Memory-dependent bistability and criticality in a stochastic Wilson-Cowan model

We introduce a new version of the stochastic Wilson-Cowan model to account for the memory created by multistep state transition processes. To solve this complex non-Markov model, we introduce effective transition rates, which explicitly decode the effect of the memory, to transform a non-Markov network into a Markov network topologically equivalent to it. Based on this technique, we analyze the phase diagram of the system in the space of parameters. Notably, we identify memory-dependent criticality and bistability, where the former is associated with a second-order (continuous) phase transition, whereas the latter arises from the presence of a first-order (discontinuous) phase transition. The corresponding two parameter regimes are separated by a memory-mediated tricritical point. At the critical point, scale-free yet memory-dependent avalanches are observed, and their exponents are consistent with the mean-field directed percolation universality class. In the memory-induced bistable region, the system can switch between up and down states when the number of neurons is not too large. Our investigation indicates that memory plays an important role in the control of neural behaviors.