Sequential Noise-Induced Escapes for Oscillatory Network Dynamics

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.

Cascading transitions in the climate system

“…The double-fold coupled system has previously been considered in another context (Brummitt et al, 2015), where it was also noted that not all subsystems undergo tipping ("hopping") in systems with more than two coupled fold cascades. Moreover, stochastically coupled multi-stable systems have been considered in networks, where different types of domino effects can occur depending on the synchrony of the transition in the different network nodes (Ashwin et al, 2017;Creaser et al, 2018). Here we only consider two coupled systems, but allow different types of bifurcations, and the systems are physically coupled in a directional way.…”

“…In the last few years, much work has been carried out to formulate statistical indicators and early warning signals of tipping points. A system close to a critical transition shows features of a "critical slowing down" (Dakos et al, 2008;Scheffer et al, 2009;Kuehn, 2011). In the vicinity of the tipping point, the system slowly loses its ability to recover from small perturbations.…”

Fast and slow domino regimes in transient network dynamics

“…For example, [14] use this to explain some phenomena in the networks of coupled oscillatory bistable units considered in [3].…”

Sequential escapes: onset of slow domino regime via a saddle connection

“…It will be interesting to understand the distributions of escape times that appear in such cases of competition between different escape routes. As noted previously, sequential escape problems are of relevance to modelling a wide range of problems ranging from epileptogenesis [25] to cell differentiation [8]. In this paper we describe a novel type of emergent behaviour in sequential escapes of coupled systems, associated with a global saddle connection bifurcation.…”

“…For the noise-free case the unstable manifolds of x QS and x SQ can be seen to lie in the basin of x QA and x AQ for β < β sc but in the basin of x AA for β > β sc . The sequential escape problem involves global dynamics: as in [2,25] we consider the initial state where x i = x Q for all i and pick a threshold h such that x S < h < x A . We define the escape time of the ith system to be…”