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Nonlinear dynamics of the heartbeat
Physica 17D (1985) 207-214 North-Holland, Amsterdam
NONLINEAR DYNAMICS OF THE HEARTBEAT II. SUBHARMONIC BIFURCATIONS OF THE CARDIAC INTERBEAT INTERVAL IN SINUS NODE DISEASE
Ary L. GOLDBERGER, School of Medicine,
University of California, San Diego, La Jolla, California, USA
Valmik BHARGAVA, schoolof Medicine, University
of California, San Diego, La Jolla, Catifornia, USA
Bruce J. WEST Center for Studies of Nonlinear L.a Jolla, California, USA
Dynamics,
La Jolla
Institute
(a@liated
with the University
of Caii/ornia,
San
Diego),
and
Arnold J. MANDELL? School of Medicine,
University of California,
San Diego, Lu Jolla, California, USA
Received 30 November 1984 Revised 16 April 1985
Changing the coupling of electronic relaxation oscillators may be associated with the emergence of complex periodic behavior. The electrocardiographic record of a patient with the “sick sinus syndrome” demonstrated periodic behavior including subharmonic bifurcations in an attractor of his interbeat interval. Such nonlinear dynamics which may emerge from alterations in the coupling of oscillating pacemakers are not predicted by traditional models in cardiac electrophysiology. An understanding of the nonlinear behavior of physical and mathematical systems may generalize to pathophysiological processes.
1. Introduction Under physiologic conditions, the sino-atrial (SA) node, the normal cardiac pacemaker, and the atrio-ventricular (AV) junction, a subsidiary pacemaker, can be modeled as a pair of coupled relaxation oscillators [l-3]. The faster intrinsic firing frequency of the SA node appears to entrain the slower subsidiary pacemaker which results in $Address for correspondence and reprints: Ary L. Goldberger, M.D., Harvard Medical School, Cardiovascular Division, Beth Israel Hospital, 330 Brookline Ave, Boston, MA 02215. USA.
1 : 1 phase-locking of these oscillators [3]. Studies of model systems have indicated that perturbation of such a network of entrained nonlinear oscillators may alter the coupling (interactions) such that a new cooperative, dynamic state emerges. For example, Gollub et al. [4, 5) observed a variety of complex periodic states in a system of variously coupled tunnel diode relaxation oscillators. Analogous alterations in the interaction of relaxation oscillators might occur in the cardiac electrophysiologic system, possibly induced by pathology in SA, atrial, and AV junctional pacemaker activity. One example of this kind of global alteration in
0167-2789/85/$03.30 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
208
A. L. Goldberger et al./ Nonlinear cardiac oscillators II
the cardiac conduction system is the so-called “sick sinus syndrome” [6, 71 which may reflect a wide variety of structural and metabolic disturbances. We hypothesize that “pathologic” coupling of pacemaker relaxation oscillators in patients with the sick sinus syndrome will be associated with periodic behavior similar to that observed in a variety of physical systems. As early as 1928 van der Pol and van der Mark [l] suggested the use of coupled relaxation oscillators as a model for the heart and built a working analog circuit to demonstrate their ideas. In other studies involving relaxation oscillators [8], they established experimentally that if a small amplitude electromotive force (emf) of frequency f is applied to an electrical relaxation oscillator of free-running frequency f,,, then if f is not too different from fO, i.e. If-&] ef, the oscillator will respond at the frequency f. As they varied the intrinsic frequency of the oscillator ( fa) they found that the system continued to vibrate with the period of the impressed emf (f). This is quite similar to the response of the AV junction to the “applied emf” of the SA node. In addition, they observed that it was also relatively easy to entrain the system to a subharmonic of the impressed frequency f, i.e. the fundamental frequency of the “entrained” oscillation in the system was observed to be f/n, n being an integer up to 100 or 200. They referred to this phenomenon as frequency demultiplication, a term that has been supplanted by subharmonic bifurcation [9]. Mathematically, this phenomenon is understood in terms of an originally stable attractor (orbit) becoming unstable and bifurcating due to the variation in the intrinsic frequency (control parameter). The first bifurcation leads from an orbit of period 2?r/f to one of twice that period, i.e. 2(2a/f) or of frequency f/2. Further variation of f0 may induce a second bifurcation to a period 4(27r/f) or frequency f/4, etc. There has been a resurgence of interest in processes associated with this type of bifurcational behavior since their richness in structure has been shown to emerge from simple discrete mappings [9, lo] having universal properties. In addition to
