http://www.mathematical-neuroscience.com The latest research articles published by The Journal of Mathematical Neuroscience 2012-05-02T00:00:00Z Neural field models describe the coarse-grained activity of populations of interacting neurons. Because of the laminar structure of real cortical tissue they are often studied in two spatial dimensions, where they are well known to generate rich patterns of spatiotemporal activity. Such patterns have been interpreted in a variety of contexts ranging from the understanding of visual hallucinations to the generation of electroencephalographic signals. Typical patterns include localized solutions in the form of traveling spots, as well as intricate labyrinthine structures. These patterns are naturally defined by the interface between low and high states of neural activity. Here we derive the equations of motion for such interfaces and show, for a Heaviside firing rate, that the normal velocity of an interface is given in terms of a non-local Biot-Savart type interaction over the boundaries of the high activity regions. This exact, but dimensionally reduced, system of equations is solved numerically and shown to be in excellent agreement with the full nonlinear integral equation defining the neural field. We develop a linear stability analysis for the interface dynamics that allows us to understand the mechanisms of pattern formation that arise from instabilities of spots, rings, stripes and fronts. We further show how to analyze neural field models with linear adaptation currents, and determine the conditions for the dynamic instability of spots that can give rise to breathers and traveling waves. http://www.mathematical-neuroscience.com/content/2/1/9 Stephen Coombes Helmut Schmidt Ingo Bojak The Journal of Mathematical Neuroscience 2012, null:9 2012-05-02T00:00:00Z doi:10.1186/2190-8567-2-9 /content/figures/2190-8567-2-9-toc.gif The Journal of Mathematical Neuroscience 2190-8567 ${item.volume} 9 2012-05-02T00:00:00Z PDF A lumped model of neural activity in neocortex is studied to identify regions of multi-stability of both steady states and periodic solutions. Presence of both steady states and periodic solutions is considered to correspond with epileptogenesis. The model, which consists of two delay differential equations with two fixed time lags, is mainly studied for its dependency on varying connection strength between populations. Equilibria are identified, and using linear stability analysis, all transitions are determined under which both trivial and non-trivial fixed points lose stability. Periodic solutions arising at some of these bifurcations are numerically studied with a two-parameter bifurcation analysis. http://www.mathematical-neuroscience.com/content/2/1/8 Sid Visser Hil Meijer Michel van Putten Stephan van Gils The Journal of Mathematical Neuroscience 2012, null:8 2012-04-25T00:00:00Z doi:10.1186/2190-8567-2-8 /content/figures/2190-8567-2-8-toc.gif The Journal of Mathematical Neuroscience 2190-8567 ${item.volume} 8 2012-04-25T00:00:00Z PDF Transient bursting behaviour of excitable cells, such as neurons, is a common feature observed experimentally, but theoretically, it is not well understood. We analyse a five-dimensional simplified model of after-depolarisation that exhibits transient bursting behaviour when perturbed with a short current injection. Using one-parameter continuation of the perturbed orbit segment formulated as a well-posed boundary value problem, we show that the spike-adding mechanism is a canard-like transition that has a different character from known mechanisms for periodic burst solutions. The biophysical basis of the model gives a natural time-scale separation, which allows us to explain the spike-adding mechanism using geometric singular perturbation theory, but it does not involve actual bifurcations as for periodic bursts. We show that unstable sheets of the critical manifold, formed by saddle equilibria of the system that only exist in a singular limit, are responsible for the spike-adding transition; the transition is organised by the slow flow on the critical manifold near folds of this manifold. Our analysis shows that the orbit segment during the spike-adding transition includes a fast transition between two unstable sheets of the slow manifold that are of saddle type. We also discuss a different parameter regime where the presence of additional saddle equilibria of the full system alters the spike-adding mechanism. http://www.mathematical-neuroscience.com/content/2/1/7 Jakub Nowacki Hinke Osinga Krasimira Tsaneva-Atanasova The Journal of Mathematical Neuroscience 2012, null:7 2012-04-24T00:00:00Z doi:10.1186/2190-8567-2-7 /content/figures/2190-8567-2-7-toc.gif The Journal of Mathematical Neuroscience 2190-8567 ${item.volume} 7 2012-04-24T00:00:00Z PDF In this paper, we analyze the invasion and extinction of activity in heterogeneous neural fields. We first consider the effects of spatial heterogeneities on the propagation of an invasive activity front. In contrast to previous studies of front propagation in neural media, we assume that the front propagates into an unstable rather than a metastable zero-activity state. For sufficiently localized initial conditions, the