We consider the so-called integrate-and-fire system $\dot{x}=f(t,x), f: R^2\to R$, in which a continuous dynamics induced by the differential equation is interrupted by "the threshold and reset behaviour", 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 from a new initial condition. The question is to describe the sequence of consecutive resets as iterations of some map, called the firing map, and the sequence of interspike-intervals (time intervals between the resets) as a sequence of displacements along a trajectory of this map. The problem appears in various applications, such as modelling of an action potential (spiking) by a neuron or electric discharges in electrical circuits, cardiac rhythms and arrhythmias. However, the dynamics of the firing map, so far has not been mathematically analyzed in details, even in the "simplest" case when the function $f$ is periodic in $t$.

Firstly, we discuss the behaviour of the displacement sequence of trajectories of an orientation preserving homeomorphism (diffeomorphism) of the circle, which covers an answer to our question for the firing map induced by a periodically driven system. We provide conditions under which the sequence is asymptotically semi-periodic or almost-periodic and characteristics of the distribution of displacements.

Secondly, in view of applications one should weaken the assumptions that the input function is periodic and continuous. We show that many of the required properties of the firing map still hold if $f$ is only locally integrable and almost periodic (either uniformly or in a sense of Stepanov).

The talk is based on my joint works with W. Marzantowicz.