Quantifying Noisy Attractors: From Heteroclinic to Excitable Networks

Abstract

Attractors of dynamical systems may be networks in phase space that can be heteroclinic (where there are dynamical connections between simple invariant sets) or excitable (where a perturbation threshold needs to be crossed to a dynamical connection between “nodes''). Such network attractors can display a high degree of sensitivity to noise both in terms of the regions of phase space visited and in terms of the sequence of transitions around the network. The two types of network are intimately related---one can directly bifurcate to the other. In this paper we attempt to quantify the effect of additive noise on such network attractors. Noise increases the average rate at which the networks are explored and can result in “macroscopic” random motion around the network. We perform an asymptotic analysis of local behavior of an escape model near heteroclinic/excitable nodes in the limit of noise $\eta\rightarrow 0^+$ as a model for the mean residence time $T$ near equilibria. The heteroclinic network case has $T$ proportional to $-\ln\eta$ while the excitable network has $T$ given by a Kramers' law, proportional to $\exp (B/\eta^2)$. There is singular scaling behavior (where $T$ is proportional to $1/\eta$) at the bifurcation between the two types of network. We also explore transition probabilities between nodes of the network in the presence of anisotropic noise. For low levels of noise, numerical results suggest that a (heteroclinic or excitable) network can approximately realize any set of transition probabilities and any sufficiently large mean residence times at the given nodes. We show that this can be well modeled in our example network by multiple independent escape processes, where the direction of first escape determines the transition. This suggests that it is feasible to design noisy network attractors with arbitrary Markov transition probabilities and residence times.