these mathematical models, period-doubling behavior has been uncovered in a variety of physical systems [ll, 121. A central question, bearing on the problem of sudden cardiac death, is whether these physical-dynamical models can apply to physiological oscillators [13, 141 as originally proposed by van der Pol and van der Mark. Their nonlinear concept of the cardiac conduction system lay fallow for many years in part because it was not apparent how this model would lead to practical consequences not anticipated by traditional electrophysiology. A stringent test of the relevance of the nonlinear viewpoint is how well it explains spontaneous pathologic variations in the normal beating of the heart. It is in response to such perturbations that the distinguishing characteristics between linear and nonlinear systems become particularly evident. Derangements in cardiac electrophysiology seem well suited to examination of these issues since the output of the system can be readily assayed with surface electrocardiography. This report of a patient with conduction system disease is the first quantitative, clinical description of emergent periodicity and subharmonic bifurcations in an attractor of the cardiac interbeat interval.
2. Case study A 60 year old man without cardiac symptoms or evidence of myocardial dysfunction underwent 24-hour ambulatory electrocardiographic monitoring because of a very slow heart rate. Analysis of the recording confirmed the clinical impression of severe conduction system disease (“sick sinus syndrome”) marked by periods of inappropriate sinus bradycardia, prolonged (> 4 s) sinus pauses, and atria1 or AV junctional extrasystoles.
3. Data analysis A continuous section of the recording encompassing 2300 consecutive beats which suggested
A.L. Goldberger et al./ Nonlinear cardiac oscillators II
209
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2.0
_I < >
1 .8
vz W t-
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I fY
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4
D
A’
C’
, 50
100
150
HEART
200
250
300
BEATS
Fig. 1. Representative moving-frame heartbeat series from patient with “sick sinus syndrome,” encompassing 342 consecutive data points ( = 500 s). Periodic state (A) is followed by subharmonic bifurcations (B, C, D) and return to relatively aperiodic steady state (S). Similar pattern is seen throughout the record with reappearance of periodic bursts (A’, C’). Note reverse bifurcation sequence in right-hand section (arrow) in which period-2 (A’) follows longer cycle length oscillation.
the greatest variability in heart rate with the patient in bed were selected for more detailed analysis. Interbeat intervals were measured with an x, y digitizer interfaced to a Tektronix 4051 computer. Because of the relatively brief duration (instability) of the periodic episodes, analysis of the data at 8-beat intervals was performed. First, as shown in fig. 1, the data were displayed as a moving-frame heartbeat series obtained by plotting the interbeat (R-R) interval versus consecutive heartbeat number. Second, graphical representation of the periodic trajectory of the interbeat interval was provided by phase plane mapping (AR-R vs. R-R) for selected periodic sequences.
4. Results Fig. 1, a moving-frame heartbeat series from a representative portion of the data, demonstrates spontaneous shifts in the dynamics of the interbeat interval (R-R) through quasi-stable steady states (S, corresponding to either an AV junctional escape rhythm or sinus rhythm), bifurcation to periodicity (A), subharmonic bifurcation to longer wavelengths (B, C, D), and return to relatively aperiodic steady state. Fig. 2 shows the actual electrocardiographic records, corresponding to data sets A-D and S. Fig. 3, a graph of the phase plane, maps the interbeat interval attractor at a steady state (S) and after bifurcation to period-2 (sequence A), the
A. L. Goldherger et ul./ Nonlineur curdiuc oscillators II
210 PERIOD 2 latrial bigeminVl
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Fig. 2. Electrocardiographic records corresponding to portions of data identified in fig. 1. Period-2 corresponds to atrial bigeminy in which two supraventricular (i.e. sinus, atrial, or AV junctional) beats are clearly coupled together. Period-4 and period-8 sequences correspond to more complex patterns of supraventricular beats. Bottom panel (S), the steady state rhythm corresponds either to AV junctional escape rhythm (with no apparent P waves) or to sinus rhythm. Record has been retouched for clarity.
