Population Dynamics of Globally Coupled Degrade-and-Fire Oscillators

TL;DR

This paper analyzes the dynamics of globally coupled degrade-and-fire oscillators inspired by bacterial genetic circuits, revealing a transition from diverse profiles to synchronized groups as coupling strength increases.

Contribution

It provides a mathematical characterization of the asymptotic behavior and phase transition in a model of pulse-coupled oscillators with global inhibitory coupling.

Findings

Trajectories are asymptotically periodic with periods depending on initial profiles.

A criterion for the existence of periodic orbits is established.

A sharp transition occurs as the coupling parameter increases, leading to synchronization.

Abstract

This paper reports the analysis of the dynamics of a model of pulse-coupled oscillators with global inhibitory coupling. The model is inspired by experiments on colonies of bacteria-embedded synthetic genetic circuits. The total population can be either of finite (arbitrary) size or infinite, and is represented by a one-dimensional profile. Profiles can be discontinuous, possibly with infinitely many jumps. Their time evolution is governed by a singular differential equation. We address the corresponding initial value problem and characterize the dynamics' main features. In particular, we prove that trajectory behaviors are asymptotically periodic, with period only depending on the profile (and on the model parameters). A criterion is obtained for the existence of the corresponding periodic orbits, which reveals the existence of a sharp transition as the coupling parameter is increased. The transition separates a regime where any profile can be obtained in the limit of large times, to a situation where only trajectories with sufficiently large groups of synchronized oscillators perdure.