On the exit time from open sets of some semi-Markov processes

TL;DR

This paper characterizes the distribution and asymptotic behavior of the first exit time from open sets for semi-Markov processes, with applications to neurophysiology and Leaky Integrate-and-Fire models.

Contribution

It provides new theoretical results on exit times for semi-Markov processes and applies these findings to develop a semi-Markov based LIF neuron model.

Findings

Asymptotic estimates for survival and distribution functions of exit times.

Conditions for absolute continuity of the exit time distribution.

Application of semi-Markov processes to realistic neuron models.

Abstract

In this paper we characterize the distribution of the first exit time from an arbitrary open set for a class of semi-Markov processes obtained as time-changed Markov processes. We estimate the asymptotic behaviour of the survival function (for large $t$) and of the distribution function (for small $t$) and we provide some conditions for absolute continuity. We have been inspired by a problem of neurophyshiology and our results are particularly usefull in this field, precisely for the so-called Leacky Integrate-and-Fire (LIF) models: the use of semi-Markov processes in these models appear to be realistic under several aspects, e.g., it makes the intertimes between spikes a r.v. with infinite expectation, which is a desiderable property. Hence, after the theoretical part, we provide a LIF model based on semi-Markov processes.