The seminar will take place at 8:30 in the Faculty Council meeting room (A1-33/34)
Firing map for integrate-and-fire models with an almost periodic drive
Consider the integrate-and-fire system , , in which the continuous dynamics is interrupted by the threshold-and-reset behaviour: if , meaning that once the value of a dynamical variable reaches a certain threshold it is immediately reset to a resting value and the system evolves again according to the differential equation. The question is to describe the time series of consecutive resets (spikes) as iterations of some map , called the firing map, and the sequence of interspike-intervals as a sequence of displacements 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 , which is smooth enough and often also periodic in . We present a complete description of the properties of the firing map arising from the most popular models: (Leaky Integrate-and-Fire) and (Perfect Integrator), when the function is only locally integrable and/or almost periodic. In particular, we prove that a Stepanov almost periodic function induces the firing map with the uniformly (Bohr) almost periodic displacement . In this way we provide a formal framework for next studying of the interspike-intervals in almost-periodically driven integrate-and-fire models.