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...

We propose a new analytical technique, cycle length analysis (CLA), which quantifies all intermittent periodic modes of a data set. CLA is especially well suited to the treatment of extended measurements of complex physical, chemical, and biological systems when subjected to variations in underlying dynamical parameters which cannot be controlled precisely or monitored accurately. We demonstrate the application of CLA using (1) a coupled oscillator model under the conditions of continuously varying control parameters and (2) clinical cardiac rhythm data.PACS numbers: 87.10.+e Dynamical systems containing many degrees of freedom often exhibit a simple periodic behavior which results from the synchronization of nonlinearly coupled motions. A good example can be found in the nearly clocklike regularity of the normal heart beat. The interaction of the millions of cells which induce the cardiac contraction yields a concerted oscillatory behavior which belies the underlying complexity of the heart. This synchrony of motion {mode locking) has been demonstrated to be a generic of models of nonlinearly coupled oscillators over wide ranges of control parameters. It has also been reported in systems as diverse as forced buckled beams [1], plasma wave oscillations [1], mechanically ventilated cats [2], and firefly emissions [3]. It is very likely that this behavior dominates the dynamics of many other complex physical, chemical, and biological oscillatory systems which have yet to be studied.Although mode locking is indicative of a complex underlying nonlinear machinery, the presence of simple mode locked states can simplify the analysis of dynamical systems due to their distinctive periodic signatures. Yet the analysis of many such systems is also complicated by fluctuations in underlying parameters which result in a continuously evolving dynamics. This, for example, is often the case for cardiac rhythms interrupted by premature heart beats (and thus shortened heart beat intervals) which can result from the interaction between the normal pacemaker of the heart and a secondary (or ectopic) pacemaker. Frequent changes in heart rate can produce a rapidly evolving set of entrainments between these competing cardiac pacemakers which are manifested in volatile transitions from one pattern of arrhythmia to another. Though these transitory patterns may be so distinctive as to be readily observed through inspection of the rhythm data, conventional techniques of time series analysis are often of little use in their characterization. The power spectrum of cardiac arrhythmia data typically contains little structure outside the low frequency regime (1//noise) [4]. This is due in large measure to the fact that the spectral decomposition of data which exhibits evolutionary or intermittent behavior reflects not only the various underlying periodicities in the data, but also the frequency components necessary to account for the transitions between intermittent states. Spectral methods are essentially global: They transform the data under th...