We use nonlinear methods to predict changes in cardiac rhythms over periods of up to several hours. The patterns of measured premature heart beats which correspond to return cycles of a 1 D map are found to be organized into a devil's staircase when plotted against heart rate. Throughout the data collections, changes in patterns with heart rate agree with the computed dynamics. This result, typical of the majority of patients examined, demonstrates that (1) simple recursions can model complex biological rhythms and (2) mode locking can play a dominant role in the patterning of arrhythmias. PACS numbers: 87.45.Bp, 42.60.Fc Coupled modulated systems can exhibit distinct dynamical behaviors. For example, a solid with two components of different natural atomic spacing can form incommensurate, commensurate, or chaotic phases [1]. In a temporal oscillating system with two natural frequencies, the orbit can be quasiperiodic, periodic (mode locked), or chaotic. In general, if the coupling strength is insufficient to induce chaotic behavior, short period return cycles are more stable than long period or quasiperiodic motions. Thus simple motions tend to overwhelm their neighboring complex ones. This phenomenon gives rise to the distinctive, self-similar structure of the "devil's staircase" as in the standard circle map [2,3]. It is the dominance of simple return cycles that facilitates the treatment of biological dynamical systems which are typically in the presence of short-term fluctuations in underlying control parameters.Some cardiac arrhythmias are known to result from the interaction of two oscillators (i.e., pacemakers) giving rise to a condition clinically termed parasystole [4]. The timing of the normal heart beat is dictated by the concerted electrical discharge of a group of specialized cells called the sinoatrial node [5]. The transmission of the electrical pulses initiated by this pacemaker coordinates the contraction of the heart muscle. Subsequent pulses can be conducted only after a period of time necessary to recharge the conductive cardiac tissue. This period of time, the refractory period, typically ranges from 0.25 to 0.50 of the time between cardiac contractions. When a secondary, or ectopic, pacemaker is present heart beats are induced by the ectopic pacemaker only when it discharges outside the refractory period following the previous sinoatrial excitation. Because the electrical pulse originating at one pacemaker can alter the local electrical environment of the other and thus the timing of its subsequent discharge, the phase relationship of the two pacemakers can be reset upon discharge of either, thus giving rise to the recursion 0n=e"-l + n-f m (e n -] ,e n -2, ... % o n -m ).
(DHere 0" is the phase within the ectopic cycle of the nth sinoatrial discharge, O is the ratio of natural periods of the sinoatrial and ectopic oscillators, and /" are the coupling terms which dictate the resetting of the nth ectopic cycle as a function of the phases of the previous mth sinoatrial discharges within the ec...