# On the Global Synchronization of Pulse-coupled Oscillators Interacting on Chain and Directed Tree Graphs

## TL;DR

This paper establishes conditions for global synchronization of pulse-coupled oscillators on chain and directed tree graphs, accommodating heterogeneity, perturbations, and any coupling strength between zero and one.

## Contribution

It introduces a novel analysis for global synchronization of heterogeneous PCOs with arbitrary initial phases on specific graph structures, without requiring large coupling strength.

## Key findings

- Global synchronization achieved under any coupling strength in [0,1]
- Heterogeneous coupling functions and frequency perturbations are tolerated
- Synchronization results apply to chain and directed tree graphs

## Abstract

Driven by increased applications in biological networks and wireless sensor networks, synchronization of pulse-coupled oscillators (PCOs) has gained increased popularity. However, most existing results address the local synchronization of PCOs with initial phases constrained in a half cycle, and results on global synchronization from any initial condition are very sparse. In this paper, we address global PCO synchronization from an arbitrary phase distribution under chain or directed tree graphs. Our results differ from existing global synchronization studies on decentralized PCO networks in two key aspects: first, our work allows heterogeneous coupling functions, and we analyze the behavior of oscillators with perturbations on their natural frequencies; secondly, rather than requiring a large enough coupling strength, our results hold under any coupling strength between zero and one, which is crucial because a large coupling strength has been shown to be detrimental to the robustness of PCO synchronization to disturbances.