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2012-11-06 Nonlinear Seminar

The seminar will take place at 8:30 in the Faculty Council meeting room (A1-33/34)

Justyna Signerska
Firing map for integrate-and-fire models with an almost periodic drive

Consider the integrate-and-fire system $$ \dot{x}=F(t,x) $$, $$ F:\mathbb{R}^2\to \mathbb{R} $$, in which the continuous dynamics is interrupted by the threshold-and-reset behaviour: $$ \lim_{t\to s^+}x(t)=x_r $$ if $$ x(s)=x_{\Theta} $$, meaning that once the value of a dynamical variable reaches a certain threshold $$ x_{\Theta} $$ it is immediately reset to a resting value $$ x_r $$ and the system evolves again according to the differential equation. The question is to describe the time series of consecutive resets (spikes) $$ t_n $$ as iterations $$ \Phi^n(t_0) $$ of some map $$ \Phi:\mathbb{R}\to \mathbb{R} $$, called the firing map, and the sequence of interspike-intervals $$ t_n-t_{n-1} $$ as a sequence of displacements $$ \Phi^n(t_0)-\Phi^{n-1}(t_0) $$ along a trajectory of this map. The problem appears in various applications, e.g. in modelling of an action potential (spiking) by a neuron.

However, so far properties of the firing map were analytically investigated only for the function $$ F $$, which is smooth enough and often also periodic in $$ t $$. We present a complete description of the properties of the firing map arising from the most popular models: $$ \dot{x}=-\sigma x +f(t) $$ (Leaky Integrate-and-Fire) and $$ \dot{x}=f(t) $$ (Perfect Integrator), when the function $$ f $$ is only locally integrable and/or almost periodic. In particular, we prove that a Stepanov almost periodic function $$ f $$ induces the firing map $$ \Phi $$ with the uniformly (Bohr) almost periodic displacement $$ \Phi-\textrm{Id} $$. In this way we provide a formal framework for next studying of the interspike-intervals in almost-periodically driven integrate-and-fire models.