Firing map of an almost periodic input function

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2011, 2011(Special): 1032-1041. doi: 10.3934/proc.2011.2011.1032

Firing map of an almost periodic input function

Wacław Marzantowicz ^1, and Justyna Signerska ^2,

1.	Faculty of Mathematics and Comp. Sci., Adam Mickiewicz University of Poznań, ul. Umultowska 87, 61-614 Poznań
2.	Institute of Mathematics of the Polish Academy of Sciences, ul. Śniadeckich 8, 00-956 Warszawa, Poland

Received July 2010 Revised February 2011 Published October 2011

Abstract
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Under a Creative Commons license

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 $ẋ$ = $f(t, x)$ is continuous and usually periodic in the time variable. In this work we show that the required properties of the firing map for the simplified 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 differential equation, rotation number, firing map, almost periodic functions, integrate-and-fire.

Mathematics Subject Classification: Primary: 37N25 42A75; Secondary: 37E45 92C20.

Citation: Wacław Marzantowicz, Justyna Signerska. Firing map of an almost periodic input function. Conference Publications, 2011, 2011 (Special) : 1032-1041. doi: 10.3934/proc.2011.2011.1032

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