Microscopic Variability Alters Macroscopic Rotation Speed in Stochastic Spiral Waves

TL;DR

This work develops a general SPDE-based theory for how microscopic variability reshapes macroscopic spiral-wave rotation in excitable media. It decomposes noise effects into an instantaneous slowdown from heterogeneity and an orbital-drift term from temporal fluctuations, yielding the central result $\overline{\Omega}_\sigma = \omega_0 + \sigma^2(\omega^{(2)} + \omega^{(2)}_{\rm od}) + O(\sigma^3)$, with coefficients given by bilinear forms on adjoint modes. The theory is validated on the Barkley model under temporal noise, where all second-order corrections yield net slowing, and generalised to spatial and spatio-temporal noise, demonstrating parameter-dependent slowdown across noise types. The findings reveal a robust mechanism by which microscopic noise and heterogeneity modulate spiral dynamics, with potential implications for cardiac tachyarrhythmias and neural wave propagation, and provide a framework for uncertainty quantification in excitable-media dynamics.

Abstract

We present a general theory for noise-induced corrections to the angular velocity of spiral waves. Stochasticity produces two second-order effects: an instantaneous term from heterogeneity that always slows rotation, and an orbital-drift term from temporal fluctuations that can either accelerate or decelerate it. For our parameters, orbital drift is weaker, producing a net slowdown. Analytical predictions match Barkley-model simulations with temporal noise. Examination of additional noise types in silico confirms angular velocity slowing. This mechanism provides a robust route by which stochasticity reshapes spiral dynamics in excitable media, with direct implications for arrhythmias and neural wave propagation.

Figures (3)

• Figure 1: Decomposition and dynamics of stochastic spiral waves. (A) Five-step decomposition of $U$. (B) Initial condition and time evolution of deterministic and stochastic (multiplicative temporal noise, $\sigma=0.4$) spirals barkley1990spiral. (C) Temporal evolution of spiral coordinates. (D) Rotation angle $\Theta(t) = \Omega(t)\cdot t$ over time for deterministic and stochastic spirals; lower panel shows the deviation from a linear fit to the deterministic spiral $\Delta \Theta(t) = \Theta(t) - \omega_0 \cdot t$.
• Figure 2: Effects of temporal noise on the angular velocity of spiral waves in the Barkley model. (left) Average of the angular difference $\Delta\Theta$ from the noiseless reference for four noise levels $\sigma$ (32 runs each); shaded regions show one standard deviation. Linear fits exclude transients. (right) Fitted angular velocities $\overline{\Omega}_\sigma$ demonstrate spiral slowing, consistent with theoretical predictions.
• Figure 3: Effects of spatio(-temporal) noise on the angular velocity of spiral waves in the Barkley model. We consider four different noise levels $\sigma$ with 32 finite-differences simulations each. The average of the angular difference $\Delta\Theta$ to the reference is plotted for each noise level. The shaded area corresponds to one standard deviation around the mean. Linear functions are fitted to the second half of the temporal evolution to exclude transient effects (left). Linear functions are fitted to the data to obtain expressions $\overline{\Omega}_\sigma$ (right). Also spatial (B) and spatio-temporal (C) noise are observed to slow down a spiral wave.