging principles. Therefore, we recall briefly, in Sections 2.1 and 2.2, several basic features of these principles, providing several examples that are closely related to the application developed in Section 3.

2.1 Periodic averaging principle

We present here an example of a slow-fast ordinary differential equation perturbed by a fast external periodic input. We have chosen this example since it readily illustrates many ideas that will be developed in the following sections. In particular, this example shows how the ratio between the time-scale separation of the system and the time-scale of the input appears as a new crucial parameter.

Example 2.1 Consider the following linear time-inhomogeneous dynamical system with <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M17">View MathML</a> two parameters:

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M18">View MathML</a>

This system is particularly handy since one can solve analytically the first ordinary differential equation, that is,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M19">View MathML</a>

where we have introduced the time-scales ratio

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M20">View MathML</a>

In this system, one can distinguish various asymptotic regimes when <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M21">View MathML</a> and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M22">View MathML</a> are small according to the asymptotic value of μ:

• Regime 1: Slow input <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M23">View MathML</a>:

First, if <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M7">View MathML</a> and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M22">View MathML</a> is fixed, then <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M26">View MathML</a> is close to <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M27">View MathML</a>, and from geometric singular perturbation theory[21,22] one can approximate the slow variable <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M28">View MathML</a> by the solution of

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M29">View MathML</a>

Now taking the limit <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M8">View MathML</a> and applying the classical averaging principle[4] for periodically driven differential equations, one can approximate <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M28">View MathML</a> by the solution of

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M32">View MathML</a>

since <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M33">View MathML</a>.

• Regime 2: Fast input <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M34">View MathML</a>:

If <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M8">View MathML</a> and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M21">View MathML</a> is fixed, then the classical averaging principle implies that <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M37">View MathML</a> is close to the solution of

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M38">View MathML</a>

so that <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M28">View MathML</a> can be approximated by

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M40">View MathML</a>

and when <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M7">View MathML</a>, one does not recover the same asymptotic behavior as in Regime 1.

• Regime 3: Time-scales matching <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M42">View MathML</a>:

Now consider the intermediate case where <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M21">View MathML</a> is asymptotically proportional to <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M22">View MathML</a>. In this case, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M37">View MathML</a> can be approximated on the fast time-scale <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M46">View MathML</a> by the periodic solution <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M47">View MathML</a> of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M48">View MathML</a>. As a consequence, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M28">View MathML</a> will be close to the solution of

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M50">View MathML</a>

since <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M51">View MathML</a>.

Thus, we have seen in this example that

1. the two limits <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M7">View MathML</a> and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M8">View MathML</a> do not commute,

2. the ratio μ between the internal time-scale separation <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M21">View MathML</a> and the input time-scale <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M22">View MathML</a> is a key parameter in the study of slow-fast systems subject to a time-dependent perturbation.

2.2 Stochastic averaging principle

Time-scales separation is a key property to investigate the dynamical behavior of non-linear multiscale systems, with techniques ranging from averaging principles to geometric singular perturbation theory. This property appears to be also crucial to understanding the impact of noise. Instead of carrying a small noise analysis, a multiscale approach based on the stochastic averaging principle[2] can be a powerful tool to unravel subtle interplays between noise properties and non-linearities. More precisely, consider a system of SDEs in <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M56">View MathML</a>:

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M57">View MathML</a>

with initial conditions <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M58">View MathML</a>, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M59">View MathML</a>, and where <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M60">View MathML</a> is called the slow variable, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M61">View MathML</a> is the fast variable, with F, G, Σ smooth functions ensuring the existence and uniqueness for the solution <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M62">View MathML</a>, and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M63">View MathML</a> a p-dimensional standard Brownian motion, defined on a filtered probability space <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M64">View MathML</a>. Time-scale separation in encoded in the small parameter ϵ, which denotes in this section a single positive real number.

In order to approximate the behavior of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M65">View MathML</a> for small ϵ, the idea is to average out the equation for the slow variable with respect to the stationary distribution of the fast one. More precisely, one first assumes that for each <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M66">View MathML</a> fixed, the frozen fast SDE,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M67">View MathML</a>

admits a unique invariant measure, denoted <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M68">View MathML</a>. Then, one defines the averaged drift vector field <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M14">View MathML</a>

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M70">View MathML</a>

(2)

and w the solution of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M71">View MathML</a> with the initial condition <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M72">View MathML</a>. Under some dissipativity assumptions, the stochastic averaging principle [2] states:

Theorem 2.1For any<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M73">View MathML</a>and<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M74">View MathML</a>,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M75">View MathML</a>

(3)

As a consequence, analyzing the behavior of the deterministic solution w can help to understand useful features of the stochastic process <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M62">View MathML</a>.

Example 2.2 In this example we consider a similar system as in Example 2.1, but with a noise term instead of the periodic perturbation. Namely, we consider <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M62">View MathML</a> the solution of the system of SDEs,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M78">View MathML</a>

with <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M79">View MathML</a> a small parameter and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M80">View MathML</a> a positive constant. From Theorem 2.1, the stochastic slow variable <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M81">View MathML</a> can be approximated in the sense of (3) by the deterministic solution w of

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M82">View MathML</a>

where <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M83">View MathML</a> is the stationary measure of the linear diffusion process,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M84">View MathML</a>

that is,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M85">View MathML</a>

Consequently, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M81">View MathML</a> can be approximated in the limit <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M87">View MathML</a> by the solution of

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M88">View MathML</a>

Applying (3) leads to the following result: for any <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M74">View MathML</a> and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M73">View MathML</a>,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M91">View MathML</a>

Interestingly, the asymptotic behavior of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M81">View MathML</a> for small ϵ is characterized by a deterministic trajectory that depends on the strength σ of the noise applied to the system. Thus, the stochastic averaging principle appears particularly interesting when unraveling the impact of noise strength on slow-fast systems.

Many other results have been developed since, extending the set-up to the case where the slow variable has a diffusion component or to infinite-dimensional settings for instance, and also refining the convergence study, providing homogenization results concerning the limit of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M93">View MathML</a> or establishing large deviation principles (see [23] for a recent monograph). However, fewer results are available in the case of non-homogeneous SDEs, that is, when the system is perturbed by an external time-dependent signal. This setting is of particular interest in the framework of stochastic learning models, and we present the main relevant mathematical results in the following section.

2.3 Double averaging principle

Combining ideas of periodic and stochastic averaging introduced previously, we present here theoretical results concerning multiscale SDEs driven by an external time-periodic input. Consider <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M94">View MathML</a> the solution of

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M95">View MathML</a>

(4)

with <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M96">View MathML</a> a τ-periodic function and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M97">View MathML</a>. Parameter <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M21">View MathML</a> represents the internal time-scale separation and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M22">View MathML</a> the input time-scale. We consider the case where both <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M21">View MathML</a> and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M22">View MathML</a> are small, that is, a strong time-scale separation between the fast variable <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M3">View MathML</a> and the slow one <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M4">View MathML</a>, and a fast periodic modulation of the fast drift <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M104">View MathML</a>.

We further denote <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M105">View MathML</a>.

Definition 2.1 We define the asymptotic time-scale ratio

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M106">View MathML</a>

(5)

Accordingly, we denote <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M107">View MathML</a> the distinguished limit when <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M7">View MathML</a>, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M8">View MathML</a> with <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M110">View MathML</a>.

The following assumption is made to ensure existence and uniqueness of a strong solution to system (4). In the following, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M111">View MathML</a> will denote the usual scalar product for vectors.

Assumption 2.1 Existence and uniqueness of a strong solution

(i) The functions F, G, and Σ are locally Lipschitz continuous in the space variable z. More precisely, for any <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M112">View MathML</a>, there exists a constant <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M113">View MathML</a> such that

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M114">View MathML</a>

(ii) There exists a constant <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M112">View MathML</a> such that

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M116">View MathML</a>

To control the asymptotic behavior of the fast variable, one further assumes the following.

Assumption 2.2 Asymptotic behavior of the fast process:

(i) The diffusion matrix Σ is bounded

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M117">View MathML</a>

and uniformly non-degenerate

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M118">View MathML</a>

(ii) There exists <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M119">View MathML</a> such that for all <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M120">View MathML</a> and for all <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M121">View MathML</a>,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M122">View MathML</a>

According to the value of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M123">View MathML</a>, the stochastic averaging principle is based on a description of the asymptotic behavior of various rescaled fast frozen processes. More precisely, under Assumptions 2.1 and 2.2, one can deduce that:

• For any fixed <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M124">View MathML</a> and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M125">View MathML</a> fixed, the law of the rescaled time-homogeneous frozen process,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M126">View MathML</a>

converges exponentially fast to a unique invariant probability measure denoted by <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M127">View MathML</a>.

• For any fixed <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M124">View MathML</a>, there exists a <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M129">View MathML</a>-periodic evolution system of measures <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M130">View MathML</a>, different from <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M131">View MathML</a> above, such that the law of the rescaled time-inhomogeneous frozen process,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M132">View MathML</a>

(6)

converges exponentially fast towards <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M133">View MathML</a>, uniformly with respect to <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M11">View MathML</a> (cf. the Appendix Theorem A.1).

• For any fixed <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M124">View MathML</a>, the law of the rescaled time-homogeneous frozen process,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M136">View MathML</a>

where <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M137">View MathML</a>, converges exponentially fast towards a unique invariant probability measure denoted by <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M138">View MathML</a>.

According to the value of μ, we introduce a vector field <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M139">View MathML</a> which will play a role similar to <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M14">View MathML</a> introduced in equation (2).

Definition 2.2 We define <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M141">View MathML</a> as follows. In the time-scale matching case, that is, when <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M142">View MathML</a>, then

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M143">View MathML</a>

(7)

Notation We may denote the periodic system of measures <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M144">View MathML</a> associated with (6) by <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M145">View MathML</a> to emphasize its relationship with F and Σ. Accordingly, we may denote <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M146">View MathML</a> by <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M147">View MathML</a>.

We are now able to present our main mathematical result. Extending Theorem 2.1, the following theorem describes the asymptotic behavior of the slow variable <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M148">View MathML</a> when <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M87">View MathML</a> with <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M110">View MathML</a>. We refer to [6] for more details about the full mathematical proof of this result.

Theorem 2.2Let<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M10">View MathML</a>. Ifwis the solution of

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M152">View MathML</a>

(8)

then the following convergence result holds, for all<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M74">View MathML</a>and<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M73">View MathML</a>:

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M155">View MathML</a>

Remark 2.1

1. The extremal cases <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M156">View MathML</a> and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M34">View MathML</a> are not covered in full rigor by Theorem 2.2. However, the study of the sequential limits <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M7">View MathML</a> followed by <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M159">View MathML</a> or <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M8">View MathML</a> followed by <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M161">View MathML</a> can be deduced from an appropriate combination of classical periodic and stochastic averaging theorems:

• Slow input: If one considers the case where the limit <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M7">View MathML</a> is taken first, so that from Theorem 2.1 with fast variable <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M163">View MathML</a> and slow variables <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M164">View MathML</a> and t (with the trivial equation <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M165">View MathML</a>), <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M148">View MathML</a> is close in probability on finite time-intervals to the solution of the following inhomogeneous ordinary differential equation:

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M167">View MathML</a>

Then taking the limit <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M8">View MathML</a>, one can apply the deterministic averaging principle to the fast periodic vector field <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M169">View MathML</a>, so that <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M170">View MathML</a> converges when <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M8">View MathML</a> to the solution of

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M172">View MathML</a>

where

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M173">View MathML</a>

• Fast input: If the limit <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M174">View MathML</a> is taken first, one first has to perform a classical averaging of the periodic drift <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M175">View MathML</a> leading to the homogeneous system of SDEs (4), but with <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M176">View MathML</a> instead of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M175">View MathML</a>. Then, an application of Theorem 2.1 on this system gives an averaged vector field

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M178">View MathML</a>

2. To study the extremal cases <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M23">View MathML</a> and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M34">View MathML</a> in full generality, one would need to consider all the possible relationships between <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M181">View MathML</a> and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M182">View MathML</a>, not only the linear one as in the present article, but also of the type <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M183">View MathML</a> for example. In this case, we believe that the regime <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M184">View MathML</a> converges to the same limit as taking the limit <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M182">View MathML</a> first and the regime <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M186">View MathML</a> corresponds to taking the limit <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M181">View MathML</a> first. The intermediate regime <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M188">View MathML</a> seems to be the only one for which the limit cannot be obtained by combining classical averaging principles. Therefore, the present article is focused on this case, in which the averaged system depends explicitly on the scaling parameter μ. Moreover, in terms of applications, this parameter can have a relatively easy interpretation in terms of the ratio of time-scales between intrinsic neuronal activity and typical stimulus time-scales in a given situation. Although the zeroth order limit (i.e., the averaged system) seems to depend only on the position of α with respect to 1, it seems reasonable to expect that the fluctuations around the limit would depend on the precise value of α. This is a difficult question which may deserve further analysis.

