Coexistence of regular and irregular dynamics in complex networks of pulse-coupled oscillators

TL;DR

This paper develops an exact stability analysis for pulse-coupled oscillator networks, revealing conditions under which synchronous regular and irregular dynamics coexist and can switch states.

Contribution

It introduces a multi-operator stability analysis for complex networks, showing inhibitory interactions stabilize synchrony regardless of network parameters.

Findings

Inhibitory interactions stabilize synchronous states

Regular and irregular dynamics coexist in complex networks

External signals can switch network between different states

Abstract

For general networks of pulse-coupled oscillators, including regular, random, and more complex networks, we develop an exact stability analysis of synchronous states. As opposed to conventional stability analysis, here stability is determined by a multitude of linear operators. We treat this multi-operator problem analytically and show that for inhibitory interactions the synchronous state is stable, independent of the parameters and the network connectivity. In randomly connected networks with strong interactions this synchronous state, displaying \textit{regular} dynamics, coexists with a balanced state that exhibits \textit{irregular} dynamics such that external signals may switch the network between qualitatively distinct states.