Stochastic switching of delayed feedback suppresses oscillations in genetic regulatory systems

TL;DR

This paper explores how stochastic switching of delays in feedback systems can suppress oscillations, revealing that fast switching between different delay states stabilizes the system.

Contribution

The study introduces a hybrid model with stochastic delays governed by a Markov chain and derives an effective delay equation in the fast switching limit, highlighting its impact on system stability.

Findings

Fast stochastic switching stabilizes oscillatory systems.

Effective delay equations incorporate all subsystem delays.

Stochastic delay switching can suppress oscillations in gene regulation models.

Abstract

Delays and stochasticity have both served as crucially valuable ingredients in mathematical descriptions of control, physical, and biological systems. In this work, we investigate how explicitly dynamical stochasticity in delays modulates the effect of delayed feedback. To do so, we consider a hybrid model where stochastic delays evolve by a continuous-time Markov chain, and between switching events, the system of interest evolves via a deterministic delay equation. Our main contribution is the calculation of an effective delay equation in the fast switching limit. This effective equation maintains the influence of all subsystem delays and cannot be replaced with a single effective delay. To illustrate the relevance of this calculation, we investigate a simple model of stochastically switching delayed feedback motivated by gene regulation. We show that sufficiently fast switching between two oscillatory subsystems can yield stable dynamics.