The case <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M189">View MathML</a> is already very rich in the sense that it combines simultaneously both the periodic and stochastic averaging principles in a new way that cannot be recovered by sequential applications of those principles. A particular role is played by the frozen periodically-forced SDE (6). The equivalent of the quasi-stationary measure <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M190">View MathML</a> of Theorem 2.1 is given by the asymptotically periodic behavior of equation (6), represented by the periodic family of measures <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M144">View MathML</a>.

3. By a rescaling of the frozen process (6), one deduces the following scaling relationships:

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M192">View MathML</a>

and

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M193">View MathML</a>

Therefore, if one knows, in the case <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M194">View MathML</a>, the averaged vector field associated with the fast process generated by a drift F and a diffusion coefficient σ, denoted <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M195">View MathML</a>, it is possible to deduce <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M139">View MathML</a> in the general case <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M197">View MathML</a> with a change <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M198">View MathML</a> and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M199">View MathML</a>.

4. It seems reasonable to expect that the above result is still valid when considering ergodic, but not necessarily periodic, time dependency of the function <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M200">View MathML</a>. In equation (7), instead of integrating <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M144">View MathML</a> over one period, one should integrate it with respect to an ergodic stationary measure. However, this extension requires non-trivial technical improvements of [5] which are beyond the scope of this paper.

2.3.1 Case of a fast linear SDE with periodic input

We present here an elementary case where one can compute explicitly the quasi-stationary time-periodic family of measures <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M202">View MathML</a>, when the equation for the fast variable is linear. Namely, we consider <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M203">View MathML</a> the solution of

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M204">View MathML</a>

with initial condition <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M205">View MathML</a>, and where <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M206">View MathML</a> is a matrix whose eigenvalues have positive real parts and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M207">View MathML</a> is a τ-periodic function.

We are interested in the large time behavior of the law of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M208">View MathML</a>, which is a time-inhomogeneous Ornstein-Uhlenbeck process. From [5] we know that its law converges to a τ-periodic family of probability measures <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M209">View MathML</a>. Due to the linearity in the previous equation, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M209">View MathML</a> is Gaussian with a time-dependent mean and a constant covariance matrix

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M211">View MathML</a>

where <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M212">View MathML</a> is the <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M213">View MathML</a>-periodic attractor of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M214">View MathML</a>, i.e.,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M215">View MathML</a>

and Q is the unique solution of the Lyapunov equation

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M216">View MathML</a>

(9)

Indeed, if one denotes <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M217">View MathML</a>, then <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M218">View MathML</a> is a solution of the classical homogeneous Ornstein-Uhlenbeck equation

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M219">View MathML</a>

whose stationary distribution is known to be a centered Gaussian measure with the covariance matrix Q solution of (9); see Chapter 3.2 of [24]. Notice that if A is self-adjoint with respect to <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M220">View MathML</a> (i.e., <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M221">View MathML</a>), then the solution is <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M222">View MathML</a>, which will be used in Appendix B.2.

Hence, in the linear case, the averaged vector field of equation (7) becomes

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M223">View MathML</a>

(10)

where <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M224">View MathML</a> is the probability density function of the Gaussian law with mean <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M225">View MathML</a> and covariance <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M226">View MathML</a>.

Therefore, due to the linearity of the fast SDE, the periodic system of measure ν is just a constant Gaussian distribution shifted by a periodic function of time <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M227">View MathML</a>. In case G is quadratic in v, this remark implies that one can perform independently the integral over time and over <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M228">View MathML</a> in formula (10) (noting that the crossed term has a zero average). In this case, contributions from the periodic input and from noise appear in the averaged vector field in an additive way.

Example 2.3 In this last example, we consider a combination between Example 2.1 and Example 2.2, namely we consider the following system of periodically forced SDEs:

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M229">View MathML</a>

As in Example 2.1 and as shown above, the behavior of this system when both <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M21">View MathML</a> and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M22">View MathML</a> are small depends on the parameter μ defined in (5). More precisely, we have the following three regimes:

• Regime 1: slow input:

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M232">View MathML</a>

• Regime 2: fast input:

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M233">View MathML</a>

• Regime 3: time-scale matching:

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M234">View MathML</a>

2.4 Truncation and asymptotic well-posedness

In some cases, Assumptions 2.1-2.2 may not be satisfied on the entire phase space <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M235">View MathML</a>, but only on a subset. Such situations will appear in Section 3 when considering learning models. We introduce here a more refined set of assumptions ensuring that Theorem 2.2 still applies.

Let us start with an example, namely the following bi-dimensional system with white noise input:

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M236">View MathML</a>

(11)

with <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M79">View MathML</a>, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M80">View MathML</a>, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M239">View MathML</a>, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M240">View MathML</a>.

For the fast drift <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M241">View MathML</a> to be non-explosive, it is necessary to have <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M242">View MathML</a> with <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M243">View MathML</a> for all time. The concern about this system comes from the fact that the slow variable w may reach l due to the fluctuations captured in the term <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M244">View MathML</a>, for instance, if κ is not large enough. Such a system may have exponentially growing trajectories. However, we claim that for small enough ϵ, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M81">View MathML</a> will remain close to its averaged limit w for a very long time, and if this limit remains below <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M246">View MathML</a>, then <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M81">View MathML</a> can be considered as well-posed in the asymptotic limit <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M87">View MathML</a>. To make this argument more rigorous, we suggest the following definition.

Definition 2.3 A stochastic differential equation with a given initial condition is asymptotically well posed in probability if for the given initial condition,

1. a unique solution exists until a random time <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M249">View MathML</a>

2. for all <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M74">View MathML</a>,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M251">View MathML</a>

We give in the following proposition sufficient conditions for system (4) to be asymptotically well posed in probability and to satisfy conclusions of Theorem 2.2.

Let us introduce the following set of additional assumptions.

Assumption 2.3 Moment conditions:

(i) There exists <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M252">View MathML</a> such that

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M253">View MathML</a>

(ii) For any <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M74">View MathML</a> and any bounded subset K of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M255">View MathML</a>,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M256">View MathML</a>

Remark 2.2 This last set of assumptions will be satisfied in all the applications of Section 3 since we consider linear models with additive noise for the equation of v, implying this variable to be Gaussian and the function G only involves quadratic moments of v; therefore, the moment conditions (i) and (ii) will be satisfied without any difficulty. Moreover, if one considers non-linear models for the variable v, then the Gaussian property may be lost; however, adding sigmoidal non-linearity has, in general, the effect of bounding the dynamics, thus making these moment assumptions reasonable to check in most models of interest.

Property 2.3If there exists a subsetof<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M255">View MathML</a>such that

1. The functionsF, G, Σsatisfy Assumptions 2.1-2.3 restricted on<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M258">View MathML</a>.

2. ℰ is invariant under the flow of<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M139">View MathML</a>, as defined in (7).

Then for any initial condition<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M260">View MathML</a>, system (4) is asymptotically well posed in probability and<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M148">View MathML</a>satisfies the conclusion of Theorem 2.2.

Proof See Appendix A.2. □

Here, we show that it applies to system (11). First, with <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M262">View MathML</a>, for some <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M263">View MathML</a>, it is possible to show that Assumptions 2.1-2.2 are satisfied on <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M264">View MathML</a>. Then, as a special case of (10), we obtain the following averaged system:

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M265">View MathML</a>

It remains to check that the solution of this system satisfies

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M266">View MathML</a>

that is, the subset <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M267">View MathML</a> is invariant under the flow of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M14">View MathML</a>.

This property is satisfied as soon as

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M269">View MathML</a>

Indeed, one can show that <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M270">View MathML</a> admits two solutions iff <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M271">View MathML</a>,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M272">View MathML</a>

and that <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M273">View MathML</a> is stable whereas <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M274">View MathML</a> is unstable. Thus, if <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M275">View MathML</a> with <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M276">View MathML</a>, then <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M277">View MathML</a> for all <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M278">View MathML</a>. In fact, the invariance property is true for all <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M279">View MathML</a>.

3 Averaging learning neural networks

In this section, we apply the temporal averaging methods derived in Section 2 on models of unsupervised learning neural networks. First, we design a generic learning model and show that one can define formally an averaged system with equation (7). However, going beyond the mere definition of the averaged system seems very difficult and we only manage to get explicit results for simple systems where the fast activity dynamics is linear. In the three last subsections, we push the analysis for three examples of increasing complexity.

In the following, we always consider that the initial connectivity is 0. This is an arbitrary choice but without consequences, because we focus on the regime where there is a single globally stable equilibrium point (see Section 3.2.3).

3.1 A generic learning neural network

We now introduce a large class of stochastic neuronal networks with learning models. They are defined as coupled systems describing the simultaneous evolution of the activity of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M280">View MathML</a> neurons and the connectivity between them. We define <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M15">View MathML</a>, the activity field of the network, and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M282">View MathML</a>, the connectivity matrix.

Each neuron variable <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M283">View MathML</a> is assumed to follow the SDE

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M284">View MathML</a>

where the function <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M285">View MathML</a> characterizes the intrinsic non-linear dynamical behavior of neuron i and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M286">View MathML</a> is the input received by neuron i. The stochastic term <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M287">View MathML</a> is added to account for internal sources of noise. In terms of notations, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M288">View MathML</a> is a standard n-dimensional Brownian motion, Σ is an <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M289">View MathML</a> matrix, possibly function of v or other variables, and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M287">View MathML</a> denotes the ith component of the vector <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M291">View MathML</a>.

The input <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M286">View MathML</a> to neuron i has mainly two components: the external input <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M293">View MathML</a> and the input coming from other neurons in the network <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M294">View MathML</a>. The latter is a priori a complex combination of post-synaptic potentials coming from many other neurons. The coefficient <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M295">View MathML</a> of the connectivity matrix accounts for the strength of a synapse <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M296">View MathML</a>. Note that neurons can be connected to themselves, i.e., <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M297">View MathML</a> is not necessarily null. Thus, we can write

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M298">View MathML</a>

where <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M299">View MathML</a> and ℋ is a function taking the history of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M283">View MathML</a> and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M301">View MathML</a> and returning a real for each time t (to take convolutions into account). In practical cases, they are often taken to be sigmoidal functions. We abusively redefine <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M302">View MathML</a> and ℋ as vector valued operators corresponding to the element-wise application of their real counterparts. We also define the function <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M303">View MathML</a> such that <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M304">View MathML</a>. Together with a slow generic learning rule, this leads to defining a stochastic learning model as the following system of SDEs.

Definition 3.1

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M305">View MathML</a>

Before applying the general theory of Section 2, let us make several comments about this generic model of neural network with learning. This model is a non-autonomous, stochastic, non-linear slow-fast system.

In order to apply Theorem 2.2, one needs Assumptions 2.1, 2.2, and 2.3 to be satisfied, restricting the space of possible functions <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M302">View MathML</a>, ℋ, ℱ, Σ, and G. In particular, Assumption 2.2(ii) seems rather restrictive since it excludes systems with multiple equilibria and suggests that the general theory is more suited to deal with rate-based networks. However, one should keep in mind that these assumptions are only sufficient, and that the double averaging principle may work as well in systems which do not satisfy readily those assumptions.

As we will show in Section 3.3, a particular form of history-dependence can be taken into account, to a certain extent. Indeed, for instance, if the function ℱ is actually a functional of the past trajectory of variable <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M307">View MathML</a> which can be expressed as the solution of an additional SDE, then it may be possible to include a certain form of history-dependence. However, purely time-delayed systems do not enter the scope of this theory, although it might be possible to derive an analogous averaging method in this framework.

The noise term can be purely additive or set by a particular function <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M308">View MathML</a> as long as it satisfies Assumption 2.2(i), meaning that it must be uniformly non-degenerate.

In the following subsection, we apply the averaging theory to various combinations of neuronal network models, embodied by choices of functions <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M302">View MathML</a>, ℋ, ℱ, Σ, and various learning rules, embodied by a choice of the function G. We will also analyze the obtained averaged system, describing the slow dynamics of the connectivity matrix in the limit of perfect time-scale separation and, in particular, study the convergence of this averaged system to an equilibrium point.

3.2 Symmetric Hebbian learning

One of the simplest, yet non-trivial, stochastic learning models is obtained when considering

• A linear model for neuronal activity, namely <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M310">View MathML</a> with l a positive constant.

• A linear model for the synaptic transmission, namely <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M311">View MathML</a> and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M312">View MathML</a>.

• A constant diffusion matrix Σ (additive noise) proportional to the identity <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M313">View MathML</a> (spatially uncorrelated noise).

