Series solution to the first-passage-time problem of a Brownian motion with an exponential time-dependent drift

Abstract

We derive the first-passage-time statistics of a Brownian motion driven by an exponential time-dependent drift up to a threshold. This process corresponds to the signal integration in a simple neuronal model supplemented with an adaptation-like current and reaching the threshold for the first time represents the condition for declaring a spike. Based on the backward Fokker-Planck formulation, we consider the survival probability of this process in a domain restricted by an absorbent boundary. The solution is given as an expansion in terms of the intensity of the time-dependent drift, which results in an infinite set of recurrence equations. We explicitly obtain the complete solution by solving each term in the expansion in a recursive scheme. From the survival probability, we evaluate the first-passage-time statistics, which itself preserves the series structure. We then compare theoretical results with data extracted from numerical simulations of the associated dynamical system, and show that the analytical description is appropriate whenever the series is truncated in an adequate order.

arXiv comment19 pages (15 pages in the journal version), 3 figures, published in J. Phys. A

References in corpus (5)

• First passage time statistics of Brownian motion with purely time dependent drift and diffusion
• Switching Time Statistics for Driven Neuron Models: Analytic Expressions versus Numerics
• Moments of the first passage time under external driving
• Onset of negative interspike interval correlations in adapting neurons
• Survival probability and first-passage-time statistics of a Wiener process driven by an exponential time-dependent drift

Cited by in corpus (5)

• Brownian motion in time-dependent logarithmic potential: Exact results for dynamics and first-passage properties
• Thermally induced passage and current of particles in a highly unstable optical potential
• Noise-induced interspike interval correlations and spike train regularization in spike-triggered adapting neurons
• Interspike interval correlations in networks of inhibitory integrate-and-fire neurons
• First-passage-time statistics of a Brownian particle driven by an arbitrary unidimensional potential with a superimposed exponential time-dependent drift