Anomalous Synchronization of Integrate-and-Fire Oscillators with Global Coupling

We study an infinite assembly of integrate and fire oscillators with global coupling. Numerical simulations reveal that the time required for the mutual synchrony of all the oscillators exhibits singular behavior as a function of the interaction strength. The origin of this anomalous synchronization is elucidated by a theoretical analysis. Synchronization is one of the most fascinating nonlinear phenomena far from equilibrium. 1) It appears in various fields, such as oscillating chemical reactions, 2) active kinetics of allosteric enzymes 3), 4) and the flashing of fireflies. 5), 6) There are basically two ways of modeling rhythmic synchronization. One is the description of nonlinear oscillators in terms of the phase variables. Since the pioneering work of Kuramoto 7) (and Winfree 8)) there have been many investigations of the dynamics of phase-coupled oscillators with distributed intrinsic frequencies. (See Ref. 9) and the earlier references cited therein.) The second way of modeling such synchronization consists of a description in terms of pulse-coupled oscillators. In this model, each element has an activity level that increases in time. When the activity exceeds a certain threshold, it fires, and the activity is reset to zero. An impulse is transmitted due to the firing and it thereby affects the activity level of other elements. Peskin 10) was the first to investigate pulse-coupled (integrate-and-fire) oscillators. He studied the mutual synchronization of a population of identical pulse-coupled oscillators. Mirollo and Strogatz 11) have studied synchronization of pulse-coupled oscillators, generalizing Peskin's model, and have proved that an assembly consisting of a finite number of oscillators becomes synchronized for almost all initial conditions within a finite time. Their results have been verified by experiments using artificial oscillators composed by electronic circuits. 12) There have been further related developments recently. 13)–17) In the present paper, we study mutual synchronization of integrate-and-fire os-cillators with global coupling. We assume that all the oscillators are identical and that the dynamics do not contain any stochasticity. We are mainly concerned with the time required for mutual synchronization in the asymptotic limit of a large population. Although integrate-and-fire oscillators have been investigated for many years,