Critical behaviour of the stochastic Wilson-Cowan model

Abstract

Spontaneous brain activity is characterized by bursts and avalanche-like dynamics, with scale-free features typical of critical behaviour. The stochastic version of the celebrated Wilson-Cowan model has been widely studied as a system of spiking neurons reproducing non-trivial features of the neural activity, from avalanche dynamics to oscillatory behaviours. However, to what extent such phenomena are related to the presence of a genuine critical point remains elusive. Here we address this central issue, providing analytical results in the linear approximation and extensive numerical analysis. In particular, we present results supporting the existence of a bona fide critical point, where a second-order-like phase transition occurs, characterized by scale-free avalanche dynamics, scaling with the system size and a diverging relaxation time-scale. Moreover, our study shows that the observed critical behaviour falls within the universality class of the mean-field branching process, where the exponents of the avalanche size and duration distributions are, respectively, −3/2 and −2. We also provide an accurate analysis of the system behaviour as a function of the total number of neurons, focusing on the time correlation functions of the firing rate in a wide range of the parameter space.Author summaryNetworks of spiking neurons are introduced to describe some features of the brain activity, which are characterized by burst events (avalanches) with power-law distributions of size and duration. The observation of this kind of noisy behaviour in a wide variety of real systems led to the hypothesis that neuronal networks work in the proximity of a critical point. This hypothesis is at the core of an intense debate. At variance with previous claims, here we show that a stochastic version of the Wilson-Cowan model presents a phenomenology in agreement with the existence of a bona fide critical point for a particular choice of the relative synaptic weight between excitatory and inhibitory neurons. The system behaviour at this point shows all features typical of criticality, such as diverging timescales, scaling with the system size and scale-free distributions of avalanche sizes and durations, with exponents corresponding to the mean-field branching process. Our analysis unveils the critical nature of the observed behaviours.

Power spectrum and critical exponents in the 2D stochastic Wilson–Cowan model

“…The distribution of avalanche sizes is found to scale as P(S) ∼ S −α , that of avalanche durations as P(T) ∼ T −τ , while the mean size of avalanches scales as �S� ∼ T γ as a function of the duration 4-8 . A good indicator of criticality is believed to be given by the scaling relation γ = τ −1 α−1 , as originally predicted in the theory of crackling noise 9,10 , as well as by the collapse of rescaled shapes of avalanches of different durations 4,7,8 . The simple branching model of avalanche propagation predicts exponents α = 3/2 , τ = 2 and γ = 2 , observed in some experimental realizations 1,11 and in models of neural dynamics, including the fully-connected stochastic Wilson-Cowan model 7 .…”

“…In Ref. 7 it was shown that there is a critical point at h = 0 and w0,0 = β −1 α . For w0,0 larger than the critical value, an attractive fixed point with � 0 > 0 exists even when the external input h → 0 .…”

“…In Refs. 7,22,23 a further simplification was considered, that w ij depends only on the type of the pre-synaptic neuron, and is given by w ij = 1 2N E ( w0,0 + ws,0 ) if j is excitatory and w ij = 1 2N I ( w0,0 − ws,0 ) if j is inhibitory. In this case, the temporal autocorrelation function of time dependent variables, like the fraction of active neurons or the firing rate, can be written in the limit of large number of neurons as 23 where X(t) is the variable considered, X 0 is its average value in time, A 1 and A 2 are constant coefficients, and τ 1 , τ 2 are the characteristic relaxation times.…”

“…In the present paper we study the 2D version of the stochastic Wilson-Cowan model 7,22 , where neurons are distributed uniformly on a 2D lattice, and connections between them decay exponentially with the distance. We can tune the network topology from 2D to fully-connected by changing the ratio between the range of the exponential decay of the connections, and the side of the lattice L. While in the fully-connected case the dynamics of the model is completely described by specifying two variables, the fraction of active excitatory and inhibitory neurons, in the 2D case one needs to define such fractions at each site of the lattice.…”

Heteroclinic cycling and extinction in May-Leonard models with demographic stochasticity

“…The ambiguity intrinsic to stochastic extensions of ODE systems is not confined to ecological models. A series of papers have debated the most appropriate way to extend the deterministic Hodgkin-Huxley equations to incorporate the effects of random gating of ion channels in neural dynamics [3,14,20,21,28,30,31] At the level of large-scale neural circuits, several distinct stochastic generalizations have been proposed that coincide with the classical deterministic Wilson-Cowan neural field equations in the mean-field limit [5,6,9,12,13].…”