Escape rates in periodically driven Markov processes

Abstract

We present an approximate analytical expression for escape rates of time-dependent driven stochastic processes with an absorbing boundary such as the driven leaky integrate-and-fire model for neural spiking. The novel approximation is based on a discrete state Markovian modeling of the full long-time dynamics with time-dependent rates. It is valid in a wide parameter regime beyond the restraining limits of weak driving (linear response) and/or weak noise. The scheme is carefully tested and yields excellent agreement with three different numerical methods based on the Langevin equation, the Fokker-Planck equation and an integral equation. r

Quantum resonant activation

“…For example, fluctuations around the MFPT, quantified by the second moment, provide a measure of the number of detections needed to collect a reliable statistics for the FPT analysis. The FPT pdf also determines the so-termed residence time and interspike pdfs, which generally are more readily available in experiments, e.g., in the context of stochastic resonance phenomena [39], and involve suitable averages over the FPT pdf [6,21,37,38]. The residence time pdf is explicitly evaluated in Sec.…”

Neuron firing in driven nonlinear integrate-and-fire models

“…Either numerical methods have to be used or in specific cases, tailored approximations may be applied in order to analyze relevant aspects of the dynamical behavior of the model [7][8][9]4]. The numerical methods either deal with the stochastic differential equations [10,11] or the equivalent Fokker-Planck equation [11][12][13]. For time dependent Ornstein-Uhlenbeck processes the first passage time density can also be obtained from an integral equation [14], which provides a convenient starting point for numerical investigations [15,13].…”

“…The numerical methods either deal with the stochastic differential equations [10,11] or the equivalent Fokker-Planck equation [11][12][13]. For time dependent Ornstein-Uhlenbeck processes the first passage time density can also be obtained from an integral equation [14], which provides a convenient starting point for numerical investigations [15,13].…”

“…For the slow subthreshold driving considered here we may follow the line of reasoning that was applied to the linear leaky integrate-and-fire model [12,13] and outlined in Ref. [18] in more detail.…”

“…Therefore, the primary task is to find the firing rate. For weak noise and slow driving frequencies the time dependent rates are given by the values of the rates at the instantaneous driving strength [18,12,13,19].…”