Approximation of the first passage time density of a Wiener process to an exponentially decaying threshold by two-piecewise linear threshold. Application to neuronal spiking activity

TL;DR

This paper introduces a method to approximate the first passage time density of a Wiener process crossing an exponentially decaying threshold by using a two-piecewise linear boundary, with applications to neuronal spike modeling.

Contribution

It provides explicit formulas for the passage time density with a piecewise linear boundary and introduces an optimal approximation method for curved thresholds in neuronal models.

Findings

Explicit first passage time density formulas derived.

Optimal linear threshold approximation minimizes boundary error.

Method accurately estimates neuronal spiking statistics.

Abstract

The first passage time density of a diffusion process to a time varying threshold is of primary interest in different fields. Here we consider a Brownian motion in presence of an exponentially decaying threshold to model the neuronal spiking activity. Since analytical expressions of the first passage time density are not available, we propose to approximate the curved boundary by means of a continuous two-piecewise linear threshold. Explicit expressions for the first passage time density towards the new boundary are provided. Then we introduce different approximating linear threshold and describe how to choose the optimal one minimizing the distance to the curved boundary and hence the error in the corresponding passage time density. Theoretical means, variances and coefficients of variation given by our method are then compared with empirical quantities from simulated data as well as other firing statistics derived under the assumption of a small amplitude of the time-dependent change in the threshold. Finally maximum likelihood and moment estimators of the parameters of the Wiener process are also proposed and compared on simulated data.