Synchronization of Pulse-Coupled Oscillators and Clocks under Minimal Connectivity Assumptions

TL;DR

This paper proves that pulse-coupled oscillators and clocks can synchronize under minimal connectivity conditions, specifically when the communication graph contains a root node, extending previous results that required stronger connectivity.

Contribution

It generalizes existing synchronization results by relaxing the connectivity assumption to a directed spanning tree, which is also shown to be necessary.

Findings

Synchronization is achieved under the existence of a root node in the communication graph.

The minimal connectivity condition for synchronization is established as necessary and sufficient.

The results apply to networks with general phase-response curves and communication topologies.

Abstract

Populations of flashing fireflies, claps of applauding audience, cells of cardiac and circadian pacemakers reach synchrony via event-triggered interactions, referred to as pulse couplings. Synchronization via pulse coupling is widely used in wireless sensor networks, providing clock synchronization with parsimonious packet exchanges. In spite of serious attention paid to networks of pulse coupled oscillators, there is a lack of mathematical results, addressing networks with general communication topologies and general phase-response curves of the oscillators. The most general results of this type (Wang et al., 2012, 2015) establish synchronization of oscillators with a delay-advance phase-response curve over strongly connected networks. In this paper we extend this result by relaxing the connectivity condition to the existence of a root node (or a directed spanning tree) in the graph. This condition is also necessary for synchronization.