  We extend a non-Tikhonov asymptotic embedding, 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 arrhythmias
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 lead to non-Tikhonov problems where known asymptotic
results do not apply. The Caricature Noble model is considered with particular
care to demonstrate the well-posedness of the corresponding boundary-value
problem. The developed method for calculation of conduction velocity
restitution 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.
