Robust formation of metachronal waves in directional chains of phase oscillators

TL;DR

This paper investigates how directional chains of phase oscillators can stably form metachronal waves, showing that asymmetry and noise influence wave stability and speed.

Contribution

It demonstrates that directional, asymmetric oscillator chains can develop stable metachronal waves influenced by perturbations and noise, advancing understanding of biological locomotion mechanisms.

Findings

Directional models exhibit instability to short wavelength perturbations under certain conditions.

Weak noise can seed and sustain metachronal wave states.

The continuum approximation captures key dynamics of the discrete oscillator chain.

Abstract

Biological systems can rely on collective formation of a metachronal wave in an ensemble of oscillators for locomotion and for fluid transport. We consider one-dimensional chains of phase oscillators with nearest neighbor interactions, connected in a loop and with rotational symmetry, so each oscillator resembles every other oscillator in the chain. Numerical integrations of the discrete phase oscillator systems and a continuum approximation show that directional models (those that do not obey reversal symmetry), can exhibit instability to short wavelength perturbations but only in regions where the slope in phase has a particular sign. This causes short wavelength perturbations to develop that can vary the winding number that describes the sum of phase differences across the loop and the resulting metachronal wave speed. Numerical integrations of stochastic directional phase oscillator models show that even a weak level of noise can seed instabilities that resolve into metachronal wave states.