The spontaneous rhythmic activity of aggregates of embryonic chick heart cells was perturbed by the injection of single current pulses and periodic trains of current pulses. The regular and irregular dynamics produced by periodic stimulation were predicted theoretically from a mathematical analysis of the response to single pulses. Period-doubling bifurcations, in which the period of a regular oscillation doubles, were predicted theoretically and observed experimentally.

Abstract.A mathematical model for the perturbation of a biological oscillator by single and periodic impulses is analyzed. In response to a single stimulus the phase of the oscillator is changed. If the new phase following a stimulus is plotted against the old phase the resulting curve is called the phase transition curve or PTC (Pavlidis, 1973). There are two qualitatively different types of phase resetting. Using the terminology of Winfree (1977Winfree ( , 1980, large perturbations give a type 0 PTC (average slope of the PTC equals zero), whereas small perturbations give a type 1 PTC. The effects of periodic inputs can be analyzed by using the PTC to construct the Poincar6 or phase advance map. Over a limited range of stimulation frequency and amplitude, the Poincar6 map can be reduced to an interval map possessing a single maximum. Over this range there are period doubling bifurcations as well as chaotic dynamics. Numerical and analytical studies of the Poincar6 map show that both phase locked and non-phase locked dynamics occur. We propose that cardiac dysrhythmias may arise from desynchronization of two or more spontaneously oscillating regions of the heart. This hypothesis serves to account for the various forms of atrioventricular (AV) block clinically observed. In particular 2 : 2 and 4 : 2 AV block can arise by period doubling bifurcations, and intermittent or variable AV block may be due to the complex irregular behavior associated with chaotic dynamics.

We simulate the effect of periodic stimulation on a strand of ventricular muscle by numerically integrating the one-dimensional cable equation using the Beeler-Reuter model to represent the transmembrane currents. As stimulation frequency is increased, the rhythms of synchronization { 1 : 1 -* 2 : 2 -* 2 : 1 -* 4: 2 -* irregular-* 3 : 1 -* 6: 2 -* irregular-* 4: 1 -* 8 : 2 -*... --~ 1 : 0} are successively encountered. We show that this sequence of rhythms can be accounted for by considering the response of the strand to premature stimulation. This involves deriving a one-dimensional finiteditierence equation or "'map" from the response to premature stimulation, and then iterating this map to predict the response to periodic stimulation. There is good quantitative agreement between the results of iteration of the map and the results of the numerical integration of the cable equation. Calculation of the Lyapunov exponent of the map yields a positive value, indicating sensitive dependence on initial conditions ("chaos"), at stimulation frequencies where irregular rhythms are seen in the corresponding numerical cable simulations. The chaotic dynamics occurs via a previously undescribed route, following two period-doubling bifurcations. Bistability (the presence of two different synchronization rhythms at a fixed stimulation frequency) is present both in the simulations and the map. Thus, we have been able to directly reduce consideration of the dynamics of a partial differential equation (which is of infinite dimension) to that of a one-dimensional map, incidentally demonstrating that concepts from the field of non-linear dynamics--such as perioddoubling bifurcations, bistability, and chaotic dynamics--can account for the phenomena seen in numerical simulations of the cable equation. Finally, we sketch out how the one-dimensional description can be extended, and point out some implications of our work for the generation of malignant ventricular arrhythmias.

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Action potentials resulting from periodic stimulation of nerve axons occur at intervals that are irregular at moderate stimulation frequencies. Histograms of the intervals are multimodal, as seen in stochastic resonance. At higher stimulation frequencies, the action potentials are suppressed entirely, leaving only subthreshold dynamics. Return maps constructed from data show that both types of response are governed by the same deterministic one-dimensional description, with an unstable subthreshold fixed point largely accounting for the irregular intervals at moderate stimulation frequencies.[S0031-9007(96)00200-1]