Sarracino, A. | bioRxiv Science

bioRxiv2022

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

The power spectrum of brain activity is composed by peaks at characteristic frequencies superimposed to a background that decays as a power law of the frequency, f-{beta}, with an exponent {beta} close to 1 (pink noise). This exponent is predicted to be connected with the exponent{gamma} related to the scaling of the average size with the duration of avalanches of activity. "Mean field" models of neural dynamics predict exponents {beta} and{gamma} equal or near 2 at criticality (brown noise), including the simple branching model and the fully connected stochastic Wilson Cowan model. We here show that a 2D version of the stochastic Wilson Cowan model, where neuron connections decay exponentially with the distance, is characterized by exponents {beta} and{gamma} markedly different from those of mean field, respectively around 1 and 1.3. The exponents and{tau} of avalanche size and duration distributions, equal to 1.5 and 2 in mean field, decrease respectively to 1.29 {+/-} 0.01 and 1.37 {+/-} 0.01. This seems to suggest the possibility of a different universality class for the model in finite dimension.

neuroscience↗

bioRxiv2021

Critical behaviour of the stochastic Wilson-Cowan model

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.

neuroscience↗