Theory of spiral wave dynamics in weakly excitable media: Asymptotic reduction to a kinematic model and applications

In a weakly excitable medium, characterized by a large threshold stimulus, the free end of an isolated broken plane wave (wave tip) can either rotate (steadily or unsteadily) around a large excitable core, thereby producing a spiral pattern, or retract, causing the wave to vanish at boundaries. An a symptotic analysis of spiral motion and retraction is carried out in this w eakly excitable large core regime starting from the free-boundary limit of the reaction-diffusion models, valid when the excited region is delimited b y a thin interface. The wave description is shown to naturally split betwee n the tip region and a far region that are smoothly matched on an intermedi ate scale. This separation allows us to rigorously derive an equation of mo tion for the wave tip, with the large scale motion of the spiral wave front , slaved to the tip. This kinematic description provides both a physical pi cture and exact predictions for a wide range of wave behavior, including (i ) steady rotation (frequency and core radius), (ii) exact treatment of the meandering instability in the free-boundary limit with the prediction that the frequency of unstable motion is half the primary steady frequency, (iii ) drift under external actions (external field .with application to axisymm etric scroll ring motion in three dimensions, and spatial-or/and time-depen dent variation of excitability), and (iv) the dynamics of multiarmed spiral waves with the prediction that steadily rotating waves with two or more ar ms are linearly unstable. Numerical simulations of FitzHugh-Nagumo kinetics are used to test several aspects of our results. In addition, we discuss t he semiquantitative extension of this theory to finite cores and pinpoint m athematical subtleties related to the thin interface limit of singly diffus ive reaction-diffusion models. [S1063-651X(99)01610-4].