Slow switching in a population of delayed pulse-coupled oscillators

TL;DR

This paper investigates slow switching phenomena in populations of delayed pulse-coupled oscillators, revealing how saddle connections and bifurcations lead to complex collective dynamics, supported by an asymptotic analytical framework.

Contribution

It introduces an asymptotic reduction method for analyzing cluster states and explains the emergence of slow switching via saddle connections near bifurcations.

Findings

Slow switching arises from saddle connections between cluster states.

Bifurcation points facilitate the formation of saddle connections.

Asymptotic theory enables analytical study of complex dynamics.

Abstract

We show that peculiar collective dynamics called slow switching arises in a population of leaky integrate-and-fire oscillators with delayed, all-to-all pulse-couplings. By considering the stability of cluster states and symmetry possessed by our model, we argue that saddle connections between a pair of the two-cluster states are formed under general conditions. Slow switching appears as a result of the system's approach to the saddle connections. It is also argued that such saddle connections easy to arise near the bifurcation point where the state of perfect synchrony loses stability. We develop an asymptotic theory to reduce the model into a simpler form, with which an analytical study of cluster states becomes possible.