N. F. Otani and R. F. Gilmour, Memory Models for the Electrical Properties of Local Cardiac Systems, Journal of Theoretical Biology 187, 409-436 (1997).
A series of related new models for the local dynamics of cardiac tissue is introduced. The models are based on a simple memory-like quantity that is used to determine the relationship among the durations and amplitudes of the stimulated action potentials. The first of these models produces period-doubling and chaos, consistent with constant pacing experiments, when standard restitution dynamics would predict stability of the primary 1:1 pattern. Analysis of the associated one-dimensional map suggests how various physiological parameters affect the period-doubling sequence. Many of these relationships have been observed in experiments. The remaining models extend the formalism of the first to account for the Hopf bifurcation of 2:2 patterns observed in experiments. One of these models reproduces the bifurcation sequence, 1:1, 2:2, Hopf bifurcation of 2:2, 2:2, and 2:1 seen in experiments as the pacing interval is decreased. The models clarify the dynamics involved in determining the amplitudes and durations of successive action potentials. Results from these models together with comparison with the experiment strongly suggest that quantities with time constants on the order of 50 and 400 msec exist and affect action potential formation in heart tissue.