Low-dimensional dynamics of populations of pulse-coupled oscillators

TL;DR

This paper demonstrates that the Winfree model of pulse-coupled oscillators evolves into the Ott-Antonsen manifold, enabling exact analysis of synchronization and chimera states in large biological oscillator populations.

Contribution

It provides the first complete mathematical analysis of the Winfree model, showing its reduction to the Ott-Antonsen manifold and exploring chimera states in pulse-coupled oscillators.

Findings

Brief pulses can synchronize heterogeneous populations.

Broad pulses are less effective at synchronization.

Multiple chimera states, including chaotic ones, are identified.

Abstract

Large communities of biological oscillators show a prevalent tendency to self-organize in time. This cooperative phenomenon inspired Winfree to formulate a mathematical model that originated the theory of macroscopic synchronization. Despite its fundamental importance, a complete mathematical analysis of the model proposed by Winfree ---consisting of a large population of all-to-all pulse-coupled oscillators--- is still missing. Here we show that the dynamics of the Winfree model evolves into the so-called Ott-Antonsen manifold. This important property allows for an exact description of this high-dimensional system in terms of a few macroscopic variables, and the full investigation of its dynamics. We find that brief pulses are capable of synchronizing heterogeneous ensembles which fail to synchronize with broad pulses, specially for certain phase response curves. Finally, to further illustrate the potential of our results, we investigate the possibility of `chimera' states in populations of identical pulse-coupled oscillators. Chimeras are self-organized states in which the symmetry of a population is broken into a synchronous and an asynchronous part. Here we derive three ordinary differential equations describing two coupled populations, and uncover a variety of chimera states, including a new class with chaotic dynamics.