Oscillatory Survival Probability: Analytical and Numerical Study of a Non-Poissonian Exit Time

Abstract

We consider the escape of a planar diffusion process from the domain of attraction $\Omega$ of a stable focus of the drift in the limit of small diffusion. The boundary $\partial\Omega$ of $\Omega$ is an unstable limit cycle of the drift, and the focus is close to the limit cycle. A new phenomenon of oscillatory decay of the peaks of the survival probability of the process in $\Omega$ emerges for a specific distance of the focus to the boundary which depends on the amplitude of the diffusion coefficient. We show that the phenomenon is due to the complex-valued second eigenvalue $\lambda_2(\Omega)$ of the Fokker--Planck operator with Dirichlet boundary conditions. The dominant oscillation frequency, $1/\mathfrak{I}m\{\lambda_2(\Omega)\}$, is independent of the relative noise strength. The density of exit points on $\partial\Omega$ is concentrated in a small arc of $\partial\Omega$ closest to the focus. When the focus approaches the boundary, the principal eigenvalue, the reciprocal of which is the mean escape time, does not necessarily decay exponentially as the amplitude of the noise strength decays. The oscillatory escape is manifested in a mathematical model of neural networks with synaptic depression. Oscillation peaks are identified in the density of the time the network spends in a specific state. This observation explains the oscillations of stochastic trajectories around a focus prior to escape and also the non-Poissonian escape times. This phenomenon has been observed and reported in experiments and simulations of neural networks.