115 research outputs found
Period-adding bifurcations and chaos in a periodically stimulated excitable neural relaxation oscillator
The response of an excitable neuron to trains of electrical spikes is relevant to the understanding
of the neural code. In this paper we study a neurobiologically motivated relaxation oscillator, with
appropriately identified fast and slow coordinates, that admits an explicit mathematical analysis.
An application of geometric singular perturbation theory shows the existence of an attracting
invariant manifold which is used to construct the Fenichel normal form for the system. This
facilitates the calculation of the response of the system to pulsatile stimulation and allows the
construction of a so-called extended isochronal map. The isochronal map is shown to have a single
discontinuity and be of a type that can admit three types of response: mode-locked, quasi-periodic
and chaotic. The bifurcation structure of the system is seen to be extremely rich and supports
period-adding bifurcations separated by windows of both chaos and periodicity. A bifurcation
analysis of the isochronal map is presented in conjunction with a description of the various routes
to chaos in this system
Period-adding bifurcations and chaos in a periodically stimulated excitable neural relaxation oscillator
This is a pre-print. The definitive version: COOMBES, S. and OSBALDESTIN, A.H., 2000. Period-adding bifurcations and chaos in a periodically stimulated excitable neural relaxation oscillator. Physical Review E, 62(3), pp.4057-4066 Part B.The response of an excitable neuron to trains of electrical spikes is relevant to the understanding
of the neural code. In this paper we study a neurobiologically motivated relaxation oscillator, with
appropriately identified fast and slow coordinates, that admits an explicit mathematical analysis.
An application of geometric singular perturbation theory shows the existence of an attracting
invariant manifold which is used to construct the Fenichel normal form for the system. This
facilitates the calculation of the response of the system to pulsatile stimulation and allows the
construction of a so-called extended isochronal map. The isochronal map is shown to have a single
discontinuity and be of a type that can admit three types of response: mode-locked, quasi-periodic
and chaotic. The bifurcation structure of the system is seen to be extremely rich and supports
period-adding bifurcations separated by windows of both chaos and periodicity. A bifurcation
analysis of the isochronal map is presented in conjunction with a description of the various routes
to chaos in this system
Metastability in a stochastic neural network modeled as a velocity jump Markov process
One of the major challenges in neuroscience is to determine how noise that is
present at the molecular and cellular levels affects dynamics and information
processing at the macroscopic level of synaptically coupled neuronal
populations. Often noise is incorprated into deterministic network models using
extrinsic noise sources. An alternative approach is to assume that noise arises
intrinsically as a collective population effect, which has led to a master
equation formulation of stochastic neural networks. In this paper we extend the
master equation formulation by introducing a stochastic model of neural
population dynamics in the form of a velocity jump Markov process. The latter
has the advantage of keeping track of synaptic processing as well as spiking
activity, and reduces to the neural master equation in a particular limit. The
population synaptic variables evolve according to piecewise deterministic
dynamics, which depends on population spiking activity. The latter is
characterised by a set of discrete stochastic variables evolving according to a
jump Markov process, with transition rates that depend on the synaptic
variables. We consider the particular problem of rare transitions between
metastable states of a network operating in a bistable regime in the
deterministic limit. Assuming that the synaptic dynamics is much slower than
the transitions between discrete spiking states, we use a WKB approximation and
singular perturbation theory to determine the mean first passage time to cross
the separatrix between the two metastable states. Such an analysis can also be
applied to other velocity jump Markov processes, including stochastic
voltage-gated ion channels and stochastic gene networks
Mechanisms explaining transitions between tonic and phasic firing in neuronal populations as predicted by a low dimensional firing rate model
Several firing patterns experimentally observed in neural populations have
been successfully correlated to animal behavior. Population bursting, hereby
regarded as a period of high firing rate followed by a period of quiescence, is
typically observed in groups of neurons during behavior. Biophysical
membrane-potential models of single cell bursting involve at least three
equations. Extending such models to study the collective behavior of neural
populations involves thousands of equations and can be very expensive
computationally. For this reason, low dimensional population models that
capture biophysical aspects of networks are needed.
