Global Analysis of Firing Maps

In this paper, we study the behavior of pulse-coupled integrate-and-fire oscillators. Each oscillator is characterized by a state evolving between two threshold values. As the state reaches the upper threshold, it is reset to the lower threshold and emits a pulse which increments by a constant value the state of every other oscillator. The behavior of the system is described by the so-called firing map: depending on the stability of the firing map, an important dichotomy characterizes the behavior of the oscillators (synchronization or clustering). The firing map is
the composition of a linear map with a scalar nonlinearity. After briefly discussing the case of the scalar firing map (corresponding to two oscillators), the stability analysis is extended to the general n-dimensional firing map (for n +1 oscillators). Different models are considered (leaky oscillators, quadratic oscillators,...), with a particular emphasis on the persistence of the dichotomy in higher dimensions.