Sequential noise-induced escapes for oscillatory network dynamics

TL;DR

This paper investigates how noise induces sequential transitions between steady and oscillatory states in coupled networks, using first passage time theory to analyze escape rates and the effects of coupling strength.

Contribution

It introduces a master equation approach to quantify sequential noise-induced escapes in oscillatory networks, highlighting its limitations at strong coupling.

Findings

Master equation accurately models sequential escapes at weak coupling.

Sequential escape patterns depend on coupling strength.

Breakdown of the master equation at strong coupling.

Abstract

It is well known that the addition of noise in a multistable system can induce random transitions between stable states. The rate of transition can be characterised in terms of the noise-free system's dynamics and the added noise: for potential systems in the presence of asymptotically low noise the well-known Kramers' escape time gives an expression for the mean escape time. This paper examines some general properties and examples of transitions between local steady and oscillatory attractors within networks: the transition rates at each node may be affected by the dynamics at other nodes. We use first passage time theory to explain some properties of scalings noted in the literature for an idealised model of initiation of epileptic seizures in small systems of coupled bistable systems with both steady and oscillatory attractors. We focus on the case of sequential escapes where a steady attractor is only marginally stable but all nodes start in this state. As the nodes escape to the oscillatory regime, we assume that the transitions back are very infrequent in comparison. We quantify and characterise the resulting sequences of noise-induced escapes. For weak enough coupling we show that a master equation approach gives a good quantitative understanding of sequential escapes, but for strong coupling this description breaks down.