\noindent The present paper uses a firing-rate model to study mechanisms that
trigger and stop transitions between tonic and phasic population firing. These
mechanisms are captured through a two-dimensional system, which can potentially
be extended to include interactions between different areas of the nervous
system with a small number of equations. The typical behavior of midbrain
dopaminergic neurons in the rodent is used as an example to illustrate and
interpret our results.
\noindent The model presented here can be used as a building block to study
interactions between networks of neurons. This theoretical approach may help
contextualize and understand the factors involved in regulating burst firing in
populations and how it may modulate distinct aspects of behavior.Comment: 25 pages (including references and appendices); 12 figures uploaded
as separate file
Birth and destruction of collective oscillations in a network of two populations of coupled type 1 neurons
We study the macroscopic dynamics of large networks of excitable type 1
neurons composed of two populations interacting with disparate but symmetric
intra- and inter-population coupling strengths. This nonuniform coupling scheme
facilitates symmetric equilibria, where both populations display identical
firing activity, characterized by either quiescent or spiking behavior, or
asymmetric equilibria, where the firing activity of one population exhibits
quiescent but the other exhibits spiking behavior. Oscillations in the firing
rate are possible if neurons emit pulses with non-zero width but are otherwise
quenched. Here, we explore how collective oscillations emerge for two
statistically identical neuron populations in the limit of an infinite number
of neurons. A detailed analysis reveals how collective oscillations are born
and destroyed in various bifurcation scenarios and how they are organized
around higher codimension bifurcation points. Since both symmetric and
asymmetric equilibria display bistable behavior, a large configuration space
with steady and oscillatory behavior is available. Switching between
configurations of neural activity is relevant in functional processes such as
working memory and the onset of collective oscillations in motor control
Synchronization in Neuronal Networks with Electrical and Chemical Coupling
Synchronized cortical activities in the central nervous systems of mammals are crucial for sensory perception, coordination, and locomotory function. The neuronal mechanisms that generate synchronous synaptic inputs in the neocortex are far from being fully understood. This thesis contributes toward an understanding of the emergence of synchronization in networks of bursting neurons as a highly nontrivial, combined effect of chemical and electrical connections. The first part of this thesis addresses the onset of synchronization in networks of bursting neurons coupled via both excitatory and inhibitory connections. We show that the addition of pairwise repulsive inhibition to excitatory networks of bursting neurons induces synchrony, in contrast to one’s expectations. Through stability analysis, we reveal the mechanism underlying this purely synergistic phenomenon and demonstrates that it originates from the transition between different types of bursting, caused by excitatory-inhibitory synaptic coupling. We also report a universal scaling law for the synchronization stability condition for large networks in terms of the number of excitatory and inhibitory inputs each neuron receives, regardless of the network size and topology. In the second part of this thesis, we show that similar effects are also observed in other models of bursting neurons, capable of switching from square-wave to plateau bursting. Finally, in the third part, we report a counterintuitive find that combined electrical and inhibitory coupling can synergistically induce robust synchronization in a range of parameters where electrical coupling alone promotes anti-phase spiking and inhibition induces anti-phase bursting. We reveal the underlying mechanism which uses a balance between hidden properties of electrical and inhibitory coupling to act together to synchronize neuronal bursting. We show that this balance is controlled by the duty cycle of the self-coupled system which governs the synchronized bursting rhythm. This work has potential implications for understanding the emergence of abnormal synchrony in epileptic brain networks. It suggests that promoting presumably desynchronizing inhibition in an attempt to prevent seizures can have a counterproductive effect and induce abnormal synchronous firing
Stochastic synchronization of neuronal populations with intrinsic and extrinsic noise
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
Mathematical frameworks for oscillatory network dynamics in neuroscience
The tools of weakly coupled phase oscillator theory have had a profound impact on the neuroscience community, providing insight into a variety of network behaviours ranging from central pattern generation to synchronisation, as well as predicting novel network states such as chimeras. However, there are many instances where this theory is expected to break down, say in the presence of strong coupling, or must be carefully interpreted, as in the presence of stochastic forcing. There are also surprises in the dynamical complexity of the attractors that can robustly appear—for example, heteroclinic network attractors. In this review we present a set of mathemat- ical tools that are suitable for addressing the dynamics of oscillatory neural networks, broadening from a standard phase oscillator perspective to provide a practical frame- work for further successful applications of mathematics to understanding network dynamics in neuroscience
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