ar X iv : 0 90 8 . 39 23 v 1 [ q - bi o . T O ] 2 7 A ug 2 00 9 Asymptotics of conduction velocity restitution

We extend a nonstandard asymptotic method proposed earlier, for calculation of conduction velocity restitution curves in ionic models of cardiac excitability. Conduction velocity restitution is the simplest nontrivial spatially extended problem in excitable media, and in the case of cardiac tissue it is an important tool for prediction of cardiac arrythmias and fibrillation. An idealized conduction velocity restitution curve requires solving a nonlinear eigenvalue problem with periodic boundary conditions, which in the cardiac case is very stiff and calls for the use of asymptotic methods. We compare asymptotics of restitution curves in four examples, two generic excitable media models, and two ionic cardiac models. The generic models include the classical FitzHugh-Nagumo model and its variation by Barkley. They are treated with standard singular perturbation techniques. The ionic models include a simplified “caricature” of the Noble (1962) model and the Beeler and Reuter (1977) model, which are treated with the nonstandard method. The Caricature Noble model is considered with particular care to demonstrate the well-posedness of the corresponding boundary-value problem. The developed method is then applied to the Beeler-Reuter model. We discuss new mathematical features appearing in cardiac ionic models and possible applications of the developed method.