http://www.mathematical-neuroscience.com The most accessed research articles published by The Journal of Mathematical Neuroscience 2012-05-02T00:00:00Z Background: Development of effective and plausible numerical tools is an imperative task for thorough studies of nonlinear dynamics in life science applications. Results: We have developed a complementary suite of computational tools for two-parameter screening of dynamics in neuronal models. We test a "brute-force" effectiveness of neuroscience plausible techniques specifically tailored for the examination of temporal characteristics, such duty cycle of bursting, interspike interval, spike number deviation in the phenomenological Hindmarsh-Rose model of a bursting neuron and compare the results obtained by calculus-based tools for evaluations of an entire spectrum of Lyapunov exponents broadly employed in studies of nonlinear systems. Conclusions: We have found that the results obtained either way agree exceptionally well, and can identify and differentiate between various fine structures of complex dynamics and underlying global bifurcations in this exemplary model. Our future planes are to enhance the applicability of this computational suite for understanding of polyrhythmic bursting patterns and their functional transformations in small networks. http://www.mathematical-neuroscience.com/content/1/1/6 Roberto Barrio Andrey Shilnikov The Journal of Mathematical Neuroscience 2011, null:6 2011-07-11T00:00:00Z doi:10.1186/2190-8567-1-6 /content/figures/2190-8567-1-6-toc.gif The Journal of Mathematical Neuroscience 2190-8567 ${item.volume} 6 2011-07-11T00:00:00Z XML A major obstacle in the analysis of many physiological models is the issue of model simplification. Various methods have been used for simplifying such models, with one common technique being to eliminate certain 'fast' variables using a quasi-steady-state assumption. In this article, we show when such a physiological model reduction technique in a slow-fast system is mathematically justified. We provide counterexamples showing that this technique can give erroneous results near the onset of oscillatory behaviour which is, practically, the region of most importance in a model. In addition, we show that the singular limit of the first Lyapunov coefficient of a Hopf bifurcation in a slow-fast system is, in general, not equal to the first Lyapunov coefficient of the Hopf bifurcation in the corresponding layer problem, a seemingly counterintuitive result. Consequently, one cannot deduce, in general, the criticality of a Hopf bifurcation in a slow-fast system from the lower-dimensional layer problem. http://www.mathematical-neuroscience.com/content/1/1/9 Wenjun Zhang Vivien Kirk James Sneyd Martin Wechselberger The Journal of Mathematical Neuroscience 2011, null:9 2011-09-23T00:00:00Z doi:10.1186/2190-8567-1-9 /content/figures/2190-8567-1-9-toc.gif The Journal of Mathematical Neuroscience 2190-8567 ${item.volume} 9 2011-09-23T00:00:00Z XML 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 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 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 Background: Physical and physiological invariance laws, in particular time invariance and local symmetry, are at the outset of an abstract model. Harmonic analysis and Lie theory are the mathematical prerequisites for its deduction. Results: The main result is a linear system of partial differential equations (referred to as the structure equations) that describe the result of signal processing in the cochlea. It is formulated for phase and for the logarithm of the amplitude. The changes of these quantities are the essential physiological observables in the description of signal processing in the auditory pathway. Conclusions: The structure equations display in a quantitative way the subtle balance for processing information on the basis of phase versus amplitude. From a mathematical point of view, the linear system of equations is classified as an inhomogeneous - equation. In suitable variables the solutions can be represented as the superposition of a particular solution (determined by the system) and a holomorphic function (determined by the incoming signal). In this way, a global picture of signal processing in the cochlea emerges. http://www.mathematical-neuroscience.com/content/1/1/5 Hans Martin Reimann The Journal of Mathematical Neuroscience 2011, null:5 2011-06-06T00:00:00Z doi:10.1186/2190-8567-1-5 /content/figures/2190-8567-1-5-toc.gif The Journal of Mathematical Neuroscience 2190-8567 ${item.volume} 5 2011-06-06T00:00:00Z XML We employ a Hodgkin-Huxley-type model of basolateral ionic currents in bullfrog saccular hair cells for studying the genesis of spontaneous voltage oscillations and their role in shaping the response of the hair cell to external mechanical stimuli. Consistent with recent experimental reports, we find that the spontaneous dynamics of the model can be categorized using conductance parameters of calcium-activated potassium, inward rectifier potassium, and mechano-electrical transduction (MET) ionic currents. The model is demonstrated for exhibiting a broad spectrum of autonomous rhythmic activity, including periodic and quasi-periodic oscillations with two independent frequencies as well as various regular and chaotic bursting patterns. Complex patterns of spontaneous oscillations in the model emerge at