Fast and slow domino regimes in transient network dynamics

Abstract

It is well known that the addition of noise to a multistable dynamical system can induce random transitions from one stable state to another. For low noise, the times between transitions have an exponential tail and Kramers' formula gives an expression for the mean escape time in the asymptotic limit. If a number of multistable systems are coupled into a network structure, a transition at one site may change the transition properties at other sites. We study the case of escape from a "quiescent" attractor to an "active" attractor in which transitions back can be ignored. There are qualitatively different regimes of transition, depending on coupling strength. For small coupling strengths, the transition rates are simply modified but the transitions remain stochastic. For large coupling strengths, transitions happen approximately in synchrony-we call this a "fast domino" regime. There is also an intermediate coupling regime where some transitions happen inexorably but with a delay that may be arbitrarily long-we call this a "slow domino" regime. We characterize these regimes in the low noise limit in terms of bifurcations of the potential landscape of a coupled system. We demonstrate the effect of the coupling on the distribution of timings and (in general) the sequences of escapes of the system.

Sequential Noise-Induced Escapes for Oscillatory Network Dynamics

“…This generalises the setting of [3] to a case of bistable nodes where one of the attractors is periodic. For this network, A ji ∈ {0, 1} is the adjacency matrix and β ≥ 0 the coupling strength: we assume that A ii = 0 for all i.…”

“…In the presence of noise, we define, analogously to [3], the escape time of the node τ to be when the realisation trajectory switches from being close to the origin (quiescent) to being close to the stable limit cycle (active). More precisely, if the noise-free system has stable limit cycles at |z| = 0 and |z| = R max separated by an unstable limit cycle at 0 < |z| = R c < R max , then the escape time for a given threshold ξ ∈ (R c , R max ) is…”

“…Coupling of such systems can promote (or decrease) escape of others on the network. However, there may also be critical values of the coupling, as identified in [3,8,9], at which the qualitative nature of the escape changes associated with bifurcations on basin boundaries of the attractors where typical transitions occur. As coupling strength increases, the escape of a node is increasingly dependent on the input from other nodes in addition to the noise perturbations.…”

“…We develop several ideas from that paper to the case where there is bi-stability between steady and oscillatory attractors. As in [3], each node when uncoupled has two attractors, a stable steady state (that can be destabilised by noise) and a more deeply stable oscillatory attractor (that is more resistant to noise). Starting with the system in the steady attractor, we say it "escapes" when it crosses a threshold to the basin of the oscillatory attractor.…”

“…In a recent paper [3] we consider sequential noise-induced escapes for a network of systems where there is escape from a "shallow" equilibrium attractor to a "deep" equilibrium attractor at each node. We develop several ideas from that paper to the case where there is bi-stability between steady and oscillatory attractors.…”