I 4 4 8.8
first subharmonic bifurcation to period-4 and an apparent second subharmonic bifurcation to period-g. The bifurcation to period-2, is represented by beat-to-beat oscillation of the R-R interval between low (L) and high (H) values (see sequences A and A’ in fig. 1) and corresponds to the well known clinical arrhythmia referred to as atria1 bigeminy (fig. 2, Panel A). Thus, over an g-beat cycle, this periodic pattern will yield the sequence L-H-L-H-L-H-L-H. For example, in fig. 1 (sections A and A’), the R-R interval oscillates between relatively low values (c 1.0 s) and relatively high values (> 2 s) on a beat-to-beat basis. The period-4 cycles repeat twice in an g-beat interval. Sequences B and C in figs. l-3 show two different period-4 cycles. Sequence B corresponds to the interbeat interval pattern: L-L-H-H-L-L-H-H, while sequence C corresponds to H-H-H-L-H-H-H-L. Over the entire 2300-beat data set, the period-4 sequence in
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A. L. Goldberger et al./ Nonlinear cardiac oscillators II
211
Fig. 3. Phase-plane graph (AR-R/R-R) with A, B, C, D, and S indicating the identical portions of data shown in figs. 1 and 2. Relatively aperiodic steady states (AV junctional rhythm or sinus rhythm) were interrupted by bifurcations to period-2 (A), a subharmonic bifurcation to period-4 (B and C), and an apparent second subharmonic bifurcation to period-8 (D). Period-4 in C includes two points close together near the abscissa.
panel B was noted 23 times ( p < 0.001, &i-square test) and the period-4 sequence in panel C was noted three times ( p < 0.001). A period-8 pattern (sequence D of figs. l-3) appeared to be present as well. However, this cycle did not ever repeat itself in 16 consecutive beats. Therefore, the identification of period-8 remains tentative due to the relative instability of these apparent cardiac bifurcation patterns. Furthermore, it was noted that period-2 was not invariably followed by period-4, and that period-4 sometimes appeared without a preceding burst of period-2. Finally, as shown in the right side of fig. 1, reverse bifurcation sequences also occurred. In this case, period-2 (sequence A’) emerged immediately following some longer cycle oscillations in R-R interval.
5. Discussion These data are consistent with the hypothesis that under pathologic conditions the cardiac electrophysiologic system behaves as if there were nonlinear coupling of multiple oscillatory pacemakers. Such nonlinear interaction between relaxation oscillators as seen in coupled tunnel diodes [3-51 can lead to the emergence of longer periods than those intrinsic to any participant and
represents the kind of subharmonic solutions obtained mathematically with the forced van der Pol equation [15-181, the earliest model of the heartbeat [l]. In a model of the cardiac conduction system consisting of two coupled anharmonic oscillators (corresponding to the SA node and the AV junction) we demonstrated a bifurcation from 1 : 1 phase locking (corresponding to sinus rhythm) to a 2 : 1 pattern which occurred when the SA oscillator was driven over a critical range of frequencies [3]. This type of phenomenon mimics the periodicity which may appear when the right atrium is driven at increasing rates with an external pacemaker. In a general mathematical model of two coupled van der Pol oscillators, Keith and Rand [19] showed that the transition from 1: 1 to 2 : 1 phase entrainment can also result from a change in the magnitude of the coupling or from a change in the ratio of the natural frequencies 01/w2. In a possibly analogous way, period-doubling behavior has been observed in the spontaneous rhythmic activity of aggregates of chick embryo heart cells perturbed by injection of current pulses [13] and in the intact canine heart stressed with norepinephrine [14]. Cardiac tamponade, a pathologic condition in which fluid accumulates in the pericardial space and compresses the heart, is associated with a subharmonic bifurcation in the
212
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et ul./
Nonlineur
curdiac oscilkutors II
frequency of cardiac mechanical oscillation as the heart swings back and forth in the pericardial fluid
A
;
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3.2
