Rate processes in a delayed, stochastically driven, and overdamped system

Abstract

A Fokker-Planck formulation of systems described by stochastic delay differential equations has been recently proposed. A separation of time scales approximation allowing this Fokker-Planck equation to be simplified in the case of multistable systems is hereby introduced, and applied to a system consisting of a particle coupled to a delayed quartic potential. In that approximation, population numbers in each well obey a phenomenological rate law. The corresponding transition rate is expressed in terms of the noise variance and the steady-state probability density. The same type of expression is also obtained for the mean first passage time from a given point to another one. The steady-state probability density appearing in these formulas is determined both from simulations and from a small delay expansion. The results support the validity of the separation of time scales approximation. However, the results obtained using a numerically determined steady-state probability are more accurate than those obtained using the small delay expansion, thereby stressing the high sensitivity of the transition rate and mean first passage time to the shape of the steady-state probability density. Simulation results also indicate that the transition rate and the mean first passage time both follow Arrhenius' law when the noise variance is small, even if the delay is large. Finally, deterministic unbounded solutions are found to coexist with the bounded ones. In the presence of noise, the transition rate from bounded to unbounded solutions increases with the delay.

Correlation times in stochastic equations with delayed feedback and multiplicative noise

“…The same expression has been obtained from a Taylor expansion of the conditional probability distribution function p(x τ ,t − τ |x,t) in [2], based on methods introduced in [20,21] under the Ito interpretation of stochastic differential equation. It can also be obtained from a Taylor expansion of the stochastic process defined byẋ (t,t − τ ) [22,23]. All methods assume that the time delay is small and lead to, under stationary conditions,…”

Experimental Investigation of a Bistable System in the Presence of Noise and Delay

“…Such stochastic delay systems with additive noise have recently been examined using Fokker-Planck equations [7,8,10,12]. When the system is linear or when the delay time is small compared to the Kramers times, exact solutions of the Fokker-Planck equation can be derived.…”

“…Various mathematical tools have been developed to study noise-activated escape processes, mostly using the Markovian assumption that the system evolves slowly compared to the time scale of the correlations of the random forces [5,6]. Recently, non-Markovian stochastic processes have also been the subject of increased interest [7][8][9][10][11][12][13].…”

Correlated Noise and Memory Effects in Neural Firing Statistics

“…The presence of this delay clearly renders the whole problem non-Markovian. We have made some progress on understanding escape times in potentials with delayed dynamics [9] for small delays. For larger delays, a new formalism is needed.…”