A plausible mechanism for the motion of untwisted scroll rings in excitable media

Periodically rotating spiral waves have been much studied in two-dimensional excitable media. In three dimensions the corresponding ‘scroll rings’ have a richer geometry and an additional dynamical aspect. Owing to the curvature of such a ring, its speed of rotation and the normal velocity of wavefronts are only quasi-periodic. These effects lead to horizontal and vertical polarizations of the wavefront's motion, which in turn lead to horizontal and vertical drift. We derive the dependence of these motions on the curvature of the ring: it agrees qualitatively with numerical experiments. The wavefront is also deformed near the center of rotation (the ring) in ways revealed by this analysis.