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.
Highlights
The hypothesis that some features of the brain activity can be interpreted as the result of a dynamics acting close to a critical point has inspired several statistical models where a critical state can be selected by the fine tuning of a parameter [14,15,16], or is self-organized [17,18,19,20]
Networks of spiking neurons are introduced to describe some features of the brain activity, which are characterized by burst events 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
Summary
The hypothesis that some features of the brain activity can be interpreted as the result of a dynamics acting close to a critical point has inspired several statistical models where a critical state can be selected by the fine tuning of a parameter [14,15,16], or is self-organized [17,18,19,20]. Other stochastic models have been proposed that can reproduce the avalanche dynamics of the neural activity, without invoking the existence of an underlying critical behaviour. Among these models of spiking neurons, a central role is played by the celebrated Wilson-Cowan model (WCM), which describes the coupled dynamics of populations of excitatory and inhibitory neurons [21,22,23,24]. The stochastic version of this model has been shown to reproduce avalanche dynamics [26] and oscillatory behaviour of the activity [27]. The underlying mechanisms responsible for such phenomenology have been identified in the noisy functionally coupled structure of the dynamics, rather than in the presence of a critical point
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