Period-Doubling Instability and Memory in Cardiac Tissue

Abstract

Theoretical studies have indicated that alternans (period-doubling instability) of action potential duration is associated with a restitution relation with a slope >or=1. However, recent experimental findings suggest that the slope of the restitution relation is not necessarily predictive of alternans. Here, we compared a return map memory model to action potential data from an ionic model and found that the memory model reproduced dynamics that could not be explained by a unidimensional restitution relation. Using linear stability analysis, we determined the onset of the alternans in the memory model and confirmed that the slope of the restitution curve was not predictive.

An Ionically Based Mapping Model with Memory for Cardiac Restitution

“…This difference illustrates the point that data from the restitution portrait provide a demanding test of any mathematical model. 9 In fact, as may be seen from the figure, the residual is not minimized by the value for τ sclose that we have chosen. The value τ sclose = 320 ms is a compromise: we accept a residual that is 15 % larger than the minimum in order to reduce the error in the leading-order asymptotics used in deriving the mapping from the idealized ionic model.…”

“…In an effort to model rate dependence, various authors (Chialvo et al, 1990;Otani and Gilmour, 1997;Fox et al, 2002a) have introduced ad hoc 2D mapping models, one variable being A n and the other a "memory" variable. It was shown in Kalb et al 2004 that, although these models exhibit the right qualitative behavior, no single set of parameter values can match all the data in the restitution portrait, such as in Figure 2(a).…”

“…Like the models described in Chialvo et al 1990, Otani and, and Fox et al 2002a, our mapping contains two variables-one is A n as above; the other, which will be denoted by C n , specifies an ion concentration in the cell in the underlying ionic model at the start of the n th action potential (see the Appendix). Under periodic stimulation (say with period B), these two variables evolve according to the iteration 5…”

“…It is instructive to compare our model to the ad hoc models of Chialvo et al 1990 andFox et al 2002a, in which the action potential duration A n and the memory variable M n evolve according to…”

Asymptotic approximation of an ionic model for cardiac restitution

“…• A fixed charge ε enters the cell during each action potential; in symbols (7) The precise form of I(t) is not important; to achieve the properties above we choose2 (8) Note that the time constants in Table 1 satisfy the following property: (9) Below in deriving a mapping to describe the behavior of (2), (4) and (6), we shall assume that (9) holds, as well as (10) Although it is not critical, we shall also assume that 1…”