Asymptotic periodicity in networks of degrade-and-fire oscillators

TL;DR

This paper demonstrates that networks of degrade-and-fire oscillators exhibit asymptotic periodicity under general coupling topologies, extending previous mean-field results and providing explicit descriptions of their dynamics.

Contribution

It proves that asymptotic periodicity persists in arbitrary coupling networks with balanced in and out weights, generalizing prior mean-field findings.

Findings

Trajectories with reasonable firing sequences are asymptotically periodic.

Periodic orbits are uniquely determined by firing sequences.

Explicit examples fully describe the dynamics.

Abstract

Networks of coupled degrade-and-fire (DF) oscillators are simple dynamical models of assemblies of interacting self-repressing genes. For mean-field interactions, which most mathematical studies have assumed so far, every trajectory must approach a periodic orbit. Moreover, asymptotic cluster distributions can be computed explicitly in terms of coupling intensity, and a massive collection of distributions collapses when this intensity passes a threshold. Here, we show that most of these dynamical features persist for an arbitrary coupling topology. In particular, we prove that, in any system of DF oscillators for which in and out coupling weights balance, trajectories with reasonable firing sequences must be asymptotically periodic, and periodic orbits are uniquely determined by their firing sequence. In addition to these structural results, illustrative examples are presented, for which the dynamics can be entirely described.