Firing map of an almost periodic input function

Pages: 1032 - 1041, Issue Special, September 2011

 Abstract        Full Text (370.1K)              

Wacław Marzantowicz - Faculty of Mathematics and Comp. Sci., Adam Mickiewicz University of Poznań, ul. Umultowska 87, 61-614 Poznań, Poland (email)
Justyna Signerska - Institute of Mathematics of the Polish Academy of Sciences, ul. Śniadeckich 8, 00-956 Warszawa, Poland (email)

Abstract: In mathematical biology and the theory of electric networks the fi ring map of an integrate-and-fi re 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 $ẋ$ = $f(t, x)$ is continuous and usually periodic in the time variable. In this work we show that the required properties of the firing map for the simplifi ed model $ẋ$ = $f(t)$ still hold if $f \in L(^1_(loc))(R)$ 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).

Keywords:  ordinary di fferential equation, integrate-and- fire, fi ring map, almost periodic functions, rotation number
Mathematics Subject Classification:  Primary: 37N25 42A75; Secondary: 37E45 92C20

Received: July 2010;      Revised: February 2011;      Published: October 2011.