Firing map of an almost periodic input function

TL;DR

This paper extends the mathematical understanding of the firing map in integrate-and-fire systems to include almost periodic stimulus functions, broadening the class of functions for which key properties hold.

Contribution

It proves that the firing map retains its essential properties when the stimulus function is almost periodic and integrable, not just continuous or periodic.

Findings

Firing map properties hold for almost periodic stimuli

Framework established for discrete dynamics of firing maps

Extends previous results beyond continuous functions

Abstract

In mathematical biology and the theory of electric networks the firing map of an integrate-and-fire system is a notion of importance. In order to prove useful properties of this map authors of previous papers assumed that the stimulus function f of the system \dot{x}= f(t,x) is continuous and usually periodic in the time variable. In this work we show that the required properties of the firing map for the simplified model \dot{x}=f(t) still hold if f \in L_{loc}^1(R) and f is an almost periodic function. Moreover, in this way we prepare a formal framework for next study of a discrete dynamics of the firing map arising from almost periodic stimulus that gives information on consecutive resets (spikes).