Phase synchronization of pulse-coupled excitable clocks

Consider a distributed network on a simple graph $G=(V,E)$ where each node has a phase oscillator revolving on $S^{1}=\mathbb{R}/\mathbb{Z}$ with natural period of 1 second. Pulse-coupling is a class of distributed time evolution rule for such networked clock system inspired by biological oscillators, where local interactions between neighboring nodes are event-triggered and do not depend on any kind of global information. In this paper, we propose a novel inhibitory pulse-coupling and prove that arbitrary phase configuration on any tree with diameter $d$ synchronizes in at most $d$ minutes with at most $120d(|V|-1)$ bits of total information exchange. Our algorithm assumes four-state internal dynamics on nodes that depend on the phase dynamics, inspired by the Greenberg-Hastings model for excitable media. Furthermore, if we allow that each node can distinguish between different neighbors, we can compose our pulse-coupling with a distributed spanning tree algorithm and obtain a scalable self-stabilizing distributed phase synchronization algorithm which runs on any connected topology in $O(|V|)$ times.