• A Hebbian learning rule with linear decay, namely <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M314">View MathML</a>. Actually, it corresponds to the tensor product: <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M315">View MathML</a>.

This model can be written as follows:

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M316">View MathML</a>

(12)

where neurons are assumed to have the same decay constant: <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M317">View MathML</a>; u is a periodic continuous input (it replaces <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M318">View MathML</a> in the previous section); <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M319">View MathML</a> with <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M320">View MathML</a> and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M63">View MathML</a> is n-dimensional Brownian noise.

The first question that arises is about the well-posedness of the system: What is the definition interval of the solutions of system (12)? Do they explode in finite time? At first sight, it seems there may be a runaway of the solution if the largest real part among the eigenvalues of W grows bigger than l. In fact, it turns out this scenario can be avoided if the following assumption linking the parameters of the system is satisfied.

Assumption 3.1 There exists <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M322">View MathML</a> such that

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M323">View MathML</a>

where <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M324">View MathML</a>.

It corresponds to making sure the external (i.e., <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M325">View MathML</a>) or internal (i.e., σ) excitations are not too large compared to the decay mechanism (represented by κ and l). Note that if <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M322">View MathML</a>, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M325">View MathML</a> and d are fixed, it is sufficient to increase κ or l for this assumption to be satisfied.

Under this assumption, the space

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M328">View MathML</a>

is invariant by the flow of the averaged system <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M14">View MathML</a>, where <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M330">View MathML</a> means W is semi-definite positive and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M331">View MathML</a> means <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M332">View MathML</a> is definite positive. Therefore, the averaged system is defined and bounded on <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M333">View MathML</a>. The slow/fast system being asymptotically close to the averaged system, it is therefore asymptotically well-defined in probability. This is summarized in the following theorem.

Theorem 3.1If Assumption 3.1 is verified for<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M334">View MathML</a>, then system (12) is asymptotically well posed in probability and the connectivity matrix<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M335">View MathML</a>, the solution of system (12), converges toWin the sense that for all<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M336">View MathML</a>,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M337">View MathML</a>

whereWis the deterministic solution of

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M338">View MathML</a>

(13)

where<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M339">View MathML</a>is the<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M213">View MathML</a>-periodic attractor of<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M341">View MathML</a>, where<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M342">View MathML</a>is supposed to be fixed.

Proof See Theorem B.1 in Appendix B.2. □

In the following, we focus on the averaged system described by (13). Its right-hand side is made of three terms: a linear and homogeneous decay, a correlation term, and a noise term. The last two terms are made explicit in the following.

3.2.1 Noise term

As seen in Section 2, in the linear case, the noise term Q is the unique solution of the Lyapunov equation (9) with <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M343">View MathML</a> and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M344">View MathML</a>. Because the noise is spatially uncorrelated and identical for each neuron and also because the connectivity is symmetric, observe that <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M345">View MathML</a> is the unique solution of the system.

In more complicated cases, the computation of this term appears to be much more difficult as we will see in Section 3.4.

3.2.2 Correlation term

This term corresponds to the auto-correlation of neuronal activity. It is only implicitly defined; thus, this section is devoted to finding an explicit form depending only on the parameters l, μ, τ, the connectivity W, and the inputs u. Actually, one can perform an expansion of this term with respect to a small parameter corresponding to a weakly connected expansion. Most terms vanish if the connectivity W is small compared to the strength of the intrinsic decaying dynamics of neurons l.

The auto-correlation term of a <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M213">View MathML</a>-periodic function can be rewritten as

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M347">View MathML</a>

With this notation, it is simple to think of v as a ‘semi-continuous matrix’ of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M348">View MathML</a>. Hence, the operator ‘⋅’ can be though of as a matrix multiplication. Similarly, the transpose operator turns a matrix <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M349">View MathML</a> into a matrix <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M350">View MathML</a>. See Appendix B.1 for details about the notations.

It is common knowledge, see [17] for instance, that this term gathers information about the correlation of the inputs. Indeed, if we assume that the input is sufficiently slow, then <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M212">View MathML</a> has enough time to converge to <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M352">View MathML</a> for all <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M353">View MathML</a>. Therefore, in the first order <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M354">View MathML</a>. This leads to <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M355">View MathML</a>. In the weakly connected regime, one can assume that <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M356">View MathML</a> leading to <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M357">View MathML</a> which is the auto-correlation of the inputs.

Actually, without the assumption of a slow input, lagged correlations of the input appear in the averaged system. Before giving the expression of these temporal correlations, we need to introduce some notations. First, define the convolution filter <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M358">View MathML</a>, where H is the Heaviside function. This family of functions is displayed for different values of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M359">View MathML</a> in Figure 4(a). Note that <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M360">View MathML</a> when <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M361">View MathML</a>, where <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M362">View MathML</a> is the Dirac distribution centered at the origin. In this asymptotic regime, the convolution filter and its iterates <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M363">View MathML</a> are equal to the identity.

We also define the filtered correlation of the inputs <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M364">View MathML</a> by

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M365">View MathML</a>

where <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M366">View MathML</a> is the kth convolution of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M367">View MathML</a> with itself and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M368">View MathML</a>. This is the correlation matrix of the inputs filtered by two different functions. It is easy to show that this is similar to computing the cross-correlation of the inputs with the inputs filtered by another function,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M369">View MathML</a>

(14)

which motivates the definition of the <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M370">View MathML</a>-temporal profile <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M371">View MathML</a>, where <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M372">View MathML</a>. This notation is deliberately similar to that of the transpose operator we use in the proofs. These functions are shown in Figure 1. We have not found a way to make them explicit; therefore, the following remarks are simply based on numerical illustrations. When <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M373">View MathML</a>, the temporal profiles are centered. The larger the difference <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M374">View MathML</a>, the larger the center of the bell. The larger the sum <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M375">View MathML</a>, the larger the standard deviation. This motivates the idea that <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M376">View MathML</a> can be thought of as the <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M374">View MathML</a> lagged correlation of the inputs. One can also say that <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M378">View MathML</a> is more blurred than <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M379">View MathML</a> in the sense that the inputs are temporally integrated over a ‘wider’ window in the first case.

thumbnailFig. 1. This shows the <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M380">View MathML</a>-temporal profiles with <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M381">View MathML</a>, i.e., the functions <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M382">View MathML</a> for <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M383">View MathML</a> and k ranging from 0 to 6. For <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M384">View MathML</a>, the temporal profile is even and this also occurs to be true for any <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M373">View MathML</a>. When <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M386">View MathML</a>, the function reaches its maximum for strictly positive values that grow with the difference <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M374">View MathML</a>. Besides, the temporal profiles are flattened when <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M375">View MathML</a> increases.

Observe that <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M389">View MathML</a>. Therefore, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M390">View MathML</a>. Thanks to Young’s inequality for convolutions, which says that <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M391">View MathML</a>, it can be proved that <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M392">View MathML</a>.

We intend to express the correlation term as an infinite converging sum involving these filtered correlations. In this perspective, we use a result we have proved in [25] to write the solution of a general class of non-autonomous linear systems (e.g., <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M393">View MathML</a>) as an infinite sum, in the case <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M194">View MathML</a>.

Lemma 3.2If<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M212">View MathML</a>is the solution, with zero as initial condition, of<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M396">View MathML</a>it can be written by the sum below which converges ifWis in<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M397">View MathML</a>for<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M322">View MathML</a>.

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M399">View MathML</a>

where<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M400">View MathML</a>.

Proof See Lemma B.2 in Appendix B.2. □

This is a decomposition of the solution of a linear differential system on the basis of operators where the spatial and temporal parts are decoupled. This important step in a detailed study of the averaged equation cannot be achieved easily in models with non-linear activity. Everything is now set up to introduce the explicit expansion of the correlation we are using in what follows. Indeed, we use the previous result to rewrite the correlation term as follows.

Property 3.3The correlation term can be written

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M401">View MathML</a>

Proof See Theorem B.3 in Appendix B.2. □

This infinite sum of convolved filters is reminiscent of a property of Hawkes processes that have a linear input-output gain [26].

The speed of inputs characterized by μ only appears in the temporal profiles <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M402">View MathML</a>. In particular, if the inputs are much slower than neuronal activity time-scale, i.e., <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M156">View MathML</a>, then <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M404">View MathML</a> and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M405">View MathML</a>. Therefore, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M406">View MathML</a> and the sums in the formula of Property 3.3 are separable, leading to <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M407">View MathML</a>, which corresponds to the heuristic result previously explained.

Therefore, the averaged equation can be explicitly rewritten

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M408">View MathML</a>

(15)

In Figure 2, we illustrate this result by comparing, for different <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M409">View MathML</a> (i.e., we choose <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M410">View MathML</a> in this example), the stochastic system and its averaged version. The above decomposition has been used as the basis for numerical computation of trajectories of the averaged system.

thumbnailFig. 2. The first two figures, (a) and (b), represent the evolution of the connectivity for original stochastic system (12), superimposed with averaged system (13), for two different values of ϵ: respectively <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M411">View MathML</a> and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M412">View MathML</a>, where we have chosen <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M413">View MathML</a>. Each color corresponds to the weight of an edge in a network made of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M414">View MathML</a> neurons. As expected, it seems that the smaller ϵ, the better the approximation. This can be seen in the picture (c) where we have plotted the precision on the y-axis and ϵ on the x-axis. The parameters used here are <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M415">View MathML</a>, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M416">View MathML</a>, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M417">View MathML</a>, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M418">View MathML</a>. The inputs have a random (but frozen) spatial structure and evolve according to a sinusoidal function.

3.2.3 Global stability of the equilibrium point

Now that we have found an explicit formulation for the averaged system, it is natural to study its dynamics. Actually, we prove in the following that if the connectivity W is kept smaller than <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M419">View MathML</a>, i.e., Assumption 3.1 is verified with <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M420">View MathML</a>, then the dynamics is trivial: the system converges to a single equilibrium point. Indeed, under the previous assumption, the system can be written <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M421">View MathML</a>, where F is a contraction operator on <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M422">View MathML</a>. Therefore, one can prove the uniqueness of the fixed point with the Banach fixed point argument and exhibit an energy function for the system.

Theorem 3.4If Assumption 3.1 is verified for<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M420">View MathML</a>, then there is a unique equilibrium point in the invariant subset<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M397">View MathML</a>which is globally, asymptotically stable.

Proof See Theorem B.4 in Appendix B.2. □

The fact that the equilibrium point is unique means that the ‘knowledge’ of the network about its environment (corresponding by hypothesis to the connectivity) eventually is unique. For a given input and any initial condition, the network can only converge to the same ‘knowledge’ or ‘understanding’ of this input.

3.2.4 Explicit expansion of the equilibrium point

When the network is weakly connected, the high-order terms in expansion (15) may be neglected. In this section, we follow this idea and find an explicit expansion for the equilibrium connectivity where the strength of the connectivity is the small parameter enabling the expansion. The weaker the connectivity, the more terms can be neglected in the expansion.

In fact, it is not natural to speak about a weakly connected learning network since the connectivity is a variable. However, we are able to identify a weak connectivity index which controls the strength of the connectivity. We say the connectivity is weak when it is negligible compared to the intrinsic leak term, i.e., <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M425">View MathML</a> is small. We show in the Appendix that this weak connectivity index depends only on the parameters of the network and can be written

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M426">View MathML</a>

In the asymptotic regime <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M427">View MathML</a>, we have <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M428">View MathML</a>. This index is the ‘small’ parameter needed to perform the expansion. We also define <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M429">View MathML</a>, which has information about the way <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M430">View MathML</a> is converging to zero. In fact, it is the ratio of the two terms of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M430">View MathML</a>.

With these, we can prove that the equilibrium connectivity <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M432">View MathML</a> has the following asymptotic expansion in <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M430">View MathML</a>.

Theorem 3.5

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M434">View MathML</a>

Proof See Theorem B.5 in Appendix B.2. □

At the first order, the final connectivity is <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M379">View MathML</a>, the filtered correlation of the inputs convolved with a bell-shaped centered temporal profile. In the case of Figure 3, this is quite a good approximation of the final connectivity.

thumbnailFig. 3. (a) shows the temporal evolution of the input to a <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M436">View MathML</a> neurons network. It is made of two spatially random patterns that are shown alternatively. (b) shows the correlation matrix of the inputs. The off-diagonal terms are null because the two patterns are spatially orthogonal. (c), (d), and (e) represent the first order of Theorem 3.5 expansion for different μ. Actually, this approximation is quite good since the percentage of error between the averaged system and the first order, computed by <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M437">View MathML</a>, have an order of magnitude of 10−4% for the three figures. These figures make it possible to observe the role of μ. If μ is small, i.e., the inputs are slow, then the transient can be neglected and the learned connectivity is roughly the correlation of the inputs; see (a). If μ increases, i.e., the inputs are faster, then the connectivity starts to encode a link between the two patterns that were flashed circularly and elicited responses that did not fade away when the other pattern appeared. The temporal structure of the inputs is also learned when μ is large. The parameters used in this figure are <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M412">View MathML</a>, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M439">View MathML</a>, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M417">View MathML</a>, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M441">View MathML</a>.

