Global analysis of a continuum model for monotone pulse-coupled oscillators

TL;DR

This paper analyzes a continuum model of pulse-coupled oscillators, establishing global stability results and a Lyapunov function that characterizes their dichotomic synchronization or asynchrony behavior.

Contribution

It introduces a novel global Lyapunov function based on total variation distance, providing a comprehensive stability analysis for continuum pulse-coupled oscillators.

Findings

Oscillators either synchronize in finite time or asynchronously spread on the circle.

The Lyapunov function decreases monotonically, indicating system stability.

Results apply to popular models like the leaky integrate-and-fire model.

Abstract

We consider a continuum of phase oscillators on the circle interacting through an impulsive instantaneous coupling. In contrast with previous studies on related pulse-coupled models, the stability results obtained in the continuum limit are global. For the nonlinear transport equation governing the evolution of the oscillators, we propose (under technical assumptions) a global Lyapunov function which is induced by a total variation distance between quantile densities. The monotone time evolution of the Lyapunov function completely characterizes the dichotomic behavior of the oscillators: either the oscillators converge in finite time to a synchronous state or they asymptotically converge to an asynchronous state uniformly spread on the circle. The results of the present paper apply to popular phase oscillators models (e.g. the well-known leaky integrate-and-fire model) and draw a strong parallel between the analysis of finite and infinite populations. In addition, they provide a novel approach for the (global) analysis of pulse-coupled oscillators.