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.

Disentangling the critical signatures of neural activity

“…The multivariate Ornstein-Uhlenbeck (mOU) is a paradigmatic model which, albeit simple enough to allow for an analytical treatment, does not account for many biological aspects. In order to show that our results generalize to more biologically sound models, we now consider the neural activity described by a Wilson–Cowan model 55 – 59 as a variant of the extrinsic model. It includes both excitatory and inhibitory synapses and non-linearities in the transfer function, and its derivation is based on arguments over the dynamics of the neurons and action potentials 55 , which makes it a general tool to model mesoscopic neural regions.…”

“…We consider a stochastic version of the Wilson–Cowan model 56 – 59 , which includes a stochastic term that accounts for the finite size of the populations. We consider N non-interacting neural populations, and each one is modeled through the activity of two sub-populations, one of excitatory neurons and one of inhibitory neurons .…”

“…Hence, we model h as the firing rate of another Wilson–Cowan model, namely In order to reproduce avalanches, as the external input h we should choose a stochastic modulation that displays bursts of activity separated by periods of silences. Importantly, it was recently shown in 59 that this model admits a critical point at , where power-law distributed avalanches will emerge independently of the size of the system. Thus, clearly, a possible choice for the external stochastic modulation would be a Wilson–Cowan unit in the critical state.…”

Homeostatic Criticality in Neuronal Networks

“…One important aspect in any SOC model is that phase transitions, and therefore criticality, exist only for zero or very small external field [21], so any homeostatic mechanism will need to self-organize the system to a state where the effective external field vanishes.…”

“…To achieve criticality, we also need h to be 0. For spin systems, zero external magnetic field is a natural condition, despite being a fine-tuning operation seldom discussed in the literature of neuronal avalanches [19,21]. Here, for integrate-and-fire neurons, this condition is not so natural: we must fine-tune θ c = I/(1 − µ) in order to achieve h c = 0.…”