Arrays of stochastic oscillators: Nonlocal coupling, clustering, and wave formation

We consider an array of units each of which can be in one of three states. Unidirectional transitions between these states are governed by Markovian rate processes. The interactions between units occur through a dependence of the transition rates of a unit on the states of the units with which it interacts. This coupling is nonlocal, that is, it is neither an all-to-all interaction (referred to as global coupling), nor is it a nearest neighbor interaction (referred to as local coupling). The coupling is chosen so as to disfavor the crowding of interacting units in the same state. As a result, there is no global synchronization. Instead, the resultant spatiotemporal configuration is one of clusters that move at a constant speed and that can be interpreted as traveling waves. We develop a mean field theory to describe the cluster formation and analyze this model analytically. The predictions of the model are compared favorably with the results obtained by direct numerical simulations.