Not only the spatial correlation is encoded in the weights, but there is also some information about the temporal correlation, i.e., two successive but orthogonal events occurring in the inputs will be wired in the connectivity although they do not appear in the spatial correlations; see Figure 3 for an example.

3.3 Trace learning: band-pass filter effect

In this section, we study an improvement of the learning model by adding a certain form of history dependence in the system and explain the way it changes the results of the previous section. Given that Theorem 2.2 only applies to an instantaneous process, we will only be able to treat the history-dependent systems which can be reformulated as instantaneous processes. Actually, this class of systems contains models which are biologically more relevant than the previous model and which will exhibit interesting additional functional behaviors. In particular, this covers the following features:

• Trace learning.

It is likely that a biological learning rule will integrate the activity over a short time. As Földiàk suggested in [27], it makes sense to consider the learning equation as being

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M442">View MathML</a>

where ∗ is the convolution and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M443">View MathML</a>. Rolls and Deco numerically show [15] that the temporal convolution, leading to a spatio-temporal learning, makes it possible to perform invariant object recognition. Besides, trace learning appears to be the symmetric part of the biological STDP rule that we detail in Section 3.4.

• Damped oscillatory neurons.

Many neurons have an oscillatory behavior. Although we cannot take this into account in a linear model, we can model a neuron by a damped oscillator, which also introduces a new important time-scale in the system. Adding adaptation to neuronal dynamics is an elementary way to implement this idea. This corresponds to modeling a single neuron without inputs by the equivalent formulations

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M444">View MathML</a>

• Dynamic synapses.

The electro-chemical process of synaptic communication is very complicated and non-linear. Yet, one of the features of synaptic communication we can take into account in a linear model is the shape of the post-synaptic potentials. In this section, we consider that each synapse is a linear filter whose finite impulse response (i.e., the post-synaptic potential) has the shape <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M445">View MathML</a>. This is a common assumption which, for instance, is at the basis of traditional rate based models; see Chapter 11 of [7].

For mathematical tractability, we assume in the following that <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M446">View MathML</a> such that <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M447">View MathML</a>, i.e., the time-scales of the neurons, those of the synapses and those of the learning windows are the same. Actually, there is a large variety of temporal scales of neurons, synapses, and learning windows, which makes this assumption not absurd. Besides, in many brain areas, examples of these time constants are in the same range (≃10 ms). Yet, investigating the impact of breaking this assumption would be necessary to model better biological networks. This leads to the following system:

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M448">View MathML</a>

(16)

where the notations are the same as in Section 3.2. The behavior of a single neuron will be oscillatory damped if <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M449">View MathML</a> is a pure imaginary number, i.e., <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M450">View MathML</a>. This is the regime on which we focus. Actually, the Hebbian linear case of Section 3.2 corresponds to <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M451">View MathML</a> in this delayed system.

To comply with the hypotheses of Theorem 2.2 (i.e., no dependence of the history of the process), we can add a variable z to the system which takes care of integrating the variable v over an exponential window. It leads to the equivalent system (in the limit <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M452">View MathML</a>)

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M453">View MathML</a>

This trick makes it possible to deal with some history-based processes where the dependence on the past is exponential.

It turns out most of the results of Section 3.2 remain true for system (16) as detailed in the following. The existence of the solution on <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M333">View MathML</a> is proved in Theorem B.6. The computations show that in the averaged system, the noise term remains identical, whereas the correlation term is to be replaced by <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M455">View MathML</a>. Besides, Lemma 3.2 can be extended to our delayed system by changing only the temporal filters; see Lemma 34. Together with Lemma C.3, this proves the result of Theorem B.8.

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M456">View MathML</a>

where

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M457">View MathML</a>

where <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M458">View MathML</a>. Observe that applying Young’s inequality to convolutions leads to <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M459">View MathML</a>. Actually, Lemma C.3 shows that <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M460">View MathML</a>, where <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M461">View MathML</a> is the Bessel function of the first kind. The value of the L1 norm of v is computed in Appendix C.3. It leads to <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M462">View MathML</a> if Δ is a pure imaginary number and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M463">View MathML</a> else.

Therefore, the averaged system can be rewritten

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M464">View MathML</a>

As before, the existence and uniqueness of a globally attractive equilibrium point is guaranteed if Assumption 3.1 is verified for <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M465">View MathML</a>; see Theorem B.9.

Besides, the weakly connected expansion of the equilibrium point we did in Section 3.2.4 can be derived in this case (see Theorem B.10). At the first order, this leads to the equilibrium connectivity

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M466">View MathML</a>

The second order is given in Theorem B.10. The main difference with the Hebbian linear case is the shape of the temporal filters. Actually, the temporal filters have an oscillatory damped behavior if Δ is purely imaginary. These two cases are compared in Figure 4.

thumbnailFig. 4. These represent the temporal filter <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M467">View MathML</a> for different parameters. (a) When <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M468">View MathML</a>, we are in the Hebbian linear case of Appendix B.2. The temporal filters are just decaying exponentials which averaged the signal over a past window. (b) When the dynamics of the neurons and synapse are oscillatory damped, some oscillations appear in the temporal filters. The number of oscillations depends on Δ. If Δ is real, then there are no oscillations as in the previous case. However, when Δ becomes a pure imaginary number, it creates a few oscillations which are even more numerous if <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M469">View MathML</a> increases.

These oscillatory damped filters have the effect of amplifying a particular frequency of the input signal. As shown in Figure 5, if Δ is a pure imaginary number, then <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M470">View MathML</a> is the cross-correlation of the band-pass filtered inputs with themselves. This band-pass filter effect can also be observed in the higher-order terms of the weakly connected expansion. This suggests that the biophysical oscillatory behavior of neurons and synapses leads to selecting the corresponding frequency of the inputs and performing the same computation as for the Hebbian linear case of the previous section: computing the correlation of the (filtered) inputs.

thumbnailFig. 5. This is the spectral profile <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M471">View MathML</a> for <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M472">View MathML</a> and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M473">View MathML</a>, where <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M474">View MathML</a> denotes the Fourier transform of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M475">View MathML</a>. When <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M476">View MathML</a>, the filter reaches its maximum for the null frequency, but if l increases beyond <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M477">View MathML</a>, the filter becomes a band-pass filter with long tails in <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M478">View MathML</a>.

3.4 Asymmetric ‘STDP’ learning with correlated noise

Here, we extend the results to temporally asymmetric learning rules and spatially correlated noise. We consider a learning rule that is similar to the spike-timing-dependent plasticity (STDP) which is closer to biological experiments than the previous Hebbian rules. It has been observed that the strength of the connection between two neurons depends mainly on the difference between the time of the spikes emitted by each neuron as shown in Figure 6; see [12].

thumbnailFig. 6. This figure represents the synapse strength modification when the post-synaptic neuron emits a spike. The y-axis corresponds to an additive or multiplicative update of the connectivity. For instance, in [28], this is <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M479">View MathML</a> for the negative part of the curve. However, we assume an additive update in this paper. The x-axis is the time at which a pre-synaptic spike reaches the synapse, relatively to the time of post-synaptic time chosen to be 0.

Assuming that the decay time of the positive and negative parts of Figure 6 are equal, we approximate this function by <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M480">View MathML</a>, where <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M481">View MathML</a>. Actually, this corresponds to <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M482">View MathML</a>. If the neuron has a spiking behavior, then the term <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M483">View MathML</a> is significant when the post-synaptic neuron i is spiking at time t, and then it counts the number of previous spikes from the pre-synaptic neuron j that might have caused the post-synaptic spike. This calculus is weighted by an exponentially decaying function. This accounts for the left part of Figure 6. The last term <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M484','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M484">View MathML</a> takes the opposite perspective. It is significant when the pre-synaptic neuron j is spiking and counts the number of previous spikes from the post-synaptic neuron i that are not likely to have been caused by the pre-synaptic neuron. The computation is also weighted by the mirrored function of an exponentially decaying function. This accounts for the right part of Figure 6. This leads to the coupled system

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M485','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M485">View MathML</a>

(17)

where the non-linear intrinsic dynamics of the neurons is represented by f. Indeed, the term <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M486">View MathML</a> is negligible when the neuron is quiet and maximal at the top of the spikes emitted by neuron i. Therefore, it records the value of the pre-synaptic membrane potential weighted by the function <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M487">View MathML</a> when the post-synaptic neuron spikes. This accounts for the positive part of Figure 6. Similarly, the negative part corresponds to <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M488">View MathML</a>.

Actually, this formulation is valid for any non-linear activity with correlated noise. However, studying the role of STDP in spiking networks is beyond the scope of this paper since we are only able to have explicit results for models with linear activity. Therefore, we will assume that the activity is linear while keeping the learning rule as it was derived in the spiking case, i.e., we assume <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M489">View MathML</a> in the system above.

We also use the trick of adding additional variables to get rid of the history-dependency. This reads

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M490">View MathML</a>

In this framework, the method exposed in Section 3.2 holds with small changes. First, the well-posedness assumption becomes

Assumption 3.2 There exists <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M322">View MathML</a> such that

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M492','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M492">View MathML</a>

where <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M493">View MathML</a> is the maximal eigenvalue of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M494">View MathML</a>.

Under this assumption, the system is asymptotically well posed in probability (Theorem B.11). And we show the averaged system is

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M495','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M495">View MathML</a>

(18)

where we have used Theorem B.12 to expand the correlation term. The noise term Q is equal to <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M496','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M496">View MathML</a>, where <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M497','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M497">View MathML</a> is the unique solution of the Lyapunov equation <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M498">View MathML</a>. Lemma D.1 gives a solution for this equation which leads to <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M499">View MathML</a>. In equation (18), the correlation matrices <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M500">View MathML</a> are given by

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M501">View MathML</a>

According to Theorem B.13, the system is also globally asymptotically convergent to a single equilibrium, which we study in the following.

We perform a weakly connected expansion of the equilibrium connectivity of system (18). As shown in Theorem B.14, the first order of the expansion is

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M502">View MathML</a>

Writing <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M503','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M503">View MathML</a>, where S is symmetric and A is skew-symmetric, leads to

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M504','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M504">View MathML</a>

According to Lemma C.1, the symmetric part is very similar to the trace learning case in Section 3.3. Applying Lemma C.2 leads to

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M505','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M505">View MathML</a>

(19)

Therefore, the STDP learning rule simply adds an antisymmetric part to the final connectivity keeping the symmetric part as the Hebbian case. Besides, the antisymmetric part corresponds to computing the cross-correlation of the inputs with its derivative. For high-order terms, this remains true although the temporal profiles are different from the first order. These results are in line with previous works underlying the similarity between STDP learning and differential Hebbian learning, where <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M506','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M506">View MathML</a>; see [29].

Figure 7 shows an example of purely antisymmetric STDP learning, i.e., <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M507','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M507">View MathML</a>. The final connectivity matrix is therefore antisymmetric as shown in Figure 7(b) and the noise has no impact on learning. It shows the network finally approximates the connectivity given in (19).

thumbnailFig. 7. Antisymmetric STDP learning for a network of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M508','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M508">View MathML</a> neurons. (a) Temporal evolution of the inputs to the network. The three neurons are successively and periodically excited. The red color corresponds to an excitation of 1 and the blue to no excitation. (b) Equilibrium connectivity. The matrix is antisymmetric and shows that neurons excite one of their neighbors and are inhibited by the other. (c) Temporal evolution of the connectivity strength. The colors correspond to those of (b). The connectivity of system (17) corresponds to the plain thin oscillatory curves. The connectivity of the averaged system (18) (with <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M509','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M509">View MathML</a>) corresponds to the plain thick lines. Note that each curve corresponds to the superposition of three connections which remain equal through learning. The dashed curves correspond to the antisymmetric part in (19). The parameters chosen for this simulation were <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M510','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M510">View MathML</a>, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M511','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M511">View MathML</a>, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M512','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M512">View MathML</a>, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M513','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M513">View MathML</a>, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M514','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M514">View MathML</a>, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M515">View MathML</a>, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M410">View MathML</a>, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M412">View MathML</a>. The system was simulated on the fast time-scale during <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M518">View MathML</a> time steps of size <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M519','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M519">View MathML</a>.

