Macroscopic coherent structures in a stochastic neural network: from interface dynamics to coarse-grained bifurcation analysis (Q2408052)

The authors consider a stochastic neural network first studied by Gong and Robinson in 2012. It consists of a one-dimensional lattice where each site is either firing or refractory or quiescent. The time is discrete, the probability of firing depends on the activity of all other sites through nonlocal spatial interactions, and the recovery from refractory to quiescence is stochastic. The network supports stationary and travelling ``bumps'' of localised activity. The authors first consider a deterministic version of the model on a spatial continuum and analytically construct stationary and travelling bumps. Then they consider the original model and compute these patterns and their stability using equation-free methods, in which a macroscopic evolution equation for these patterns is assumed to exist, even though it is not explicitly available. Their numerical results suggest that moderate to long refractory times support travelling waves, while short refractory times support localised, meandering bumps.