Global stability of synchronous and out-of-phase oscillations in central pattern generators

TL;DR

This paper analyzes coupled oscillator arrays modeling central pattern generators, demonstrating how diffusive and repulsive couplings produce synchronous or out-of-phase oscillations, with stability proven via contraction analysis, and applied to a leech heartbeat model.

Contribution

It provides a mathematical framework for understanding how different coupling types influence oscillation patterns in CPG models, with stability proofs and a biological application.

Findings

Repulsive coupling induces out-of-phase oscillations.

Diffusive coupling leads to synchronous oscillations.

Global stability of oscillation states is proven.

Abstract

Coupled arrays of Andronov-Hopf oscillators are investigated. These arrays can be diffusively or repulsively coupled, and can serve as central pattern generator models in animal locomotion and robotics. It is shown that repulsive coupling generates out-of-phase oscillations, while diffusive coupling generates synchronous oscillations. Specifically, symmetric solutions and their corresponding amplitudes are derived, and contraction analysis is used to prove global stability and convergence of oscillations to either symmetric out-of-phase or synchronous states, depending on the coupling constant. Next, the two mechanisms are used jointly by coupling multiple arrays. The resulting dynamics is analyzed, in a model inspired by the CPG-motorneuron network that controls the heartbeat of a medicinal leech.