Bulletin of Mathematical Biology

, Volume 69, Issue 2, pp 459–482 | Cite as

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
Original Article

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.

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Copyright information

© Society for Mathematical Biology 2007

Authors and Affiliations

  • David G. Schaeffer
    • 1
    • 4
  • John W. Cain
    • 1
  • Daniel J. Gauthier
    • 2
    • 3
    • 4
  • Soma S. Kalb
    • 3
  • Robert A. Oliver
    • 3
  • Elena G. Tolkacheva
    • 2
  • Wenjun Ying
    • 1
  • Wanda Krassowska
    • 3
    • 4
  1. 1.Department of MathematicsDuke UniversityDurhamUSA
  2. 2.Department of PhysicsDuke UniversityDurhamUSA
  3. 3.Department of Biomedical EngineeringDuke UniversityDurhamUSA
  4. 4.Center for Nonlinear and Complex SystemsDuke UniversityDurhamUSA

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