Our data indicate that perturbation of the intact human electrophysiologic system may be also associated with the emergence of periodic behavior of sinus node and ectopic (non-sinus) pacemakers, manifested by atria1 bigeminy (period-2) or more complex, quasi-periodic rhythms (period-4, period8). The clinical term “sick sinus syndrome” is a misnomer since it suggests impairment of a single physiologic element. Phase plane mapping of the electrocardiogram in this case (fig. 3) is more suggestive of a new pathophysiologic mechanism emerging from the nonlinear interactions of multiple perturbed pacemakers. The graph of the restricted time series (fig. 1) suggests that the overall heartbeat time trace is that of an attractor undergoing bifurcations (and reverse bifurcations) due to spontaneous changes in physiologic parameters modulating the interaction of the SA node and subsidiary pacemakers. The global nature of this perturbation is supported by electrophysiologic studies in patients with the sick sinus syndrome documenting a high prevalence of conduction abnormalities involving the AV junction as well as the His-Purkinje system [21-231, in addition to the SA node. Cardiac perturbations, therefore, may be associated with the appearance of bifurcations which are not predicted by traditional models. On the basis of conventional analysis, the electrocardiographic records in fig. 2 would be interpreted as showing depressed sinus node automaticity or conduction with frequent ectopic beat activity due to AV junctional or atria1 automaticity and possibly reentry. The pattern changes seen in the sequential panels of fig. 2 can be explained by implicating a number of- such different mechanisms occurring at different times and at different local sites. This level of analysis, however, does not account for the period-doubling sequences apparent in the long time trace in fig. 1 and the phase plane maps in fig. 3. Bifurcation behavior of this kind is most consistent with perturbation of a system of coupled
2.6 2
24
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= H
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0.6
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BEATS
Fig. 4. Moving-frame heartbeat series from another patient (an 83 year old man) during cardiopulmonary resuscitation (CPR) after cardiac arrest. Periodicity and visually obvious bifurcations (corresponding to multiple ectopic supraventrkular and ventricular beats) are noted at the beginning of the record, which was interrupted during resuscitation, after which a return to a relatively aperiodic state (sinus rhythm) occurred.
nonlinear oscillators. The precise factor(s) (e.g., neural, metabolic) modulating the global interaction of these nonlinear pacemakers cannot be identified in the present case. From a clinical viewpoint, patients with sick sinus syndrome commonly develop atria1 fibrillation [7] which may represent a new temporal and spatial bifurcation in the electrodynamics of the heartbeat. The suggestion, however, that fibrillation represents cardi,ac chaos [24] emerging at the end-point of a series of subharmonic bifurcations is not supported in the present case. In this patient with sick sinus syndrome, atria1 fibrillation did not appear after any of the bifurcation’ sequences. Furthermore, recent [25] as well as previous [26-291 data based on epicardial mapping studies and spectral analysis support the contention that fibrillation may be a relatively periodic, not a chaotic process, associated with some type(s) of organized nonlinear traveling waves. The transition to fibrillatory activity of the atria or ventricles, therefore, appears to represent a further bifurcation to pathologic periodicity rather than to chaos [25]. The relevance of bifurcational behavior to the problem of sudden cardiac death is suggested in
A. L. Goldberger et al./ Nonlineur curdiac oscillators II CPR
213
band stability [31] associated with l/f-like scaling, to periodic, pathologic states with the capacity for further bifurcations to subharmonic, quasi-periodic, and perhaps in the case of flutter [l] and fibrillation [25] to travelling wave solutions.
Im Fig. 5. Representative electrocardiographic record from data set in fig. 4. During cardiopulmonary resuscitation (CPR) multiple ectopic (non-sinus) beats were seen (top panel) followed by resumption of sinus rhythm. Deflections marked with x in top panel are artifacts caused by external cardiac compressions.
