^{1,a)}, Elena G. Tolkacheva

^{2}, David G. Schaeffer

^{3}, Daniel J. Gauthier

^{4}and Wanda Krassowska

^{5}

### Abstract

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, , where is the APD following a DI given by . The number of variables previous to determines the degree of memory in the mapping model. Recent experiments have shown that mapping models should contain at least three variables 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 analysis of the different RCs in the RP for mapping models with an arbitrary amount of memory. We determine the number of variables necessary to represent the different RCs in the RP. We also present a graphical visualization of these RCs. Our analysis reveals that the dynamic and S1–S2 RCs reside on two-dimensional surfaces, and therefore provide limited information for mapping models with more than two variables. However, constant-BCL restitution is a feature of the RP that depends on higher dimensions and can possibly be used to determine a lower bound on the dimensionality of cardiac dynamics.

Mathematical models that describe cardiac dynamics are important tools in understanding lethal cardiac arrhythmias, such as ventricular fibrillation. One approach to modeling is to attempt to reproduce the physiological ionic fluxes and changes in concentration across the cell membrane. However, analysis of such multidimensional models is very difficult. In contrast, mapping models, which relate the duration of the action potential (APD) to a number of previous diastolic intervals (DIs) and/or APDs, lend themselves to analysis. In fact, much of the practical theory regarding the onset of instability of heart rhythms is based on analysis of the most simple mapping where APD is a function of only the previous DI. This relationship between APD and preceding DI is measured experimentally using various pacing protocols, yielding so-called restitution curves (RCs). Recent experiments have demonstrated that the most simple mapping model is inadequate to describe the different RCs, and that memory (dependence on more previous APDs/DIs) must be included in mapping models. However, it is not well understood how to interpret different experimental RCs for mapping models with higher degrees of memory. In this paper, we examine mapping models with an arbitrary amount of memory to interpret the different RCs measured experimentally and to derive expressions for their slopes. We use a graphical visualization to aid in this presentation. Our analysis demonstrates the limitations and advantages of existing protocols and RCs, which may be useful in the design of new protocols that take into account the multidimensional nature of memory in cardiac tissue.

The authors gratefully acknowledge the financial support of the National Institutes of Health under Grant No. 1R01-HL-72831 and the National Science Foundation under Grant No. PHY-0243584.

I. INTRODUCTION

II. THE RESTITUTION PORTRAIT

III. RESTITUTION CURVES FOR HIGHER-DIMENSIONAL MODELS

A. The dynamic restitution curve

B. The S1–S2 restitution curve

C. Transient CB responses

IV. VISUALIZATION

A. The dynamic restitution curve

1. Graphical determination of

B. The S1–S2 restitution curve

1. Graphical determination of

C. CB transient

D. Simultaneous projection onto plane

V. DISCUSSION

### Key Topics

- Eigenvalues
- 29.0
- Cardiac dynamics
- 23.0
- Numerical modeling
- 15.0
- Surface dynamics
- 9.0
- Graphical methods
- 8.0

## Figures

An RP produced by iteration of the mapping (A4) given in the Appendix. BCL was decreased from in increments. At each BCL, perturbations were applied at . CB–S responses do not fall on a single line. The line drawn through the CB–S responses, which is a least-squares fit to the responses, emphasizes that the CB responses do not fall on the S1–S2 RC segment. This can be seen more clearly in a close-up for a single BCL in Fig. 2 .

An RP produced by iteration of the mapping (A4) given in the Appendix. BCL was decreased from in increments. At each BCL, perturbations were applied at . CB–S responses do not fall on a single line. The line drawn through the CB–S responses, which is a least-squares fit to the responses, emphasizes that the CB responses do not fall on the S1–S2 RC segment. This can be seen more clearly in a close-up for a single BCL in Fig. 2 .

Details of the RP shown in Fig. 1 for a single BCL .

For mapping models of the form (4) , the duration of the action potential is a function of previous DIs and APDs as shown in the figure. The BCLs are shown above the action potentials.

For mapping models of the form (4) , the duration of the action potential is a function of previous DIs and APDs as shown in the figure. The BCLs are shown above the action potentials.

