An ionically based mapping model with memory for cardiac restitution

David G. Schaeffer, John W. Cain, Daniel J. Gauthier, Soma S. Kalb, Robert A. Oliver, Elena G. Tolkacheva, Wenjun Ying, Wanda Krassowska

Research output: Contribution to journalArticlepeer-review

22 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)459-482
Number of pages24
JournalBulletin of mathematical biology
Issue number2
StatePublished - Feb 2007

Bibliographical note

Funding Information:
The Support of the National Institutes of Health under grant 1R01-HL-72831 and the National Science Foundation under grants PHY-0243584 and DMS-9983320 is gratefully acknowledged.


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