4 Discussion

We have applied temporal averaging methods on slow/fast systems modeling the learning mechanisms occurring in linear stochastic neural networks. When we make sure the connectivity remains small, the dynamics of the averaged system appears to be simple: the connectivity always converges to a unique equilibrium point. Then, we performed a weakly connected expansion of this final connectivity whose terms are combinations of the noise covariance and the lagged correlations of the inputs: the first-order term is simply the sum of the noise covariance and the correlation of the inputs.

• As opposed to the former input/ouput vision of the neurons, we have considered the membrane potential v to be the solution of a dynamical system. The consequence of this modeling choice is that not only the spatial correlations, but also the temporal correlations are learned. Due to the fact we take the transients into account, the activity never converges but it lives between the representation of the inputs. Therefore, it links concepts together.

The parameter μ is the ratio of the time-scales between the inputs and the activity variable. If <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M156">View MathML</a>, the inputs are infinitely slow and the activity variable has enough time to converge towards its equilibrium point. When μ grows, the dynamics becomes more and more transient, it has no time to converge. Therefore, if the inputs are extremely slow, the network only learns the spatial correlation of the inputs. If the inputs are fast, it also learns the temporal correlations. This is illustrated in Figure 3.

This suggests that learning associations between concepts, for instance, learning words in two different languages, may be optimized by presenting two words to be associated circularly with a certain frequency. Indeed, increasing the frequency (with a fixed duration of exposition to each word) amounts to increasing μ. Therefore, the network learns better the temporal correlations of the inputs and thus strengthens the link between these two concepts.

• According to the model of resonator neuron [30], Section 3.3 suggests that neurons and synapses with a preferred frequency of oscillation will preferably extract the correlation of the inputs filtered by a band pass filter centered on the intrinsic frequency of the neurons.

Actually, it has been observed that the auditory cortex is tonotopically organized, i.e., the neurons are arranged by frequency [31]. It is traditionally thought that this is achieved thanks to a particular connectivity between the neurons. We exhibit here another mechanism to select this frequency which is solely based on the parameters of the neurons: a network with a lot of different neurons whose intrinsic frequencies are uniformly spread is likely to perform a Fourier-like operation: decomposing the signal by frequency.

In particular, this emphasizes the fact that the network does not treat space and time similarly. Roughly speaking, associating several pictures and associating several sounds are therefore two different tasks which involve different mechanisms.

• In this paper, the original hierarchy of the network has been neglected: the network is made of neurons which receive external inputs. A natural way to include a hierarchical structure (with layers for instance), without changing the setup of the paper, is therefore to remove the external input to some neurons. However, according to Theorem 3.5 (and its extensions Theorems B.10 and B.14), one can see that these neurons will be disconnected from the others at the first order (if the noise is spatially uncorrelated). Linear activities imply that the high level neurons disconnect from others, which is a problem. In fact, one can observe that the second-order term in Theorem 3.5 is not null if the noise matrix Σ is not diagonal. In fact, this is the noise between neurons which will recruit the high level neurons to build connections from and to them.

It is likely that a significant part of noise in the brain is locally induced, e.g., local perturbations due to blood vessels or local chemical signals. In a way, the neurons close to each other share their noise and it seems reasonable to choose the matrix Σ so that it reflects the biological proximity between neurons. In fact, Σ specifies the original structure of the network and makes it possible for close-by neurons to recruit each other.

Another idea to address hierarchy in networks would be to replace the synaptic decay term <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M521">View MathML</a> by another homeostatic term [32] which would enforce the emergence of a strong hierarchical structure.

• It is also interesting to observe that most of the noise contribution to the equilibrium connectivity for STDP learning (see Theorem B.14) vanishes if the learning is purely skew-symmetric, i.e., <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M507','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M507">View MathML</a>. In fact, it is only the symmetric part of learning, i.e., the Hebbian mechanism, that writes the noise in the connectivity.

• We have shown that there is a natural analogous STDP learning for spiking neurons in our case of linear neurons. This asymmetric rule converges to a final connectivity which can be decomposed into symmetric and skew-symmetric parts. The first one is similar to the symmetric Hebbian learning case, emphasizing that the STDP is nothing more than an asymmetric Hebbian-like learning rule. The skew-symmetric part of the final connectivity is the cross-correlation between the inputs and their derivatives.

This has an interesting signification when looking at the spontaneous activity of the network post-learning. In fact, if we assume that the inputs are generated by an autonomous system <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M523','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M523">View MathML</a>, then according to the bottom equation in formula (19), the spontaneous activity is governed by

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M524','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M524">View MathML</a>

In a way, the noise terms generate random patterns which tend to be forgotten by the network due to the leak term <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M525','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M525">View MathML</a>. The only drift is due to <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M526','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M526">View MathML</a> which is the expectation of the vector field defining the dynamics of inputs with a measure being the scalar product between the activity variable and the inputs. In other words, if the activity is close to the inputs at a given time <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M527','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M527">View MathML</a>, i.e., <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M528','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M528">View MathML</a> is large, then the activity will evolve in the same direction as what this input would have done. The network has modeled the temporal structure of the inputs. The spontaneous activity predicts and replays the inputs the network has learned.

There are still numerous challenges to carry on in this direction.

First, it seems natural to look for an application of these mathematical methods to more realistic models. The two main limitations of the class of models we study in Section 3 are (i) the activity variable is governed by a linear equation and (ii) all the neurons are assumed to be identical. The mathematical analysis in this paper was made possible by the assumption that the neural network has a linear dynamics, which does not reflect the intrinsic non-linear behavior of the neurons. However, the cornerstone of the application of temporal averaging methods to a learning neural network, namely Property 3.3, is similar to the behavior of Poisson processes [26] which has useful applications for learning neural networks [19,20]. This suggests that the dynamics studied in this paper might be quite similar to some non-linear network models. Studying more rigorously the extension of the present theory to non-linear and heterogeneous models is the next step toward a better modeling of biologically plausible neural networks.

Second, we have shown that the equilibrium connectivity was made of a symmetric and antisymmetric term. In terms of statistical analysis of data sets, the symmetric part corresponds to classical correlation matrices. However, the antisymmetric part suggests a way to improve the purely correlation-based approach used in many statistical analyses (e.g., PCA) toward a causality-oriented framework which might be better suited to deal with dynamical data.

Appendix A: Stochastic and periodic averaging

A.1 Long-time behavior of inhomogeneous Markov processes

In order to construct the averaged vector field <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M529','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M529">View MathML</a> in the time-scale matching case (<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M42">View MathML</a>), one needs to understand properly the long-time behavior of the rescaled inhomogeneous frozen process

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M531','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M531">View MathML</a>

(20)

Under regularity and dissipativity conditions, [5] proves the following general result about the asymptotic behavior of the solution of

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M532','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M532">View MathML</a>

where <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M533','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M533">View MathML</a> and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M534','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M534">View MathML</a> are τ-periodic.

The first point of the following theorem gives the definition of evolution systems of measures, which generalizes the notion of invariant measures in the case of inhomogeneous Markov processes. The exponential estimate of 2. in the following theorem is a key point to prove the averaging principle of Theorem 2.2.

Theorem A.1 ([5])

1. There exists a uniqueτ-periodic family of probability measures<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M535','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M535">View MathML</a>such that for all functionsϕcontinuous and bounded,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M536','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M536">View MathML</a>

Such a family is called evolution systems of measures.

2. Furthermore, under stronger dissipativity condition, the convergence of the law ofXtoμis exponentially fast. More precisely, for any<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M537','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M537">View MathML</a>, there exist<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M538','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M538">View MathML</a>and<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M539','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M539">View MathML</a>such that for allϕin the space ofp-integrable functions with respect to<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M540','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M540">View MathML</a>, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M541','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M541">View MathML</a>,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M542','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M542">View MathML</a>

A.2 Proof of Property 2.3

Property A.2If there exists a smooth subsetof<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M255">View MathML</a>such that

1. The functionsF, G, Σsatisfy Assumptions 2.1-2.3 restricted on<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M258">View MathML</a>.

2. ℰ is invariant under the flow of<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M139">View MathML</a>, as defined in (7).

Then for any initial condition<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M260">View MathML</a>, system (4) is asymptotically well posed in probability and<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M148">View MathML</a>satisfies the conclusion of Theorem 2.2.

Proof The idea of the proof is to truncate the original system, replacing G by a smooth truncation which coincides with G on ℰ and which is close to 0 outside ℰ. More precisely, for <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M548','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M548">View MathML</a>, we introduce <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M549','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M549">View MathML</a> a regular function (locally Lipschitz) such that <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M550','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M550">View MathML</a> if <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M551','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M551">View MathML</a> or <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M552','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M552">View MathML</a> and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M553','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M553">View MathML</a> if <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M554','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M554">View MathML</a>. We define

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M555','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M555">View MathML</a>

Then, we introduce <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M556','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M556">View MathML</a> the solution of the auxiliary system

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M557','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M557">View MathML</a>

with the same initial condition as <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M558','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M558">View MathML</a>.

Let <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M559','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M559">View MathML</a> be three positive reals. Let us introduce a few more notations. We will need to consider a subset of ℰ defined by

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M560','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M560">View MathML</a>

We also introduce the following stopping times:

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M561','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M561">View MathML</a>

Finally, we define <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M562','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M562">View MathML</a> and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M563','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M563">View MathML</a>.

Let us remark at this point that in order to prove that <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M564','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M564">View MathML</a> (which is our aim), it is sufficient to work on the bounded stopping time <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M565','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M565">View MathML</a>, since <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M566','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M566">View MathML</a>. In other words, the realizations of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M567','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M567">View MathML</a> which stay longer than T inside ℰ are not problematic. Therefore, we introduce <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M568','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M568">View MathML</a>.

Our first claim is that on finite time intervals <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M569','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M569">View MathML</a>, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M570','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M570">View MathML</a> is a good approximation of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M567','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M567">View MathML</a> inside ℰ as long as one chooses β sufficiently small. To prove our claim, we proceed in two steps, first working inside <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M572','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M572">View MathML</a> and then in <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M573','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M573">View MathML</a>:

1. For any <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M574','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M574">View MathML</a>, one controls the difference between <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M567','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M567">View MathML</a> and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M576','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M576">View MathML</a> on <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M577','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M577">View MathML</a> since one controls the difference between the drifts. By an application of Lemma A.3 below (we need here the moment Assumption 2.3(i)), there exists a constant C (which may depend on <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M578','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M578">View MathML</a>) such that

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M579','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M579">View MathML</a>

(21)

We conclude by an application of the Markov inequality, implying

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M580','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M580">View MathML</a>

(22)

2. One needs now to control the situation outside <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M577','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M577">View MathML</a>, that is, on <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M582','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M582">View MathML</a>. The idea is that while one does not control the difference between G and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M583','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M583">View MathML</a> anymore, one can still choose β sufficiently small such that <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M572','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M572">View MathML</a> becomes arbitrary close to ℰ, hence implying that <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M585','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M585">View MathML</a> and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M586','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M586">View MathML</a> are arbitrary close with high probability, namely

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M587','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M587">View MathML</a>

(23)

With <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M588','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M588">View MathML</a> and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M589','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M589">View MathML</a>, one obtains that for sufficiently small β,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M590','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M590">View MathML</a>

(24)

Let us denote <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M591','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M591">View MathML</a>. Then, one can split the calculus of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M592','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M592">View MathML</a> according to the event <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M593','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M593">View MathML</a>:

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M594','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M594">View MathML</a>

where we have used the Cauchy-Schwarz inequality and the moment Assumption 2.3(ii) (yielding the constant <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M595','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M595">View MathML</a>) in the second line.

So, we deduce by the Markov inequality that <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M596','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M596">View MathML</a> is arbitrary small in probability.

From the combination of 1. and 2., we deduce that one can choose β small enough such that

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M597','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M597">View MathML</a>

(25)

We can now proceed to the application of Theorem 2.2 to the truncated system. As <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M598','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M598">View MathML</a> remains in <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M258">View MathML</a>, one can extend smoothly F and Σ outside ℰ so that <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M600','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M600">View MathML</a> satisfies Assumptions 2.1-2.2. Therefore, one can apply Theorem 2.2 to the auxiliary system: for all <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M336">View MathML</a>,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M602','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M602">View MathML</a>

where w is defined by (8). As a consequence, there exists <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M603','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M603">View MathML</a> such that for all ϵ with <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M604','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M604">View MathML</a>,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M605','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M605">View MathML</a>

Then, as <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M606','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M606">View MathML</a>, one deduces that for all ϵ with <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M604','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M604">View MathML</a>,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M608','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M608">View MathML</a>

that is to say,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M609','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M609">View MathML</a>

We know by assumption 2. of the statement of Property 2.3, for all <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M610','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M610">View MathML</a>, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M611','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M611">View MathML</a>, so we conclude the proof by observing that for all <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M74">View MathML</a>,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M613','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M613">View MathML</a>

 □

In the following lemma, we show that the solutions of two SDEs, whose drifts are close on a subset of the state space, remain close on a finite time interval. The difficulty here lies in the fact that we deal with only locally Lipschitz coefficients.

