Firing map of an almost periodic input function
Wacław Marzantowicz - Faculty of Mathematics and Comp. Sci., Adam Mickiewicz University of Poznań, ul. Umultowska 87, 61-614 Poznań, Poland (email) 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 $ẋ$ = $f(t, x)$ 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, integrate-and-fire, firing map, almost
periodic functions, rotation number
Received: July 2010; Revised: February 2011; Published: October 2011. |