Coupled three-state oscillators

TL;DR

This paper studies coupled stochastic three-state oscillators, revealing how different transition rules lead to either stable states or coherent oscillations, with the latter arising from a Hopf bifurcation.

Contribution

It compares two types of three-state oscillators and demonstrates how deterministic delays induce coherent oscillations via a Hopf bifurcation.

Findings

Deterministic delays cause coherent oscillations.

Poissonian transitions lead to stable states.

Hopf bifurcation explains the transition to oscillations.

Abstract

We investigate globally coupled stochastic three-state oscillators, which we consider as general models of stochastic excitable systems. We compare two situations:in the first case the transitions between the three states of each unit 1->2->3->1 are determined by Poissonian waiting time distributions. In the second case only transition 1->2 is Poissonian whereas the others are deterministic with a fixed delay. When coupled the second system shows coherent oscillations whereas the first remains in a stable stationary state. We show that the coherent oscillations are due to a Hopf-bifurcation in the dynamics of the occupation probabilities of the discrete states and discuss the bifurcation diagram.