Lemma A.3Supposexandyare solutions, with identical initial conditions in<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M614','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M614">View MathML</a>, of the following stochastic differential equations in<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M615','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M615">View MathML</a>:

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M616','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M616">View MathML</a>

(26)

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M617','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M617">View MathML</a>

(27)

Let<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M74">View MathML</a>be a fixed time. We define

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M619','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M619">View MathML</a>

We make the following assumptions:

1. Approximation assumption:

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M620','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M620">View MathML</a>

2. Local Lipschitz assumption: for all<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M621','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M621">View MathML</a>with<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M622','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M622">View MathML</a>, there exists a constant<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M623','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M623">View MathML</a>such that

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M624','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M624">View MathML</a>

3. Boundedness assumption: there exists<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M252">View MathML</a>and<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M626','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M626">View MathML</a>such that

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M627','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M627">View MathML</a>

and if<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M628','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M628">View MathML</a>, then there exists<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M629','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M629">View MathML</a>such that<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M630','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M630">View MathML</a>.

Under the above assumptions, there exists a constantC (depending on the quantities defined above, but not onξ) such that

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M631','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M631">View MathML</a>

(28)

Proof Although the Lipschitz constant is not bounded on ℋ, we can use the boundedness assumption to show that the probability of reaching a level R before time T will be very small for large R, and then use the classical strategy inside <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M632','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M632">View MathML</a> where everything works as if the coefficients were globally Lipschitz. A similar strategy is used in [33] to prove a strong convergence theorem for the Euler scheme without the global Lipschitz assumption. We adapt here the ideas of their proof to our setting.

Therefore, we introduce the following stopping times:

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M633','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M633">View MathML</a>

We also denote <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M634','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M634">View MathML</a>.

Splitting the following expectation according to the value of ρ, and applying the Young inequality,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M635','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M635">View MathML</a>

we obtain, for any <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M636','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M636">View MathML</a>,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M637','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M637">View MathML</a>

Then we use the boundedness assumption and the Markov inequality to deduce that

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M638','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M638">View MathML</a>

Now, we can focus on the supremum of the error before time ρ. We first apply the Cauchy-Schwarz inequality

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M639','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M639">View MathML</a>

Then, we use the local Lipschitz and the boundedness assumptions, together with the Doob inequality (the first inequality) to deal with the stochastic integral: for any <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M640','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M640">View MathML</a>,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M641','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M641">View MathML</a>

We then apply the Gronwall lemma

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M642','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M642">View MathML</a>

(29)

Finally, we choose d small enough such that

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M643','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M643">View MathML</a>

and R large enough such that

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M644','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M644">View MathML</a>

yielding

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M645','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M645">View MathML</a>

 □

Appendix B: Proofs of Section 3

B.1 Notations and definitions

Throughout the paper, lower-case normal letters are constants, lower-case bold letters are vectors or vector-valued functions, and upper-case bold letters are matrices.

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M646','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M646">View MathML</a> are parameters of the network. We also define <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M647','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M647">View MathML</a> for Section 3.3 and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M648','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M648">View MathML</a>, a fixed noise matrix, for Section 3.4. We write <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M649','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M649">View MathML</a>.

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M650','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M650">View MathML</a> is the number of neurons in the network.

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M651','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M651">View MathML</a> is the field of membrane potential in the network.

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M652','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M652">View MathML</a> is the field of inputs to the network. We write

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M653','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M653">View MathML</a>

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M654','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M654">View MathML</a> is the tensor product between u and v, which simply means <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M655','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M655">View MathML</a>.

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M656','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M656">View MathML</a> is the connectivity of the network. Throughout the paper, we assume <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M657','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M657">View MathML</a>.

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M658','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M658">View MathML</a> is the scalar product between two vectors <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M659','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M659">View MathML</a>.

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M660','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M660">View MathML</a> for <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M661','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M661">View MathML</a> is the <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M662','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M662">View MathML</a> norm of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M663','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M663">View MathML</a>, i.e., <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M664','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M664">View MathML</a>. And similarly for the connectivity matrices of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M665','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M665">View MathML</a> with a double sum.

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M666','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M666">View MathML</a>.

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M667','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M667">View MathML</a> is the transpose of the matrix <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M668','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M668">View MathML</a>.

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M669','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M669">View MathML</a> is the cross-correlation matrix of two compactly supported and differentiable functions from ℝ to <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M615','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M615">View MathML</a>, i.e.,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M671','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M671">View MathML</a>

H is the Heaviside function, i.e.,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M672','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M672">View MathML</a>

• The real functions

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M673','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M673">View MathML</a>

(30)

are integrable on ℝ.

B.1.1 Notations for the Appendix

The computations involve a lot of convolutions and, for readability of the Appendix, we introduce some new notations. Indeed, we rewrite the time-convolution between u and g, an integrable function on ℝ,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M674','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M674">View MathML</a>

This suggests one should think of v as a semi-continuous matrix of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M675','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M675">View MathML</a> and of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M676','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M676">View MathML</a> as a continuous matrix of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M677','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M677">View MathML</a>, such that <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M678','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M678">View MathML</a> and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M679','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M679">View MathML</a>. Indeed, in this framework the convolution with g is nothing but the continuous matrix multiplication between v and a continuous Toeplitz matrix generated row by row by g. Hence, the operator ‘⋅’ can be though of as a matrix multiplication.

Therefore, it is natural to define <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M680','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M680">View MathML</a>, where <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M681','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M681">View MathML</a> is the transpose of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M682','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M682">View MathML</a>, i.e., the continuous Toeplitz matrix generated row by row by <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M683','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M683">View MathML</a> and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M684','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M684">View MathML</a>. Thus, for g and h, two integrable functions on ℝ, we can rewrite

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M685','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M685">View MathML</a>

where <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M686','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M686">View MathML</a> and ℋ are their associated continuous matrices. More generally, the bold curved letters <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M682','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M682">View MathML</a>, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M688','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M688">View MathML</a>, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M689','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M689">View MathML</a> represent these continuous Toeplitz matrices which are well defined through their action as convolution operators with g, v, and w. The previous formulation naturally expresses the symmetry of relation (14).

B.2 Hebbian learning with linear activity

In this part, we consider system (12).

B.2.1 Application of temporal averaging theory

Theorem B.1If Assumption 3.1 is verified for<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M334">View MathML</a>, then system (12) is asymptotically well posed in probability and the connectivity matrix<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M335">View MathML</a>, the solution of system (12), converges toW, in the sense that for all<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M336">View MathML</a>,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M693','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M693">View MathML</a>

whereWis the deterministic solution of

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M694','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M694">View MathML</a>

where<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M339">View MathML</a>is the<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M213">View MathML</a>-periodic attractor of<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M341">View MathML</a>, where<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M282">View MathML</a>is supposed to be fixed.

Proof We are going to apply Property 2.3. For <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M322">View MathML</a>, consider the space

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M700','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M700">View MathML</a>

First, since <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M701','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M701">View MathML</a> is strictly positive for W in <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M397">View MathML</a>, Assumptions 2.1-2.2 are satisfied on <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M703','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M703">View MathML</a>. Then, we only need to compute the averaged vector field <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M14">View MathML</a> and show that <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M397">View MathML</a> is invariant under the flow of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M14">View MathML</a>.

1. Computation of the averaged vector field <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M14">View MathML</a>:

The fast variable is linear, the averaged vector field is given by (10). This reads

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M708','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M708">View MathML</a>

where <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M709','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M709">View MathML</a> is the probability density function of the Gaussian law with mean v and covariance Q. And Q is the unique solution of (9), with <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M710','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M710">View MathML</a>. This leads to <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M711','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M711">View MathML</a>.

Therefore,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M712','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M712">View MathML</a>

The integral term in the equation above is the correlation matrix of the <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M213">View MathML</a>-periodic function <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M714','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M714">View MathML</a>. To rewrite this term, we define <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M715','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M715">View MathML</a> such that <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M716','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M716">View MathML</a>. <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M212">View MathML</a> can be seen as a matrix gathering the history of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M212">View MathML</a>, i.e., each column of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M212">View MathML</a> corresponds to the vector <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M720','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M720">View MathML</a> for a given <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M721','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M721">View MathML</a>. It turns out

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M722','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M722">View MathML</a>

Therefore,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M723','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M723">View MathML</a>

According to the results in Section 2, the solutions of a differential system with such a right-hand side are close to that of the initial system (12). Hence, we focus exclusively on it and try to unveil the properties of its solutions which will be retrospectively extended to those of the initial system (12).

2. Invariance of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M397">View MathML</a> under the flow of (13):

Here we assume that <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M725','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M725">View MathML</a> and we want to prove that the trajectory of W is in <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M397">View MathML</a>, too.

(a) Symmetry:

It is clear that each term in <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M14">View MathML</a> is symmetric. Their sum is therefore symmetric and so is <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M728','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M728">View MathML</a>.

(b) Inequality <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M330">View MathML</a>:

The correlation term <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M730','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M730">View MathML</a> is a Gramian matrix and is therefore positive. Because <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M701','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M701">View MathML</a> is assumed to be positive, therefore, its inverse is also positive. Therefore, if <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M732','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M732">View MathML</a> is an eigenvector of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M330">View MathML</a> associated with a null eigenvalue, then <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M734','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M734">View MathML</a>. Thus, the trajectories of (13) remain positive.

(c) Inequality <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M735','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M735">View MathML</a>:

The argument here is that of the inward pointing subspace. We intend to find a condition under which the flow <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M14">View MathML</a> is pointing inward the space <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M737','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M737">View MathML</a>. Roughly speaking, this will be done by defining a real valued function g strictly negative on the subspace and positive outside and then showing that its gradient (or differential) on the border goes in the opposite direction of the flow, i.e., <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M738','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M738">View MathML</a> for <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M739','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M739">View MathML</a>.

For all <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M740','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M740">View MathML</a> such that <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M741','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M741">View MathML</a>, define a family of positive numbers <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M742','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M742">View MathML</a> whose supremum is written <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M743','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M743">View MathML</a> and a family of functions <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M744','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M744">View MathML</a> such that

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M745','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M745">View MathML</a>

Observe that the differential of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M746','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M746">View MathML</a> at W applied to J is <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M747','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M747">View MathML</a>. For <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M748','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M748">View MathML</a>, i.e., <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M749','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M749">View MathML</a>, compute

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M750','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M750">View MathML</a>

Therefore, for <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M762','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M762">View MathML</a>

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M763','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M763">View MathML</a>

where

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M764','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M764">View MathML</a>

(31)

Now write <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M765','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M765">View MathML</a> with <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M322">View MathML</a>. Equation (31) becomes

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M767','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M767">View MathML</a>

When there exists p such that <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M768','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M768">View MathML</a> (which corresponds to Assumption 3.1), then their exists a ball of radius pl on which the dynamics is pointing inward. It means any matrix W whose maximal eigenvalue is <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M769','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M769">View MathML</a> will see this eigenvalue (and those which are sufficiently close to it, i.e., for which <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M770','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M770">View MathML</a> is sufficiently small) decreasing along the trajectories of the system. Therefore, the space <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M397">View MathML</a> is invariant by the flow of the system iff Assumption 3.1 is satisfied.

• Upper bound of A:

Applying Cauchy-Schwarz leads to

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M751','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M751">View MathML</a>

However, for <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M120">View MathML</a>

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M753','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M753">View MathML</a>

Therefore, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M754','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M754">View MathML</a>.

• Upper bound of B:

Observe that for J a positive definite matrix whose eigenvalues are the <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M755','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M755">View MathML</a>, then the spectrum of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M756','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M756">View MathML</a> is <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M757','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M757">View MathML</a>. Therefore, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M758','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M758">View MathML</a>. Therefore, if <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M759','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M759">View MathML</a>, then <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M760','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M760">View MathML</a>.

Using the previous observation and Cauchy-Schwarz leads to

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M761','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M761">View MathML</a>

The trajectories of system (13) with the initial condition in <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M397">View MathML</a> are defined on <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M333">View MathML</a> and remain bounded. Indeed, if <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M725','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M725">View MathML</a>, the connectivity will stay in <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M397">View MathML</a>, in particular <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M776','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M776">View MathML</a> along the trajectories, more precisely <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M701','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M701">View MathML</a> is a strictly positive constant since <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M778','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M778">View MathML</a>. Because <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M212">View MathML</a> is also bounded by <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M780','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M780">View MathML</a>, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M781','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M781">View MathML</a> is bounded. The right-hand side of system (13) is the sum a bounded term and a linear term multiplied by a negative constant; therefore, the system remains bounded and it cannot explode in finite time: it is defined on <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M333">View MathML</a>. □

B.2.2 An expansion for the correlation term

We first state a useful lemma.