small values of the conductance of Ca2+-activated potassium currents. These patterns are significantly affected by thermal fluctuations of the MET current. We show that self-sustained regular voltage oscillations lead to enhanced and sharply tuned sensitivity of the hair cell to weak mechanical periodic stimuli. While regimes of chaotic oscillations are argued to result in poor tuning to sinusoidal driving, chaotically oscillating cells do provide a high sensitivity to low-frequency variations of external stimuli. http://www.mathematical-neuroscience.com/content/1/1/11 Alexander Neiman Kai Dierkes Benjamin Lindner Lijuan Han Andrey Shilnikov The Journal of Mathematical Neuroscience 2011, null:11 2011-10-31T00:00:00Z doi:10.1186/2190-8567-1-11 /content/figures/2190-8567-1-11-toc.gif The Journal of Mathematical Neuroscience 2190-8567 ${item.volume} 11 2011-10-31T00:00:00Z XML We study the neural field equations introduced by Chossat and Faugeras to model the representation and the processing of image edges and textures in the hypercolumns of the cortical area V1. The key entity, the structure tensor, intrinsically lives in a non-Euclidean, in effect hyperbolic, space. Its spatio-temporal behaviour is governed by nonlinear integro-differential equations defined on the Poincaré disc model of the two-dimensional hyperbolic space. Using methods from the theory of functional analysis we show the existence and uniqueness of a solution of these equations. In the case of stationary, i.e. time independent, solutions we perform a stability analysis which yields important results on their behavior. We also present an original study, based on non-Euclidean, hyperbolic, analysis, of a spatially localised bump solution in a limiting case. We illustrate our theoretical results with numerical simulations.AMS subject classifications: 30F45, 33C05, 34A12, 34D20, 34D23, 34G20, 37M05, 43A85, 44A35, 45G10, 51M10, 92B20, 92C20. http://www.mathematical-neuroscience.com/content/1/1/4 Gregory Faye Pascal Chossat Olivier Faugeras The Journal of Mathematical Neuroscience 2011, null:4 2011-06-06T00:00:00Z doi:10.1186/2190-8567-1-4 /content/figures/2190-8567-1-4-toc.gif The Journal of Mathematical Neuroscience 2190-8567 ${item.volume} 4 2011-06-06T00:00:00Z XML Pituitary cells of the anterior pituitary gland secrete hormones in response to patterns of electrical activity. Several types of pituitary cells produce short bursts of electrical activity which are more effective than single spikes in evoking hormone release. These bursts, called pseudo-plateau bursts, are unlike bursts studied mathematically in neurons (plateau bursting) and the standard fast-slow analysis used for plateau bursting is of limited use. Using an alternative fast-slow analysis, with one fast and two slow variables, we show that pseudo-plateau bursting is a canard-induced mixed mode oscillation. Using this technique, it is possible to determine the region of parameter space where bursting occurs as well as salient properties of the burst such as the number of spikes in the burst. The information gained from this one-fast/two-slow decomposition complements the information obtained from a two-fast/one-slow decomposition. http://www.mathematical-neuroscience.com/content/1/1/12 Wondimu Teka Joel Tabak Theodore Vo Martin Wechselberger Richard Bertram The Journal of Mathematical Neuroscience 2011, null:12 2011-11-08T00:00:00Z doi:10.1186/2190-8567-1-12 /content/figures/2190-8567-1-12-toc.gif The Journal of Mathematical Neuroscience 2190-8567 ${item.volume} 12 2011-11-08T00:00:00Z XML We extend the theory of noise-induced phase synchronization to the case of a neural master equation describing the stochastic dynamics of an ensemble of uncoupled neuronal population oscillators with intrinsic and extrinsic noise. The master equation formulation of stochastic neurodynamics represents the state of each population by the number of currently active neurons, and the state transitions are chosen so that deterministic Wilson-Cowan rate equations are recovered in the mean-field limit. We apply phase reduction and averaging methods to a corresponding Langevin approximation of the master equation in order to determine how intrinsic noise disrupts synchronization of the population oscillators driven by a common extrinsic noise source. We illustrate our analysis by considering one of the simplest networks known to generate limit cycle oscillations at the population level, namely, a pair of mutually coupled excitatory (E) and inhibitory (I) subpopulations. We show how the combination of intrinsic independent noise and extrinsic common noise can lead to clustering of the population oscillators due to the multiplicative nature of both noise sources under the Langevin approximation. Finally, we show how a similar analysis can be carried out for another simple population model that exhibits limit cycle oscillations in the deterministic limit, namely, a recurrent excitatory network with synaptic depression; inclusion of synaptic depression into the neural master equation now generates a stochastic hybrid system. http://www.mathematical-neuroscience.com/content/1/1/2 Paul Bressloff Yi Ming Lai The Journal of Mathematical Neuroscience 2011, null:2 2011-05-03T00:00:00Z doi:10.1186/2190-8567-1-2 /content/figures/2190-8567-1-2-toc.gif The Journal of Mathematical Neuroscience 2190-8567 ${item.volume} 2 2011-05-03T00:00:00Z XML