We investigate a two-dimensional mapping model of a paced, isolated cardiac cell that relates the duration of the action potential to the two preceding diastolic intervals as well as the preceding action potential duration. The model displays rate-dependent restitution and hence memory. We derive a criterion for the stability of the 1:1 response pattern displayed by the model. This criterion can be written in terms of experimentally measured quantities-the slopes of restitution curves obtained via different pacing protocols. In addition, we analyze the two-dimensional mapping model in the presence of closed-loop feedback control. The control is initiated by making small adjustments to the pacing interval in order to suppress alternans and stabilize the 1:1 pattern. We find that the domain of control does not depend on the functional form of the map, and, in the general case, is characterized by a combination of the slopes. We show that the gain gamma necessary to establish control may vary significantly depending on the value of the slope of the so-called standard restitution curve (herein denoted as S12), but that the product gammaS12 stays approximately in the same range.

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A generic feature of cardiac muscle is that the duration of an action potential depends on the long-term history of previous action potentials, known as cardiac 'memory'. Even though memory is known to be an important physiological response, there have only been limited studies of its effect on cardiac dynamics. Here, we investigate a map-based model of paced myocardium in the presence of closed-loop feedback control. The model relates the duration of an action potential to the preceding diastolic interval as well as the preceding action potential duration and thus has some degree of memory. We find that the range of parameters over which control is effective can be enlarged or reduced by memory, a prediction that is independent of the specific functional form of the map. Our work suggests that modifying the degree of memory (e.g., pharmacological agents) with some form of feedback control may be an effective strategy for the maintenance of normal cardiac function.

We demonstrate that a traveling pulse solution, emerging from the concatenation of two unstable kinks, can be stable. By means of stability analysis and numerical simulations, we show the stability of neuronal pulses (action potentials) with increasing refractory periods, which decompose into two (radiationally) unstable kinks in the limit. These action potentials are solutions of an ultrarefractory version of the FitzHugh-Nagumo system.