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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... more
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.
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Predicting cardiac alternans is a crucial step toward detection and prevention of ventricular fibrillation, a heart rhythm disorder that kills hundreds of thousands of people in the US each year. According to the theory of dynamical... more
Predicting cardiac alternans is a crucial step toward detection and prevention of ventricular fibrillation, a heart rhythm disorder that kills hundreds of thousands of people in the US each year. According to the theory of dynamical systems, cardiac alternans is mediated by a period-doubling bifurcation, which is associated with variations in a characteristic eigenvalue. Thus, knowing the eigenvalues would allow one to predict the onset of alternans. The existing criteria for alternans either adopt unrealistically simple assumptions and thus produce erroneous predictions or rely on complicated intrinsic functions, which are not possible to measure experimentally. In this work, we present a model-independent technique to estimate a system’s eigenvalues without requirements on the knowledge of the underlying dynamic model. The method is based on principal components analysis of a pseudo-state space; therefore, it allows one to compute the dominant eigenvalues of a system using the tim...
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Research Interests: Engineering, Physics, Mechanics, Biology, Medicine, and 15 moreNumerical Simulation, Humans, Computer Simulation, Mathematical Sciences, Animals, Physical sciences, Restitution, Heart rate, Conduction Velocity, Action potential, Biological clocks, Action Potentials, Axons, Nerve Conduction Velocity, and Action Potential Duration
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Research Interests: Mathematics, Applied Mathematics, Bifurcation theory, Mathematical Analysis, Bifurcation, and 9 moreSteady state, Degeneration, Numerical Analysis and Computational Mathematics, Spatial Dependence, Hopf Bifurcation, Saddle-node bifurcation, Periodic Solution, Standing Wave, and Period-doubling bifurcation
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Restitution, the characteristic shortening of action potential duration (APD) with increased heart rate, has been studied extensively because of its purported link to the onset of fibrillation. Restitution is often represented in the form... more
Restitution, the characteristic shortening of action potential duration (APD) with increased heart rate, has been studied extensively because of its purported link to the onset of fibrillation. Restitution is often represented in the form of mapping models where APD is a function of previous diastolic intervals (DIs) and/or APDs, An+1=F(Dn,An,Dn−1,An−1,…), where An+1 is the APD following a DI given by Dn. The number of variables previous to Dn determines the degree of memory in the mapping model. Recent experiments have shown that mapping models should contain at least three variables (Dn,An,Dn−1) to reproduce a restitution portrait (RP) that is qualitatively similar to that seen experimentally, where the RP shows three different types of restitution curves (RCs) [dynamic, S1–S2, and constant-basic cycle length (BCL)] simultaneously. However, an interpretation of the different RCs has only been presented in detail for mapping models of one and two variables. Here we present an analy...
Research Interests: Mathematics, Applied Mathematics, Biophysics, Cardiology, Medicine, and 15 moreChaos, Computers, Humans, Animals, Restitution, Atrial Fibrillation, Heart rate, Heart, Nonlinear Dynamics and Chaos, Cardiac Arrhythmias, Numerical Analysis and Computational Mathematics, Lower Bound, Higher Dimensions, Action Potentials, and Action Potential Duration
Research Interests: Mathematics, Applied Mathematics, Medicine, Electrocardiography, Chaos, and 12 moreNumerical Simulation, Karma, Iterated Function Systems, Phase Space, Mathematical Model, Nonlinear Dynamic System, Numerical Analysis and Computational Mathematics, Action (Physics), Cardiac Disease, Action Potential Duration, Ventricular Fibrillation, and Cell Membrane
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ABSTRACT
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... Authors: Dobrovolny, Hana; Oliver, Robert; Sau, Soma; Tolkacheva, Elena; Schaeffer, David; Krassowska, Wanda; Gauthier, Daniel. Affiliation: AA(Department of Physics, Duke University) AB(Department of Biomedical Engineering, Duke... more
... Authors: Dobrovolny, Hana; Oliver, Robert; Sau, Soma; Tolkacheva, Elena; Schaeffer, David; Krassowska, Wanda; Gauthier, Daniel. Affiliation: AA(Department of Physics, Duke University) AB(Department of Biomedical Engineering, Duke University) AC(Department of ...
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We investigate, both experimentally and theoretically, the period-doubling bifurcation to alternans in heart tissue. Previously, this phenomenon has been modeled with either smooth or border-collision dynamics. Using a modification of... more
We investigate, both experimentally and theoretically, the period-doubling bifurcation to alternans in heart tissue. Previously, this phenomenon has been modeled with either smooth or border-collision dynamics. Using a modification of existing experimental techniques, we find a hybrid behavior: Very close to the bifurcation point, the dynamics is smooth, whereas further away it is border-collision-like. The essence of this behavior is