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

Massimiliano  Tamborrino

doi:10.3934/mbe.2016011

Article Contents

2016, Volume 13, Issue 3: 613-629. Doi: 10.3934/mbe.2016011

This issue Previous Article A new firing paradigm for integrate and fire stochastic neuronal models Next Article

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

Massimiliano Tamborrino^1,

1.
Johannes Kepler University, Altenbergerstraße 69, 4040 Linz

Received: May 31, 2015

Revised: July 31, 2015

Published: December 2015

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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. First, we introduce different approximating linear thresholds. Then, we 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 compared with empirical quantities from simulated data. Moreover, a further comparison with firing statistics derived under the assumption of a small amplitude of the time-dependent change in the threshold, is also carried out. Finally, maximum likelihood and moment estimators of the parameters of the model are derived and applied on simulated data.

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Mathematics Subject Classification: Primary: 60G40, 62Mxx, 60J65, 60J70; Secondary: 62F10.

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