Lemma B.2If<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M212">View MathML</a>is the solution, with zero as initial condition, of<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M396">View MathML</a>, it can be written by the sum below which converges ifWis in<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M397">View MathML</a>for<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M322">View MathML</a>.

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M787','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M787">View MathML</a>

where<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M400">View MathML</a>.

Proof It can be proven as a trivial rewriting of the variation of parameters formula for linear systems. A more general approach, which extends to delayed systems, was developed by Galtier and Touboul [25]; see the first example for the proof of this lemma. □

This is useful to find the next result.

Property B.3The correlation term can be written

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M789','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M789">View MathML</a>

Proof We can use Lemma 3.2 with <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M790','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M790">View MathML</a> and compute the cross product <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M730','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M730">View MathML</a>.

Therefore, consider <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M792','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M792">View MathML</a> instead of u. A change of variable shows that <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M793','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M793">View MathML</a>. Therefore,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M794','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M794">View MathML</a>

 □

B.2.3 Global stability of the single equilibrium point

Theorem B.4If Assumption 3.1 is verified for<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M420">View MathML</a>, then there is a unique equilibrium point in the invariant subset<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M397">View MathML</a>which is globally, asymptotically stable.

Proof For this proof, define <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M797','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M797">View MathML</a>.

First, we compute the differential of F and show it is a bounded operator. Second, we show it implies the existence and uniqueness of an equilibrium point under some condition. Then, we find an energy for the system which says the fixed point is a global attractor. Finally, we show the stability condition is the same as Assumption 3.1 for <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M420">View MathML</a>.

1. We compute the differential of each term in F: The differential of F at W is the sum of these two terms.

• Formally write the second term <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M799','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M799">View MathML</a>. To find its differential, compute <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M800','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M800">View MathML</a> and keep the terms at the first order in J. Before computing the whole sum, observe that

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M801','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M801">View MathML</a>

This leads to

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M802','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M802">View MathML</a>

• Write <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M803','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M803">View MathML</a>. We can write <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M804','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M804">View MathML</a> and use the chain rule to compute the differential of Q at W, which gives <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M805','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M805">View MathML</a>. Therefore,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M806','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M806">View MathML</a>

2. We want to compute the norm of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M807','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M807">View MathML</a> for <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M808','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M808">View MathML</a>. First, observe that for three square matrices A, B, and C,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M809','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M809">View MathML</a>

for <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M732','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M732">View MathML</a> the vectors of the canonical basis of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M615','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M615">View MathML</a>. This leads to <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M812','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M812">View MathML</a>. Therefore, because <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M813','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M813">View MathML</a>,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M814','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M814">View MathML</a>

Therefore,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M815','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M815">View MathML</a>

This inequality is true for all J with <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M816','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M816">View MathML</a>; therefore, it is also true for the operator norm

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M817','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M817">View MathML</a>

Therefore, F is a k-Lipschitz operator where <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M818','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M818">View MathML</a>. This means <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M819','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M819">View MathML</a>.

3. The equilibrium points of system (15) necessarily verify the equation <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M820','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M820">View MathML</a>. If

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M821','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M821">View MathML</a>

(32)

then <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M822','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M822">View MathML</a> is a contraction map from <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M397">View MathML</a> to itself. Therefore, the Banach fixed point theorem says that there is a unique fixed point which we write <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M432">View MathML</a>.

4. We now show that, under assumption (32), <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M825','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M825">View MathML</a> is an energy function for the system <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M826','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M826">View MathML</a> (which is a rescaled version of system (15)).

Indeed, compute the derivative of this energy along the trajectories of the system

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M827','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M827">View MathML</a>

The energy is lower-bounded, takes its minimum for <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M828','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M828">View MathML</a> and the decreases along the trajectories of the system. Therefore, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M432">View MathML</a> is globally asymptotically stable if assumption (32) is verified.

5. Observe that if Assumption 3.1 is verified for <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M420">View MathML</a>, then <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M831','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M831">View MathML</a>. Therefore, Assumption 3.1 implies that (32) is also true. This concludes the proof. □

B.2.4 Explicit expansion of the equilibrium point

Recall the notations <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M832','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M832">View MathML</a> and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M833','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M833">View MathML</a>.

Theorem B.5

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M834','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M834">View MathML</a>

Actually, it is possible to compute recursively the nth term of the expansion above, although their complexity explodes.

Proof Define <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M835','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M835">View MathML</a> the smallest value in <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M836','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M836">View MathML</a> such that Assumption 3.1 is valid. This implies

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M837','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M837">View MathML</a>

The weak connectivity index <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M430">View MathML</a> controls the ratio of the connection over the strength of intrinsic dynamics. Indeed, these two variables are of the same order because

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M839','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M839">View MathML</a>

We want to approximate the equilibrium <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M432">View MathML</a>, i.e., the solution of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M841','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M841">View MathML</a>, in the regime <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M842','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M842">View MathML</a>. Define <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M843','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M843">View MathML</a> such that <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M844','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M844">View MathML</a>. We abusively write <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M845','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M845">View MathML</a> such that

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M846','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M846">View MathML</a>

Recalling <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M833','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M833">View MathML</a> leads to

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M848','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M848">View MathML</a>

Now, we write a candidate <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M849','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M849">View MathML</a>, then we chose the terms <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M850','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M850">View MathML</a> so that the first mth orders in <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M851','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M851">View MathML</a> vanish. This implies that <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M852','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M852">View MathML</a>, where <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M853','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M853">View MathML</a>. Then, we use the fact that the minimal absolute value of the eigenvalues of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M854','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M854">View MathML</a> is larger than <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M855','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M855">View MathML</a>. Indeed, it means

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M856','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M856">View MathML</a>

i.e., <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M857','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M857">View MathML</a>.

Thus, we need to find the <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M858','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M858">View MathML</a> such that the first mth orders in <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M851','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M851">View MathML</a> vanish. Therefore, we need to expand all the terms in <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M860','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M860">View MathML</a>. The first term is obvious. In the following, we write the second term <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M861','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M861">View MathML</a> associated to the correlations and look for an explicit expression of the <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M862','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M862">View MathML</a> such that <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M863','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M863">View MathML</a>. Second, we write the third term <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M864','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M864">View MathML</a> associated to the noise and look for an explicit expression of the <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M865','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M865">View MathML</a> such that <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M866','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M866">View MathML</a>.

• Finding the <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M862','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M862">View MathML</a>:

First, observe that

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M868','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M868">View MathML</a>

(33)

This leads to

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M869','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M869">View MathML</a>

The ath term in the power expansion in <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M430">View MathML</a> verifies <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M871','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M871">View MathML</a>. More precisely, this reads

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M872','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M872">View MathML</a>

This equation is scary but it reduces to simple expressions for small <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M873','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M873">View MathML</a>.

• Finding the <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M865','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M865">View MathML</a>:

Using equation (33) leads to

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M875','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M875">View MathML</a>

The ath term in the power expansion in <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M430">View MathML</a> verifies <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M877','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M877">View MathML</a>. More precisely, this reads

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M878','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M878">View MathML</a>

Therefore,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M879','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M879">View MathML</a>

Therefore, it is easy to compute <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M880','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M880">View MathML</a> for <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M873','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M873">View MathML</a>. By definition <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M882','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M882">View MathML</a>, which leads to the result. □

B.3 Trace learning with damped oscillators and dynamic synapses

Theorem B.6If Assumption 3.1 is verified for<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M334">View MathML</a>, then system (16) is asymptotically well posed in probability and the connectivity matrix<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M335">View MathML</a>, solution of system (16), converges toWin the sense that for all<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M336">View MathML</a>,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M886','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M886">View MathML</a>

whereWis the deterministic solution of

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M887','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M887">View MathML</a>

where<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M339">View MathML</a>is the<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M213">View MathML</a>-periodic attractor of<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M890','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M890">View MathML</a>, where<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M891','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M891">View MathML</a>is supposed to be fixed. And<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M892','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M892">View MathML</a>is a noise matrix described below.

Proof First, let us recall the instantaneous reformulation of (16)

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M893','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M893">View MathML</a>

Starting from this system, the structure of the proof of Theorem 3.1 remains unchanged. The correlation term is to be replaced by <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M894','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M894">View MathML</a>. The noise term we are looking for is <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M892','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M892">View MathML</a> in the Lyapunov equation (see (9)) below

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M896','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M896">View MathML</a>

Because the learning rule is symmetric, then the space of symmetric matrices is invariant and we can restrict this section to the symmetric case. It is easy to show that this Lyapunov equation has a unique solution, because the sum of two eigenvalues of the drift matrix is never null (provided W stays in <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M397">View MathML</a>). This leads to the system

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M898','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M898">View MathML</a>

One solution of equation (a) is <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M899','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M899">View MathML</a>. Equation (c) defines <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M892','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M892">View MathML</a> properly. Indeed, because W is symmetric, so is <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M901','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M901">View MathML</a> and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M902','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M902">View MathML</a>. Similarly, equation (b) defines <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M497','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M497">View MathML</a> but it remains to be checked that this definition is that of a symmetric matrix. In fact, it works because W is assumed symmetric and the noise has no off-diagonal terms. Indeed, in this case, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M904','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M904">View MathML</a>. This solution is thus the unique solution of the Lyapunov equation.

Therefore,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M905','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M905">View MathML</a>

Thus, this application of Theorem 2.2 to the instantaneous system with <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M906','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M906">View MathML</a>, leads to the previous averaged equation. To recover the initial case (16), we can let <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M452">View MathML</a>. We see that the function <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M14">View MathML</a> tends to

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M909','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M909">View MathML</a>

which we will rewrite <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M14">View MathML</a> for simplicity in the following. Thus, this definition of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M14">View MathML</a> defines the averaged system for the original equation (16).

In the derivation of the condition under which <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M912','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M912">View MathML</a> remain smaller than lp, the upper bound of the term A changes as follows. Define <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M913','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M913">View MathML</a> so that <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M914','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M914">View MathML</a> for all <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M278">View MathML</a>. Because we assume <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M916','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M916">View MathML</a>, the variation of parameters formula for linear retarded differential equations with constant coefficients (see Chapter 6 of [34]) reads <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M917','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M917">View MathML</a> where the resolvent U is the solution of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M918','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M918">View MathML</a>. We use Corollary 1.1 of Chapter 6 of [34], which is based on Grönwall’s lemma, to claim that <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M919','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M919">View MathML</a>. Therefore,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M920','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M920">View MathML</a>

Then, we used Young’s inequality for convolution to get <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M921','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M921">View MathML</a>.

Therefore, the upper bound of A remains unchanged.

Therefore, the polynomial P remains the same and Assumption 3.1 is still relevant to this problem. □

Lemma B.7If<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M212">View MathML</a>is the solution, with zero as initial condition, of<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M923','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M923">View MathML</a>, it can be written by the sum below which converges ifWis in<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M397">View MathML</a>for<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M322">View MathML</a>.

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M926','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M926">View MathML</a>

where<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M927','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M927">View MathML</a>and<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M928','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M928">View MathML</a>are convolution operators respectively generated by the functions<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M929','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M929">View MathML</a>and<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M930','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M930">View MathML</a>detailed below

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M931','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M931">View MathML</a>

whereHis the Heaviside function, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M932','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M932">View MathML</a>. If Δ is a pure imaginary number, the expression above still holds with the hyperbolic functionsshandchbeing turned into classical trigonometric functions sin and cos and Δ being replaced by its modulus.

IfWis in<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M397">View MathML</a>for<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M322">View MathML</a>, then this expansion converges.

Proof See the second example of [25]. □

Using Lemma C.3, on can redefine

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M935','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M935">View MathML</a>

where <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M688','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M688">View MathML</a> is the convolution operator generated by <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M937','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M937">View MathML</a> (see Appendix C for details). Observe that applying Young’s inequality for convolutions leads to <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M459">View MathML</a>.

Therefore, we can rewrite Theorem 3.3 into

Theorem B.8

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M939','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M939">View MathML</a>

Proof Similar to that of Theorem 3.3. □

Theorem B.9If Assumption 3.1 is verified for<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M940','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M940">View MathML</a>, there is a unique equilibrium point which is globally, asymptotically stable.

Proof Similar to the proof of Theorem B.4. □

With the same definitions for <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M941','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M941">View MathML</a> and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M942','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M942">View MathML</a>, we can show

Theorem B.10The weakly connected expansion of the equilibrium point is

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M943','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M943">View MathML</a>

Proof Define <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M843','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M843">View MathML</a> so that

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M945','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M945">View MathML</a>

So, the expansion will be in orders of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M946','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M946">View MathML</a> with <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M947','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M947">View MathML</a>.

