Generalized eikonal equation in excitable media

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Abstract
Numerical simulations show that, in excitable media, the standard eikonal equation describing the dependence of a wave front's local velocity on its curvature fails badly in the presence of significant dispersion [Pertsov et al. Phys. Rev. Lett. 78, 2656 (1997)]. Here we derive a corrected eikonal equation, valid in an unrestricted frequency range, which includes highly dispersive conditions. The derivation, which uses a finite-renormalization technique, is applied to diffusion-reaction equations with generic reactivity functions and two diffusivities of arbitrary ratio. In the important case of equal diffusivities α, we obtain at low curvature, the following contribution to the speed: [-1+(ω/c)(∂c/∂ω)](α/r), where 1/r is the curvature, ω is the frequency, and c=c(ω) is the speed of a plane wave with that frequency. In the single-diffusivity case there is a further contribution (ε/c)(∂c/∂ε)(α/r), where ε is the ratio of time scales for diffusing and nondiffusing variables; ε is not restricted to a small range. Both cases yield excellent agreement with numerical simulations. Our various formulas are compared with the classical results of Zykov (Biofizika 25, 888 (1980) [Biophysics 25, 906 (1980)]) and of Keener [SIAM J. Appl. Math. 46, 1039 (1986)].