Phase models and clustering in networks of oscillators with delayed coupling

TL;DR

This paper develops a phase model for networks of oscillators with delayed coupling, analyzing cluster solutions' existence and stability, and applies it to neuronal networks with numerical validation.

Contribution

It introduces a phase reduction approach incorporating delays as phase shifts and provides model-independent stability criteria for cluster solutions.

Findings

Time delay can lead to multiple stable clustering states.

Analytical results match numerical simulations.

Cluster solutions depend on delay parameters.

Abstract

We consider a general model for a network of oscillators with time delayed, circulant coupling. We use the theory of weakly coupled oscillators to reduce the system of delay differential equations to a phase model where the time delay enters as a phase shift. We use the phase model to study the existence and stability of cluster solutions. Cluster solutions are phase locked solutions where the oscillators separate into groups. Oscillators within a group are synchronized while those in different groups are phase-locked. We give model independent existence and stability results for symmetric cluster solutions. We show that the presence of the time delay can lead to the coexistence of multiple stable clustering solutions. We apply our analytical results to a network of Morris Lecar neurons and compare these results with numerical continuation and simulation studies.