[Paper Review] Orbital movement of spiral waves

VN Biktashev, Dwight Barkley+1 more|arXiv (Cornell University)|Jul 28, 2009

Micro and Nano Robotics3 citations

TL;DR

This paper proposes a novel mechanism for spiral wave dynamics in excitable media: orbital motion around localized inhomogeneities due to a balance between repulsion at short range and attraction at longer range. Using asymptotic theory and numerical simulations, it predicts stable circular orbits with fixed radii determined by the medium's properties, while drift speed depends on inhomogeneity strength—offering a generic, experimentally observable alternative to classical pinning.

ABSTRACT

Spiral waves in active media react to small perturbations as particle-like objects. Here we apply the asymptotic theory to the interaction of spiral waves with a localized inhomogeneity, which leads to a novel prediction: drift of the spiral rotation centre along circular orbits around the inhomogeneity. The stationary orbits have alternating stability and fixed radii, determined by the properties of the bulk medium and the type of inhomogeneity, while the drift speed along an orbit depends on the strength of the inhomogeneity. Direct simulations confirm the validity and robustness of the theoretical predictions and show that these unexpected effects should be observable in experiment.

Motivation & Objective

• To identify and theoretically predict a new type of spiral wave dynamics: stable orbital motion around localized inhomogeneities in excitable media.
• To explain how the interplay between repulsion at short distances and attraction at longer distances leads to the formation of stationary circular orbits.
• To demonstrate that orbital motion is robust and observable in experiments, contrasting it with classical pinning and meandering.
• To show that the radii of stable orbits are determined solely by the unperturbed medium's properties, while drift speed depends on inhomogeneity strength.
• To establish that orbital motion is a generic phenomenon, not limited to specific models or inhomogeneity shapes, and may be mistaken for meandering in experiments.

Proposed method

• Application of asymptotic theory to derive the equation of motion for the spiral wave rotation center under small perturbations.
• Use of the response function (RF) to quantify the sensitivity of spiral wave position to spatial perturbations in the medium parameters.
• Derivation of the drift force $ F(d) $, defined as the drift velocity per unit inhomogeneity strength, from the response function and perturbation distribution.
• Numerical solution of the reaction-diffusion equation (1) to simulate spiral wave dynamics under localized inhomogeneities.
• Analysis of the radial component $ F_r(d) $ to identify radii where $ F_r(d) = 0 $, indicating potential stationary orbits.
• Comparison of theoretical predictions with direct numerical simulations (DNS) to validate the existence and stability of orbital motion.

Experimental results

Research questions

• RQ1 Can spiral waves exhibit stable orbital motion around a localized inhomogeneity, and if so, under what conditions?
• RQ2 How do the radii of stationary orbits depend on the properties of the unperturbed medium and the type of inhomogeneity?
• RQ3 What determines the drift speed along an orbital path, and how does it vary with inhomogeneity strength?
• RQ4 How does orbital motion differ from classical pinning and meandering in terms of stability and dynamics?
• RQ5 Is orbital motion robust and observable in numerical simulations and real experiments, and can it be distinguished from other spiral wave behaviors?

Key findings

• Stable circular orbits for spiral wave rotation centers exist at discrete radii where the radial component of the drift force $ F_r(d) $ vanishes.
• The radii of these stationary orbits are determined solely by the properties of the unperturbed medium and the inhomogeneity type, not by the inhomogeneity strength.
• Orbital motion is characterized by alternating stability: for a given inhomogeneity sign, only certain orbits are stable, depending on the sign of the perturbation.
• The drift speed along an orbit is proportional to the inhomogeneity strength $ eta $, with linear dependence confirmed in simulations up to $ |eta / b_0| = 0.9 $.
• Numerical simulations confirm that trajectories between stable orbits converge to the nearest stable orbit, with the spiral entering orbital motion regardless of initial position.
• Orbital motion can be mistaken for meandering in experiments, but differs fundamentally: it arises from external inhomogeneity, not internal instabilities, and its frequency $ u $ depends on inhomogeneity strength.