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

TL;DR

This paper derives an analytical series solution for the first-passage-time problem of a Brownian motion with an exponential time-dependent drift, relevant for neuronal signal processing, and validates it against numerical simulations.

Contribution

It provides a recursive series solution for the first-passage-time statistics of a Brownian motion with exponential drift, extending previous methods.

Findings

Analytical series solution matches numerical simulations when truncated appropriately.

The method effectively characterizes first-passage times in neuronal models with adaptation.

The recursive scheme offers a practical approach for complex time-dependent stochastic processes.

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.