asymptotic velocity of the resulting pulled front is given by the linear spreading velocity, which is determined by linearizing about the unstable state within the leading edge of the front. One of the characteristic features of these so-called pulled fronts is their sensitivity to perturbations inside the leading edge. This means that standard perturbation methods for studying the effects of spatial heterogeneities or external noise fluctuations break down. We show how to extend a partial differential equation method for analyzing pulled fronts in slowly modulated environments to the case of neural fields with slowly modulated synaptic weights. The basic idea is to rescale space and time so that the front becomes a sharp interface whose location can be determined by solving a corresponding local Hamilton-Jacobi equation. We use steepest descents to derive the Hamilton-Jacobi equation from the original nonlocal neural field equation. In the case of weak synaptic heterogenities, we then use perturbation theory to solve the corresponding Hamilton equations and thus determine the time-dependent wave speed. In the second part of the paper, we investigate how time-dependent heterogenities in the form of extrinsic multiplicative noise can induce rare noise-driven transitions to the zero-activity state, which now acts as an absorbing state signaling the extinction of all activity. In this case, the most probable path to extinction can be obtained by solving the classical equations of motion that dominate a path integral representation of the stochastic neural field in the weak noise limit. These equations take the form of nonlocal Hamilton equations in an infinite-dimensional phase space. http://www.mathematical-neuroscience.com/content/2/1/6 Paul Bressloff The Journal of Mathematical Neuroscience 2012, null:6 2012-03-26T00:00:00Z doi:10.1186/2190-8567-2-6 /content/figures/2190-8567-2-6-toc.gif The Journal of Mathematical Neuroscience 2190-8567 ${item.volume} 6 2012-03-26T00:00:00Z PDF We consider a coupled, heterogeneous population of relaxation oscillators used to model rhythmic oscillations in the pre-Botzinger complex. By choosing specific values of theparameter used to describe the heterogeneity, sampled from the probability distribution of the values of that parameter, we show how the effects of heterogeneity can be studied in a computationally efficient manner. When more that one parameter is heterogeneous, full or sparse tensor product grids are used to select appropriate parameter values. The method allows us to effectively reduce the dimensionality of the model,and it provides a means for systematically investigating the effects of heterogeneity in coupled systems, linking ideas from uncertainty quantification to those for the study of network dynamics. http://www.mathematical-neuroscience.com/content/2/1/5 Carlo Laing Yu Zou Ben Smith Ioannis Kevrekidis The Journal of Mathematical Neuroscience 2012, null:5 2012-03-14T00:00:00Z doi:10.1186/2190-8567-2-5 /content/figures/2190-8567-2-5-toc.gif The Journal of Mathematical Neuroscience 2190-8567 ${item.volume} 5 2012-03-14T00:00:00Z PDF We present a novel extension of fast-slow analysis of clustered solutions to coupled networks of three cells, allowing for heterogeneity in the cells' intrinsic dynamics. In the model on which we focus, each cell is described by a pair of first-order differential equations, which are based on recent reduced neuronal network models for respiratory rhythmogenesis. Within each pair of equations, one dependent variable evolves on a fast time scale and one on a slow scale. The cells are coupled with inhibitory synapses that turn on and off on the fast time scale. In this context, we analyze solutions in which cells take turns activating, allowing any activation order, including multiple activations of two of the cells between successive activations of the third. Our analysis proceeds via the derivation of a set of explicit maps between the pairs of slow variables corresponding to the non-active cells on each cycle. We show how these maps can be used to determine the order in which cells will activate for a given initial condition and how evaluation of these maps on a few key curves in their domains can be used to constrain the possible activation orders that will be observed in network solutions. Moreover, under a small set of additional simplifying assumptions, we collapse the collection of maps into a single 2D map that can be computed explicitly. From this unified map, we analytically obtain boundary curves between all regions of initial conditions producing different activation patterns. http://www.mathematical-neuroscience.com/content/2/1/4 Jonathan Rubin David Terman The Journal of Mathematical Neuroscience 2012, null:4 2012-03-12T00:00:00Z doi:10.1186/2190-8567-2-4 /content/figures/2190-8567-2-4-toc.gif The Journal of Mathematical Neuroscience 2190-8567 ${item.volume} 4 2012-03-12T00:00:00Z PDF Rapid action potential generation - spiking - and alternating intervals of spiking and quiescence - bursting - are two dynamic patterns commonly observed in neuronal activity. In computational models of neuronal systems, the transition from spiking to bursting often exhibits complex bifurcation