Border Collision Bifurcations of Stroboscopic Maps in Periodically Driven Spiking Models

In this work we consider a general nonautonomous hybrid system based on the integrate-and-fire model, widely used as simplified version of neuronal models and other types of excitable systems. Our assumptions are that the system is monotonic, possesses an attracting subthreshold equilibrium point, and is forced by means of a periodic pulsatile (square wave) function. In contrast to classical methods, in our approach we use the stroboscopic map (time-$T$ return map) instead of the so-called firing map.

It becomes a discontinuous map potentially defined in an infinite number of partitions. By applying theory for piecewise-smooth systems, we avoid relying on particular computations, and we develop a novel approach that can be easily extended to systems with other topologies (expansive dynamics) and higher dimensions. More precisely, we rigorously study the bifurcation structure in the two-dimensional parameter space formed by the amplitude of the pulse and the ratio between $T$ and the duration of the pulse (duty cycle). We show that it is covered by regions of existence of periodic orbits given by period adding structures.

The period adding structures completely describe not only all the possible spiking asymptotic dynamics but also the behavior of the firing rate, which is a devil's staircase as a function of the parameters.