Orbital Motion of Spiral Waves in Excitable Media

Spiral waves in active media react to small perturbations as particlelike objects. Here we apply asymptotic theory to the interaction of spiral waves with a localized inhomogeneity, which leads to a novel prediction: drift of the spiral rotation center along circular orbits around the inhomogeneity. The stationary orbits have fixed radii and alternating stability, 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 numerical simulations confirm the validity and robustness of the theoretical predictions and show that these unexpected effects should be observable in experiment.

Figure 1

(a) Response function for a spiral wave. The spiral is visualized with a gray-scale plot of the $u$ field. The yellow point indicates the spiral tip and the dashed white line shows its trajectory as the spiral rotates. Superimposed is the response function in term of the $u$ component $|W_u|$ (red) and the $v$ component $|W_v|$ (blue). (b) Enhanced visualization of one component of the response function. $\mathrm{Re}(W_v)$ is plotted with medium gray zero (periphery), light gray positive, and dark gray negative. (c) Drift force. The radial $F_r$ (solid red) and azimuthal $F_a$ (dashed blue) components of the drift force, as functions of the distance $d$, calculated by (5). The vertical dash-dotted lines show zeros of the radial component, corresponding to the radii at which stationary orbital motion of the spiral wave is possible.

Figure 2

Orbital motion of a spiral. (a) Simulation of a spiral wave in the presence of an inhomogeneity. The spiral starts near the inhomogeneity (${\delta }_b=-0.02$, green disk), is repelled from it, and launches into a stable circular clockwise orbit [19]. The $u$ field (red) and $v$ field (blue) of the final spiral are shown. The preceding tip trajectory along the stable orbit is shown by a white line. The preceding centers of rotation $\overrightarrow{R}$ are indicated both for this orbit (yellow) and the preceding evolution away from the inhomogeneity (blue). (b) Theoretical vector field (black arrows, nonlinearly scaled for visualization) and predicted trajectories (blue open circles) for the center of a spiral wave near an inhomogeneity (green disk). Actual trajectories for the spiral center from a DNS (red filled circles), with ${\delta }_b=-0.001$. Black dash-dotted circles indicate stationary orbits as predicted by theory. Only every 20th position of the center is shown on both theoretical and DNS trajectories. (c) Same for ${\delta }_b=0.003$.

Figure 3

Angular speed $\Omega$ of orbital motion of the spiral wave as a function of the amplitude of the parametric inhomogeneity ${\delta }_b$: theoretical prediction vs measurements from direct numerical simulations.

Figure 4

Orbital motion of the spiral due to perturbation in parameter $ϵ$. (a) Drift force components as functions of distance for this inhomogeneity. The notation is the same as in Fig. 1c. (b),(c) Comparison of theoretical predictions and simulations, for (b) ${\delta }_{ϵ}=-0.001$ and (c) ${\delta }_{ϵ}=0.001$. The notation is the same as in Figs. 2b, 2c . Shown are pieces of trajectories of the same temporal length $t=0\dots 500$.