  Many features of the sequence of action potentials produced by repeated
stimulation of a cardiac patch can be modeled by a 1D mapping, but not the full
behavior observed in the restitution portrait: in particular, not (i) distinct
slopes for dynamic and S1-S2 restitution (rate dependence) and not (ii) long
transients in the approach to steady state (accomodation). To address these
shortcomings, \emph{ad hoc} 2D mappings, where the second variable is a
``memory'' variable, have been proposed; it seems that these models exhibit
some, but not all, of the relevant behavior. In this paper we introduce a new
2D mapping and determine a set of parameters for it that gives a rather
accurate description of the full restitution portrait found for one animal. The
changes in the mapping, compared to previous models, result from requiring that
the mapping can be derived as an asymptotic limit of a simple ionic model.
Among other benefits, one can interpret the parameters in the mapping in terms
of the ionic model. The ionic model is an extension of a two-current model that
adds a third dependent variable, a generalized concentration. The simplicity of
the ionic model and the physiological basis for the mapping contribute to the
usefulness of these ideas for describing restitution data in a variety of
contexts. The fitting procedure is straightforward and can easily be applied to
obtain a mathematical model for data from other experiments, including
experiments on different species. Uniqueness of the parameter choice is also
discussed.