Therefore,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M948','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M948">View MathML</a>

Actually, it is possible to compute recursively the nth terms, although their complexity explodes. Therefore, it is easy to compute <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M880','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M880">View MathML</a> for <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M873','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M873">View MathML</a>. By definition <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M951','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M951">View MathML</a>, which leads to the result. □

B.4 STDP learning with linear neurons and correlated noise

Consider the following n-dimensional stochastic differential system:

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M952','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M952">View MathML</a>

where u is a continuous input in <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M615','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M615">View MathML</a>, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M954','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M954">View MathML</a>, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M955','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M955">View MathML</a>, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M648','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M648">View MathML</a> and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M63">View MathML</a> is n-dimensional Brownian noise, and for all <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M958','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M958">View MathML</a>, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M959','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M959">View MathML</a> where H is the Heaviside function. Recall the well-posedness Assumption 3.2

Assumption B.1 There exists <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M322">View MathML</a> such that

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M961','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M961">View MathML</a>

Theorem B.11If Assumption 3.2 is verified for<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M334">View MathML</a>, then system (17) is asymptotically well posed in probability and the connectivity matrix<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M335">View MathML</a>, the solution of system (17), converges toWin the sense that for all<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M336">View MathML</a>,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M965','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M965">View MathML</a>

whereWis the deterministic solution of

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M966','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M966">View MathML</a>

where<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M339">View MathML</a>is the<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M213">View MathML</a>-periodic attractor of<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M969','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M969">View MathML</a>, where<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M342">View MathML</a>is supposed to be fixed. And<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M901','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M901">View MathML</a>is described below.

Proof We recall the instantaneous reformulation of the original system (17)

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M972','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M972">View MathML</a>

With this linear expression, the structure of the proof of Theorem 3.1 remains unchanged. The correlation term is to be replaced by <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M973','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M973">View MathML</a>. The noise term we are looking for is <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M901','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M901">View MathML</a> in the Lyapunov equation (see (9)) below

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M975','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M975">View MathML</a>

This leads to the system

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M976','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M976">View MathML</a>

(34)

The matrix <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M497','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M497">View MathML</a> is the solution of a Lyapunov equation (see equation (a)). Lemma D.1 gives an explicit solution: <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M978','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M978">View MathML</a>. Equation (b) leads to

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M979','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M979">View MathML</a>

We see that it does not depend on <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M980','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M980">View MathML</a>, which, once Theorem 2.2 is applied, can be considered null so that the average system <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M14">View MathML</a> corresponds to the original system (17).

Therefore,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M982','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M982">View MathML</a>

(35)

We show that for W already in <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M397">View MathML</a>, it will stay forever in <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M397">View MathML</a>:

1. Inequality <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M330">View MathML</a>:

Decomposing the connectivity as <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M986','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M986">View MathML</a> leads to <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M987','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M987">View MathML</a>. By hermiticity of S and A, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M988','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M988">View MathML</a> and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M989','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M989">View MathML</a> are real numbers. This means we only have to show that the eigenvalues of S remain positive along the dynamics. Taking the symmetric part of equation (35) leads to

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M990','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M990">View MathML</a>

Suppose we take an initial condition <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M991','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M991">View MathML</a>. It is clear that if <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M992','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M992">View MathML</a> and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M892','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M892">View MathML</a> are always positive, then S will remain positive. This would prove the result. Therefore, focus on

• Proving <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M994','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M994">View MathML</a>:

According to the first point of Lemma C.1, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M995','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M995">View MathML</a>. Therefore, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M996','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M996">View MathML</a> is a Gramian matrix and therefore it is positive.

• Proving <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M997','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M997">View MathML</a>:

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M892','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M892">View MathML</a> is the covariance matrix of the random value z, therefore, it is positive-semi-definite.

2. Inequality <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M735','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M735">View MathML</a>:

For all <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1000','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1000">View MathML</a> such that <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M741','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M741">View MathML</a>, define a family of positive numbers <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M742','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M742">View MathML</a> whose supremum is written <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M743','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M743">View MathML</a> and a family of functions <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M744','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M744">View MathML</a> such that

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1005','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1005">View MathML</a>

Because g is linear, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1006','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1006">View MathML</a>. For <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M748','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M748">View MathML</a>, i.e., <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1008','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1008">View MathML</a>, compute

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1009','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1009">View MathML</a>

• Upper bound of A:

Cauchy-Schwarz leads to

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1010','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1010">View MathML</a>

As before, we can use Young’s inequality for convolutions to find an upper bound of A which reads

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1011','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1011">View MathML</a>

• Upper bound of B:

According to Proposition 11.9.3 of [35] the solution of the Lyapunov equation (a) in system (34) can be rewritten

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1012','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1012">View MathML</a>

because <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1013','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1013">View MathML</a> is not singular due to the fact <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1014','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1014">View MathML</a>.

Observe that for A a positive matrix whose eigenvalues are the <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M755','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M755">View MathML</a>, then the spectrum of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1016','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1016">View MathML</a> is <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1017','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1017">View MathML</a>. Therefore, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1018','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1018">View MathML</a>. Therefore, if <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1019','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1019">View MathML</a>, then <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1020','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1020">View MathML</a>. This leads to

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1021','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1021">View MathML</a>

Then we apply the same arguments to say that

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1022','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1022">View MathML</a>

The rest of the proof is identical to the Hebbian case. Assumption 3.1 is changed to Assumption 3.2 for <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M397">View MathML</a> to be invariant by the flow <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M14">View MathML</a>. □

Define

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1025','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1025">View MathML</a>

such that <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1026','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1026">View MathML</a>.

In this framework, one can prove

Theorem B.12The correlation term can be written

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1027','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1027">View MathML</a>

Proof Similar to that of Theorem 3.3. □

Theorem B.13If Assumption 3.2 is verified for<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M420">View MathML</a>, there is a unique equilibrium point which is globally, asymptotically stable.

Proof Similar to the previous case. □

Now, we proceed as before by defining

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1029','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1029">View MathML</a>

Theorem B.14

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1030','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1030">View MathML</a>

Proof First, we need to work on the noise term <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1031','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1031">View MathML</a>. Recall <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M497','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M497">View MathML</a> is the solution of the Lyapunov equation <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1033','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1033">View MathML</a>. Lemma D.1 says that

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1034','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1034">View MathML</a>

is a well-defined solution. We now use the fact that <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1035','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1035">View MathML</a> to show that

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1036','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1036">View MathML</a>

and therefore

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1037','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1037">View MathML</a>

Thus, writing <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1038','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1038">View MathML</a> and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1039','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1039">View MathML</a>, the noise term is

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1040','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1040">View MathML</a>

Define <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M843','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M843">View MathML</a> such that <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M844','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M844">View MathML</a>. We improperly write <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1043','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1043">View MathML</a> such that

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1044','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1044">View MathML</a>

This leads to

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1045','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1045">View MathML</a>

We are looking for <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M862','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M862">View MathML</a> and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M865','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M865">View MathML</a> in the expansions <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1048','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1048">View MathML</a> and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1049','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1049">View MathML</a>. Recall

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1050','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1050">View MathML</a>

Therefore,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1051','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1051">View MathML</a>

Leading to

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1052','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1052">View MathML</a>

This equation is scary but it reduces to simple expressions for small <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M873','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M873">View MathML</a>.

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1054','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1054">View MathML</a>

Recall that <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1055','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1055">View MathML</a> to get the result. □

Appendix C: Properties of the convolution operators <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M676','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M676">View MathML</a>, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1057','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1057">View MathML</a>, and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M688','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M688">View MathML</a>

Recall <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M676','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M676">View MathML</a>, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1057','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1057">View MathML</a>, and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M688','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M688">View MathML</a> are convolution operators respectively generated by <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M487">View MathML</a>, v, and w defined in (30). Their Fourier transforms are respectively

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1063','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1063">View MathML</a>

C.1 Algebraic properties

Lemma C.1

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1064','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1064">View MathML</a>

Proof Compute

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1065','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1065">View MathML</a>

Therefore, if <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1066','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1066">View MathML</a>, then <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1067','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1067">View MathML</a>, and if <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1068','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1068">View MathML</a>, then <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1069','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1069">View MathML</a>. The result follows. □

Lemma C.2

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1070','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1070">View MathML</a>

where<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1071','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1071">View MathML</a>is the time-differentiation operator, i.e., <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1072','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1072">View MathML</a>.

Proof<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M676','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M676">View MathML</a> and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1074','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1074">View MathML</a> are two convolution operators respectively generated by <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M959','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M959">View MathML</a> and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1076','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1076">View MathML</a>. The Fourier transform of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M487">View MathML</a> is <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1078','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1078">View MathML</a>. Therefore, the Fourier transform of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1079','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1079">View MathML</a> is

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1080','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1080">View MathML</a>

Because <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1081','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1081">View MathML</a>, taking the inverse Fourier transform of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1082','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1082">View MathML</a> gives the result. □

Lemma C.3

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1083','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1083">View MathML</a>

Besides, if<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1084','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1084">View MathML</a>, <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1085','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1085">View MathML</a>is a convolution operator generated by

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1086','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1086">View MathML</a>

where<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M461">View MathML</a>is the Bessel function of the first kind. If<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1088','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1088">View MathML</a>, the formula above holds if one replaces<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M461">View MathML</a>by<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1090','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1090">View MathML</a>, the modified Bessel function of the first kind.

Proof We want to compute <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1091','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1091">View MathML</a>. Compute the Fourier transform of <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1092','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1092">View MathML</a>, where <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1093','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1093">View MathML</a> is the result of k convolutions of v with itself

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1094','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1094">View MathML</a>

This proves the first result.

Then observe that

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1095','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1095">View MathML</a>

The last integral can be analytically computed with the help of Bessel functions. In fact, it gives different results depending on the nature of Δ (whether it is real or imaginary).

• If <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1088','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1088">View MathML</a>, then defining <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1090','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1090">View MathML</a>, the modified Bessel function of the first kind, leads to

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1098','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1098">View MathML</a>

• If <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1084','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1084">View MathML</a>, then defining <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M461">View MathML</a>, the Bessel function of the first kind, leads to

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1101','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1101">View MathML</a>

This concludes the proof. □

C.2 Signed integral

1. <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1102','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1102">View MathML</a>.

2. For <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1103','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1103">View MathML</a>, compute

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1104','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1104">View MathML</a>

3. Similarly,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1105','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1105">View MathML</a>

C.3 L1 norm

• For <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1106','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1106">View MathML</a>, i.e., <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1107','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1107">View MathML</a>, then

1. <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1108','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1108">View MathML</a> and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1109','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1109">View MathML</a>.

2. <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1110','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1110">View MathML</a> and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1111','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1111">View MathML</a>.

3. <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1112','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1112">View MathML</a> and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1113','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1113">View MathML</a>.

• For <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1114','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1114">View MathML</a>, i.e., Δ is a pure imaginary, we rewrite <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1115','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1115">View MathML</a> and observe that

1. <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1108','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1108">View MathML</a> and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1109','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1109">View MathML</a>.

2. <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1118','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1118">View MathML</a> which changes sign on <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M333">View MathML</a>. Therefore,

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1120','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1120">View MathML</a>

3. <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1121','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1121">View MathML</a> which also changes sign on <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M333">View MathML</a>. We have not found a way to compute <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1123','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1123">View MathML</a> and write the result elegantly.

Appendix D: Solution of a Lyapunov equation

Lemma D.1The solution of the following Lyapunov equation

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1124','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1124">View MathML</a>

where<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1125','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1125">View MathML</a>is

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1126','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1126">View MathML</a>

(36)

Proof First, observe that if <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1127','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1127">View MathML</a> and <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1128','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1128">View MathML</a>, then <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1129','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1129">View MathML</a>. Therefore, X is well defined by equation (36).

Observe that <a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1130','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1130">View MathML</a>. Assuming X is defined by equation (36), then based on the fact L commutes with any matrix (because it is a scalar matrix),

<a onClick="popup('http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1131','MathML',630,470);return false;" target="_blank" href="http://www.mathematical-neuroscience.com/content/2/1/13/mathml/M1131">View MathML</a>

 □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

GW developed the theory of temporal averaging presented in this paper. MG applied this theory to learning neural networks and did the numerical simulations. Both authors read and approved the final manuscript.

Acknowledgements

MG thanks Olivier Faugeras for his support. MG was partially funded by the ERC advanced grant NerVi nb227747, by the IP project BrainScaleS #269921 and by the région PACA, France. GW thanks L. Ryzhik from Stanford University, Department of Mathematics for his hospitality during 2010-2011 where part of this work was achieved.

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