Revisiting asymptotic periodicity in networks of degrade-and-fire oscillators

TL;DR

This paper investigates the long-term behavior of networks of degrade-and-fire oscillators, revealing how interaction topology influences periodicity and extending conditions for asymptotic periodicity, especially in unidirectional chains.

Contribution

It revisits and refines conditions for asymptotic periodicity in degrade-and-fire oscillator networks and analyzes the dynamics of unidirectional chains, demonstrating the optimality of these conditions.

Findings

Updated conditions for asymptotic periodicity are proven to be optimal.

Interaction topology significantly impacts the global dynamics of the system.

Analysis of unidirectional chains illustrates the influence of network structure on behavior.

Abstract

Networks of degrade-and-fire oscillators are elementary models of populations of synthetic gene circuits with negative feedback, which show elaborate phenomenology while being amenable to mathematical analysis. In addition to thorough investigation in various examples of interaction graphs, previous studies have obtained conditions on interaction topology and strength that ensure that asymptotic behaviors are periodic (assuming that the so-called firing sequence is itself periodic and involves all nodes). Here, we revisit and extend these conditions and we analyse the dynamics in a case of unidirectional periodic chain. This example shows in particular that the updated conditions for asymptotic periodicity are optimal. Altogether, our results provide a novel instance of direct impact of the topology of interactions in the global dynamics of a collective system.