An Ionically Based Mapping Model with Memory for Cardiac Restitution

Abstract

Many features of the sequence of action potentials produced by repeated stimulation of a patch of cardiac muscle can be modeled by a 1D mapping, but not the full behavior included in the restitution portrait. Specifically, recent experiments have found that (i) the dynamic and S1-S2 restitution curves are different (rate dependence) and (ii) the approach to steady state, which requires many action potentials (accommodation), occurs along a curve distinct from either restitution curve. Neither behavior can be produced by a 1D mapping. To address these shortcomings, ad hoc 2D mappings, where the second variable is a "memory" variable, have been proposed; these models exhibit qualitative features of the relevant behavior, but a quantitative fit is not possible. In this paper we introduce a new 2D mapping and determine a set of parameters for it that gives a quantitatively accurate description of the full restitution portrait measured from a bullfrog ventricle. The mapping can be derived as an asymptotic limit of an idealized ionic model in which a generalized concentration acts as a memory variable. This ionic basis clarifies how the present model differs from previous models. The ionic basis also provides the foundation for more extensive cardiac modeling: e.g., constructing a PDE model that may be used to study the effect of memory on propagation. The fitting procedure for the mapping is straightforward and can easily be applied to obtain a mathematical model for data from other experiments, including experiments on different species.

Asymptotic approximation of an ionic model for cardiac restitution

“…In this paper we complement the modeling of Schaeffer et al [17] by showing that the mapping of that paper arises as an asymptotic limit of an idealized ionic model. We begin in Section 2 by recalling from Schaeffer et al [17] both the ionic model and the mapping.…”

“…Moreover, evolution towards the steady state is very slow, much slower than other time scales in the data. See Schaeffer et al [17] and the references in that paper for a more thorough discussion of memory issues.…”

“…In Schaeffer et al [17], we introduced a 2D mapping and chose parameters that gave a quantitatively accurate description of the full restitution behavior, including memory, measured from a bullfrog ventricle. The present paper is concerned with this model, but we study its theoretical foundations rather than apply it to fitting experimental data.…”

“…We begin in Section 2 by recalling from Schaeffer et al [17] both the ionic model and the mapping. In Section 3 we derive the mapping from the ionic model as the leading term in an asymptotic expansion.…”

“…These equations involve ten positive parameters, values of which can be obtained by fitting the model with experimental data. For example, the values listed in Table 1 were obtained in Schaeffer et al [17] from experiments with a bullfrog ventricle.…”