Firing map of an almost periodic input function

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).