structure. One type of transition involves the torus canard, which we show arises in a broad array of well-known computational neuronal models with three different classes of bursting dynamics: sub-Hopf/fold cycle bursting, circle/fold cycle bursting, and fold/fold cycle bursting. The essential features that these models share are multiple time scales leading naturally to decomposition into slow and fast systems, a saddle-node of periodic orbits in the fast system, and a torus bifurcation in the full system. We show that the transition from spiking to bursting in each model system is given by an explosion of torus canards. Based on these examples, as well as on emerging theory, we propose that torus canards are a common dynamic phenomenon separating the regimes of spiking and bursting activity. http://www.mathematical-neuroscience.com/content/2/1/3 John Burke Mathieu Desroches Anna Barry Tasso Kaper Mark Kramer The Journal of Mathematical Neuroscience 2012, null:3 2012-02-21T00:00:00Z doi:10.1186/2190-8567-2-3 /content/figures/2190-8567-2-3-toc.gif The Journal of Mathematical Neuroscience 2190-8567 ${item.volume} 3 2012-02-21T00:00:00Z PDF We study synaptic plasticity in a complex neuronal cell model where NMDA-spikes can arise in certain dendritic zones. In the context of reinforcement learning two kinds of plasticity rules are derived, zone reinforcement (ZR) and cell reinforcement (CR), which both optimize the expected reward by stochastic gradient ascent. For ZR the synaptic plasticity response to the external reward signal is modulated exclusively by quantities which are local to the NMDA spike initiation zone in which the synapse is situated. CR in addition uses nonlocal feedback from the soma of the cell, provided by mechanisms such as the backpropagating action potential. Simulation results show that, compared to ZR, the use of nonlocal feedback in CR can drastically enhance learning performance. We suggest that the availability of nonlocal feedback for learning is a key advantage of complex neurons over networks of simple point neurons, which have previously been found to be largely equivalent with regard to computational capability. http://www.mathematical-neuroscience.com/content/2/1/2 Mathieu Schiess Robert Urbanczik Walter Senn The Journal of Mathematical Neuroscience 2012, null:2 2012-02-15T00:00:00Z doi:10.1186/2190-8567-2-2 /content/figures/2190-8567-2-2-toc.gif The Journal of Mathematical Neuroscience 2190-8567 ${item.volume} 2 2012-02-15T00:00:00Z PDF We describe a phenomenological model of seizure initiation, consisting of a bistable switch between stable fixed point and stable limit-cycle attractors. We determine a quasi-analytic formula for the exit time problem for our model in the presence of noise. This formula--which we equate to seizure frequency--is then validated numerically, before we extend our study to explore the combined effects of noise and network structure on escape times. Here, we observe that weakly connected networks of 2, 3 and 4 nodes with equivalent first transitive components all have the same asymptotic escape times. We finally extend this work to larger networks, inferred from electroencephalographic recordings from 35 patients with idiopathic generalised epilepsies and 40 controls. Here, we find that network structure in patients correlates with smaller escape times relative to network structures from controls. These initial findings are suggestive that network structure may play an important role in seizure initiation and seizure frequency. http://www.mathematical-neuroscience.com/content/2/1/1 Oscar Benjamin Thomas Fitzgerald Peter Ashwin Krasimira Tsaneva-Atanasova Fahmida Chowdhury Mark Richardson John Terry The Journal of Mathematical Neuroscience 2012, null:1 2012-01-06T00:00:00Z doi:10.1186/2190-8567-2-1 /content/figures/2190-8567-2-1-toc.gif The Journal of Mathematical Neuroscience 2190-8567 ${item.volume} 1 2012-01-06T00:00:00Z XML We introduce a test for robustness of heteroclinic cycles that appear in neural microcircuits modelled as coupled dynamical cells. Robust heteroclinic cycles (RHCs) can appear as robust attractors in Lotka-Volterra-type winnerless competition models as well as in more general coupled and/or symmetric systems. It has been previously suggested that RHCs may be relevant to a range of neural activities, from encoding and binding tospatio-temporal sequence generation. The robustness or otherwise of such cycles depends both on the coupling structure and the internal structure of the neurons. We verify that RHCs can appear in systems of three identical cells, but only if we require perturbations to preserve some invariant subspaces for the individual cells. On the other hand, heteroclinic attractors can appear robustly in systems of four or more identical cells for some symmetric coupling patterns, without restriction on the internal dynamics of the cells. http://www.mathematical-neuroscience.com/content/1/1/13 Peter Ashwin Ozkan Karabacak Thomas Nowotny The Journal of Mathematical Neuroscience 2011, null:13 2011-11-28T00:00:00Z doi:10.1186/2190-8567-1-13 /content/figures/2190-8567-1-13-toc.gif The Journal of Mathematical Neuroscience 2190-8567 ${item.volume} 13 2011-11-28T00:00:00Z PDF