Condition for alternans and stability of the 1:1 response pattern in a "memory" model of paced cardiac dynamics

We analyze a mathematical model of paced cardiac muscle consisting of a map relating the duration of an action potential to the preceding diastolic interval as well as the preceding action potential duration, thereby containing some degree of "memory." The model displays rate-dependent restitution so that the dynamic and S1-S2 restitution curves are different, a manifestation of memory in the model. We derive a criterion for the stability of the 1:1 response pattern displayed by this model. It is found that the stability criterion depends on the slope of both the dynamic and S1-S2 restitution curves, and that the pattern can be stable even when the individual slopes are greater or less than one. We discuss the relation between the stability criterion and the slope of the constant-BCL restitution curve. The criterion can also be used to determine the bifurcation from the 1:1 response pattern to alternans. We demonstrate that the criterion can be evaluated readily in experiments using a simple pacing protocol, thus establishing a method for determining whether actual myocardium is accurately described by such a mapping model. We illustrate our results by considering a specific map recently derived from a three-current membrane model and find that the stability of the 1:1 pattern is accurately described by our criterion. In addition, a numerical experiment is performed using the three-current model to illustrate the application of the pacing protocol and the evaluation of the criterion.