Analysis of an Iterative Difference Equation Model of the Cardiac Cell Membrane

An iterative difference equation (DE) model based on a small set of functional characteristics derived from the modified Beeler-Reuter ionic model (MBR) of a space-clamped membrane of the ventricular myocyte was studied. These characteristics were expressed as functions of a common independent variable, the diastolic interval (DIA), to define threshold, Thr (DIA), latency, L(DIA), action potential duration, APD(DIA), and delayed excitability recovery, R (DIA), owing to a sub-threshold response. Memory effects, in the form of a dependence on the immediately preceeding action potential waveform, were not included in the current DE formulation. The entrainment bifurcation structure of the DE model was found to be very sensitive to the form of APD (DIA) but, with a suitable choice of parameter values, the model reproduced the successive stable rhythms of synchronization {1:1 → 2:2 → 2:1 → 4:2} found in the MBR model with decreasing basic cycle lengths. The fine details of the transition between stable rhythms were seen to depend on the specific forms of APD(DIA) and R(DIA) chosen, and on the initial conditions at the onset of pacing. The minor deficiencies of the DE model are attributable to the memory-free APD (DIA) used, and to uncertainties in the description of the MBR model behavior owing to the slow convergence toward a given stable rhythm. Overall, the iterative DE formulation based on the transient properties of an ionic model, as derived by premature stimulation, it is also adequate to describe the stable entrainment patterns of the ionic model.