Emergence of one- and two-cluster states in populations of globally pulse-coupled oscillators

TL;DR

This paper investigates how populations of globally pulse-coupled phase oscillators form stable one- and two-cluster states depending on the shape of their phase-response-curves, revealing conditions for their emergence and stability.

Contribution

It demonstrates the existence and stability conditions of one- and two-cluster states in pulse-coupled oscillators based on the shape of their phase-response-curves, a novel analysis for this class of systems.

Findings

Stable homoclinic connections of the one-cluster state exist for certain PRC shapes.

Coexistence of stable one- and two-cluster states depends on the PRC maximum location.

The stability of cluster states is linked to the PRC's maximum position relative to the spike.

Abstract

The subject of this paper is a system of phase-oscillators, which are globally pulse-coupled via excitatory interaction. The appearance and stability of one- and two-cluster-states is investigated for a family of unimodal phase-response-curves (PRC). The PRCs and their derivatives are assumed to be zero at the spiking point. We show that there exist stable homoclinic connections of the one-cluster state for PRCs with the maximum located shortly before the spiking point and coexisting stable two-clusters states when the maximum of the PRC is located shortly after the spike.