Responses during the application of a perturbation to determine the S1–S2 RC. All variables previous to are at the steady-state for the BCL . The S2 response is given by .

Responses during the application of a perturbation to determine the S1–S2 RC. All variables previous to are at the steady-state for the BCL . The S2 response is given by .

Visualization of the dynamic RC. (A) The intersection of the surfaces and forms the 3D–DRC (light gray trace). Note that the surfaces and are at a shallow angle to one another. (B) The projection (black) of the 3D–DRC (gray) onto the plane gives the conventional dynamic RC. The vector is tangent to the 3D–DRC at the fixed point for the BCL . is scaled in magnitude for viewing purposes. (C) The surface intersected by the hatched plane defined by (25) , where . The stars in (A) and (B) indicate the steady-state response for the BCL as determined by this intersection. The particular form of used for this visualization is given as (A5) in the Appendix.

Visualization of the dynamic RC. (A) The intersection of the surfaces and forms the 3D–DRC (light gray trace). Note that the surfaces and are at a shallow angle to one another. (B) The projection (black) of the 3D–DRC (gray) onto the plane gives the conventional dynamic RC. The vector is tangent to the 3D–DRC at the fixed point for the BCL . is scaled in magnitude for viewing purposes. (C) The surface intersected by the hatched plane defined by (25) , where . The stars in (A) and (B) indicate the steady-state response for the BCL as determined by this intersection. The particular form of used for this visualization is given as (A5) in the Appendix.

Visualization of the S1–S2 RC. (A) The shaded surface is the surface . The outlined plane is the plane . The intersection of these two surfaces (light gray trace) forms the 3D–SRC. (B) The projection (black) of the 3D–SRC (gray) onto the plane yields the conventional S1–S2 RC. The stars indicate the value of the S1–S2 RC when . The specific form of is given as (A7) in the Appendix.

Visualization of the S1–S2 RC. (A) The shaded surface is the surface . The outlined plane is the plane . The intersection of these two surfaces (light gray trace) forms the 3D–SRC. (B) The projection (black) of the 3D–SRC (gray) onto the plane yields the conventional S1–S2 RC. The stars indicate the value of the S1–S2 RC when . The specific form of is given as (A7) in the Appendix.

Visualization of the CB responses. The shaded surface is the surface . The two planes outlined in black correspond to the two eigendirections of the matrix given in (33) . The intersection of each of these planes with the surface (gray curves) governs the behavior of the transient CB responses. The filled black circles show the CB–D responses that occur when the BCL is changed from . The filled white circles show CB–S responses that occur after perturbations are applied at BCL . The intersection of the two gray curves occurs at the steady-state value for the BCL . The specific form of is given as (A8) in the Appendix.

Visualization of the CB responses. The shaded surface is the surface . The two planes outlined in black correspond to the two eigendirections of the matrix given in (33) . The intersection of each of these planes with the surface (gray curves) governs the behavior of the transient CB responses. The filled black circles show the CB–D responses that occur when the BCL is changed from . The filled white circles show CB–S responses that occur after perturbations are applied at BCL . The intersection of the two gray curves occurs at the steady-state value for the BCL . The specific form of is given as (A8) in the Appendix.

Eigendirections and associated eigenvalues shown in the plane. The CB–D responses (filled circles) and CB–S responses (open circles) are also shown in this plot, illustrating how the directions of the two eigenvectors determine the CB responses.

Eigendirections and associated eigenvalues shown in the plane. The CB–D responses (filled circles) and CB–S responses (open circles) are also shown in this plot, illustrating how the directions of the two eigenvectors determine the CB responses.

Superposition of projections of different RCs onto the plane. Responses to perturbations of BCL are shown on the S1–S2 RC. The CB–D responses show the transient that occurs in decreasing the BCL from . The CB–S responses show the recovery from the perturbations.

Superposition of projections of different RCs onto the plane. Responses to perturbations of BCL are shown on the S1–S2 RC. The CB–D responses show the transient that occurs in decreasing the BCL from . The CB–S responses show the recovery from the perturbations.

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