Escape from an attractor generated by recurrent exit

Lou Zonca,David Holcman

doi:10.1103/physrevresearch.3.023115

Abstract

Kramer's theory of activation over a potential barrier consists in computing the mean exit time from the boundary of a basin of attraction of a randomly perturbed dynamical system. Here we report that for some systems, crossing the boundary is not enough, because stochastic trajectories return inside the basin with a high probability a certain number of times before escaping far away. This situation is due to a shallow potential. We compute the mean and distribution of escape times and show how this result explains the large distribution of interburst durations in neuronal networks.

Highlights

In Kramers’ theory [1,2,3,4], the escape time over a potential barrier consists in computing the mean first passage time (MFPT) of a dynamical system perturbed by a small noise to the boundary of a basin of attraction
In the limit of small noise, a trajectory escapes a basin of attraction with probability 1 [13], but the escape time is exponentially long depending on the topology of the noiseless dynamics [14,15] and its behavior at the the distribution of exit points peaks at a bdoisutanndcaeryO
We show here that for some shallow two-dimensional dynamical systems, trajectories can first exit the basin of attraction, make excursions outside before coming back inside the domain, a behavior that occurs several times before eventually escaping far away

Summary

INTRODUCTION

In Kramers’ theory [1,2,3,4], the escape time over a potential barrier consists in computing the mean first passage time (MFPT) of a dynamical system perturbed by a small noise to the boundary of a basin of attraction. Recurrent returns inside a basin of attraction can be quantified by the Green’s function of the inner domain used in the additive properties of the MFPT [22] We show here that for some shallow two-dimensional dynamical systems, trajectories can first exit the basin of attraction, make excursions outside before coming back inside the domain, a behavior that occurs several times before eventually escaping far away This situation is peculiar and specific to dimensions greater than two and these recurrent entries need to be taken into account in computing the final escape time. We apply these results to explain the origin of long interburst durations found in neuronal network models [24]

RECURRENT ESCAPE PATTERNS

CHARACTERIZING THE ESCAPE TIME

INTERBURST DURATIONS IN A FIRING EXCITATORY NEURONAL NETWORK

Attractor A2

CONCLUDING REMARKS