Synchronization as Aggregation: Cluster Kinetics of Pulse-Coupled Oscillators

Abstract

We consider models of identical pulse-coupled oscillators with global interactions. Previous work showed that under certain conditions such systems always end up in sync, but did not quantify how small clusters of synchronized oscillators progressively coalesce into larger ones. Using tools from the study of aggregation phenomena, we obtain exact results for the time-dependent distribution of cluster sizes as the system evolves from disorder to synchrony.PACS numbers: 05.45. Xt, 05.70.Ln In one of the first experiments on firefly synchronization, the biologists John and Elizabeth Buck captured hundreds of male fireflies along a tidal river near Bangkok and then released them at night, fifty at a time, in their darkened hotel room [1]. As they looked on in wonder, they observed that "centers of synchrony began to build up slowly among the fireflies on the wall. In one area we would notice that a pair had begun to pulse in unison; in another part of the room a group of three would be flashing together, and so on." Synchronized groups continued to emerge and grow, until as many as a dozen fireflies were blinking on and off in concert. The Bucks realized that the fireflies were phase shifting each other with their flashes, driving themselves into sync.Here we study stylized models of oscillators akin to the fireflies, in which synchrony builds up stepwise, in expanding clusters. By borrowing techniques used to analyze aggregation phenomena in polymer physics, materials science, and related subjects [2, 3], we give the first analytical description of how these synchronized clusters emerge, coalesce, and grow. We hasten to add, however, that the models we discuss are not even remotely realistic descriptions of fireflies; they are merely intended as tractable first steps toward understanding how clusters evolve en route to synchrony.Our work is part of a broader interdisciplinary effort [4,5]. Oscillators coupled by sudden pulses have been used to model sensor networks [6][7][8][9][10], earthquakes [11,12], economic booms and busts [13], firing neurons [14,15], and cardiac pacemaker cells [16]. Diverse forms of collective behavior can occur in these pulse-coupled systems, depending on how the oscillators are connected in space. Systems with local coupling often display waves [17,18] or self-organized criticality [11,19,20], with possible relevance to neural computation [15] and epilepsy [21]. In contrast, systems with global coupling, where every oscillator interacts equally with every other, tend to fall into perfect synchrony. Rigorous convergence results have been proven for this case [20,[22][23][24][25]. But the techniques used previously have not revealed much about the transient dynamics leading up to synchrony-the opening and middle game, as opposed to the end game. Aggregation theory offers a new set of tools to explore this prelude to synchrony.We introduce a toy model, which we call scrambler oscillators. It consists of N 1 identical integrate-andfire oscillators coupled all to all. Each oscillator has a ...

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