  We describe an asymptotic approach to gated ionic models of single-cell
cardiac excitability. It has a form essentially different from the Tikhonov
fast-slow form assumed in standard asymptotic reductions of excitable systems.
This is of interest since the standard approaches have been previously found
inadequate to describe phenomena such as the dissipation of cardiac wave fronts
and the shape of action potential at repolarization. The proposed asymptotic
description overcomes these deficiencies by allowing, among other non-Tikhonov
features, that a dynamical variable may change its character from fast to slow
within a single solution. The general asymptotic approach is best demonstrated
on an example which should be both simple and generic. The classical model of
Purkinje fibers (Noble, 1962) has the simplest functional form of all cardiac
models but according to the current understanding it assigns a physiologically
incorrect role to the Na current. This leads us to suggest an ``Archetypal
Model'' with the simplicity of the Noble model but with a structure more
typical to contemporary cardiac models. We demonstrate that the Archetypal
Model admits a complete asymptotic solution in quadratures. To validate our
asymptotic approach, we proceed to consider an exactly solvable ``caricature''
of the Archetypal Model and demonstrate that the asymptotic of its exact
solution coincides with the solutions obtained by substituting the
``caricature'' right-hand sides into the asymptotic solution of the generic
Archetypal Model. This is necessary, because, unlike in standard asymptotic
descriptions, no general results exist which can guarantee the proximity of the
non-Tikhonov asymptotic solutions to the solutions of the corresponding
detailed ionic model.
