Stable synchrony in globally coupled integrate-and-fire oscillators

A model of integrate-and-fire oscillators is studied. In the special case of identical oscillators, the model was first proposed and analyzed by Mirollo and Strogatz [SIAM J. Appl. Math., 50 (1990), pp. 1645-1662]. We assume, as in Mirollo and Strogatz's model, that each oscillator x_i evolves according to a map f_i. Our main results are to demonstrate that the concavity structure of f_i plays an important role in determining whether Peskin's second conjecture holds true. Specifically, the following statements are proved. First, the system of convex oscillators (i.e., f _i″ < 0 for all i), in general, synchronizes when the oscillators are not quite identical. Second, the system of a certain class of concave oscillators (i.e., f_i″ > 0 for all i) will not achieve synchrony for initial conditions in a set of positive measure when the oscillators are nearly identical. Third, the system of concave oscillators may achieve synchrony under certain sufficient conditions, provided that the oscillators are not quite nonidentical and that its concavity is small.