# Synchronization of Oscillators via Active Media

## TL;DR

This paper investigates how oscillators synchronize through active media, revealing complex dynamics, stability conditions, and robustness, with applications to biological neuron models.

## Contribution

It introduces a scalar phase model for oscillator coupling via active media and analyzes synchronization patterns, stability, and robustness in various configurations.

## Key findings

- Phase locking occurs at multiple m:n patterns with one excitable and one oscillatory cell.
- Synchronization is the only attractor when two oscillatory cells are coupled via an excitable cell.
- Long chains of excitable cells enhance robustness to oscillator frequency differences.

## Abstract

In this paper, we study pairs of oscillators that are indirectly coupled via active (excitable) cells. We introduce a scalar phase model for coupled oscillators and excitable cells. We first show that one excitable and one oscillatory cell will exhibit phase locking at a variety of $m:n$ patterns. We next introduce a second oscillatory cell and show that the only attractor is synchrony between the oscillators. We will also study the robustness to heterogeneity when excitable cell fires or is quiescent. We next examine the dynamics when the oscillators are coupled via two excitable cells. In this case, the dynamics are very complicated with many forms of bistability and, in some cases, chaotic behavior. We also apply weak coupling analysis to this case and explain some of the degeneracies observed in the bifurcation diagram. We also look at pairs of oscillators coupled via long chains of excitable cells and show that small differences in the frequency of the oscillators makes their locking more robust. Finally, we demonstrate many of the same phenomena seen in the phase model for a gap-junction coupled system of Morris-Lecar neurons.