Unconventional criticality in the stochastic Wilson-Cowan model

Abstract

The Wilson-Cowan model serves as a classic framework for comprehending the collective neuronal dynamics within networks comprising both excitatory and inhibitory units. Extensively employed in literature, it facilitates the analysis of collective phases in neural networks at a mean-field level, i.e., when considering large fully connected networks. To study fluctuation- induced phenomena, the dynamical model alone is insufficient; to address this issue, we need to work with a stochastic rate model that is reduced to the Wilson-Cowan equations in a mean-field approach. Throughout this thesis, we analyze the resulting phase diagram of the stochastic Wilson-Cowan model near the active to quiescent phase transitions. We unveil eight possible types of transitions that depend on the relative strengths of excitatory and inhibitory couplings. Among these transitions are second-order and first-order types, as expected, as well as three transitions with a surprising mixture of behaviors. The three bona fide second- order phase transitions belong to the well-known directed percolation universality class, the tricritical directed percolation universality class, and a novel universality class called “Hopf tricritical directed percolation", which presents an unconventional behavior with the breakdown of some scaling relations. The discontinuous transitions behave as expected and the hybrid transitions have different anomalies in scaling across them. Our results broaden our knowledge and characterize the types of critical behavior in excitatory and inhibitory networks and help us understand avalanche dynamics in neuronal recordings. From a more general perspective, these results contribute to extending the theory of non-equilibrium phase transitions into quiescent or absorbing states.