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

TL;DR

This paper studies an array of stochastic units with nonlocal coupling that prevents global synchronization, leading to traveling wave clusters, and develops a mean field theory to describe this phenomenon.

Contribution

It introduces a model of nonlocally coupled stochastic oscillators that form traveling wave clusters, with analytical mean field theory and numerical validation.

Findings

Clusters form and move at constant speed

Model predictions agree with numerical simulations

Nonlocal coupling prevents global synchronization

Abstract

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 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 \emph{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.