Acknowledgements
This work was supported in part grants from the W.M. Keck Foundation, the Academic Senate of the University of California, San Diego, and the La Jolla Institute.
fig. 4, a moving-frame heartbeat series from another
patient, in this case during the course of a cardiopulmonary arrest. Evidence for emergent periodicity and subharmonic bifurcations is present at the beginning of the record at which time the electrocardiogram (fig. 5) showed a complex combination of ventricular and supraventricular extrasystoles. A return to a relatively aperiodic state (sinus rhythm) was observed after resuscitation. Although the brevity of this record (reflecting the patient’s clinical instability) precludes detailed analysis, the interbeat interval graph in fig. 4 is reminiscent of the dynamics in fig. 1. In both cases, the appearance of frequent ectopic activity is associated with oscillation in R-R intervals as well as the suggestion of additional frequency bifurcations. These periodicities might not be discerned by simple inspection of the electrocardiographic records in figs. 2 and 5 which suggest a highly erratic sequence of depolarization pulses. Finally, the marked periodicity of the heartbeat under a variety of pathologic conditions contrasts with its physiological dynamics. Fourier analysis of interbeat interval variability in healthy subjects has demonstrated a broadband l/f-like spectral pattern with an additional spike noted at the breathing frequency [30]. We have proposed, therefore, that the emergence of certain arrhythmias may be viewed as a bifurcation from normal sinus rhythm, a steady state with broad-
References PI B. van der Pol and J. van der Mark, Phil. Mag. Suppl. 6 (1928) 768.
PI CR. Katholi, F. Urthaler, J. Macy Jr. and T.N. James, Comp. Biomed. Res. 10 (1977) 529. [31 B.J. West, A.L. Goldberger, G. Rovner and V. Bhargava, Nonlinear dynamics of the heartbeat. I. The AV junction: passive conduit or active oscillator? Physica 17D (1985) 198 (previous paper). [41 J.P. Gollub, T.O. Brunner and B.G. Danly, Science 200 (1978) 48. [51 J.P. Gollub, Lect. Notes Phys. 132 (1980) 162. [61 M.I. Ferrer, J. Am. Med. Assoc. 206 (1968) 635. 171 J.J. Rubenstein, C.L. Schulman, P.M. Yurchak and R.W. DeSanctis, Circulation 46 (1972) 5. I81 B. van der Pol and J. van der Mark, Nature 120 (1927) 363. [91 M.J. Feigenbaum, J. Stat. Phys. 21 (1979) 669. 1101 R.M. May, Nature 261 (1976) 459. [ill A.J. Lichtenberg and M.A. Lieberman, Regular and Stochastic Motion (Springer, New York, 1983). 1121 P.S. Linsay, Phys. Rev. Lett. 19 (1981) 1349. [131 M.R. Guevara, L. Glass and A. Shrier, Science 214 (1981) 1350. 1141 A.L. Ritzenberg, D.R. Adam and R.J. Cohen, Nature 307 (1984) 159. [151 M.L. Cartwright and J.E. Littlewood, J. Lond. Math. Sot. 20 (1945) 180. 1161 J. Guckenheimer, Physica 1D (1980) 227. 1171 M. Levy, Lect. Notes Math. 819 (1980). 1181 J.E. Flaherty and F.C. Hoppensteadt, Stud. App. Math. 58 (1978) 5. 1191 W.L. Keith and R.H. Rand, J. Math. Biol. 20 (1984) 133. 1201 A.L. Goldberger, R. Shabetai, V. Bhargava, B.J. West and A.J. Mandell, Am. Heart J. 107 (1984) 1297.
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[21] K.M. Rosen, H.S. Loeb, M.Z. Sirmo, S.H. Rahimtoola and
[22] [23] [24] [25] [26]
R.M. Gunnar, Circulation 43 (1971) 836. O.S. Narula, Circulation 44 (1971) 1096. D.G. Wyse, M.H. McAnulty, S.H. Rahimtoola and E.S. Murphy, Circulation 60 (1979) 413. J.M. Smith and R.J. Cohen, Proc. Natl, Acad. USA 81 (1984) 233. A.L. Goldberger, V. Bhargava, B.J. West and A.J. Mandell, Clin. Res. 32 (1984) 169A. R.E. Ideker, C.J. Klein, L. Harrison, W.H. Smith, J. Kassell, K.A. Reimer, A.G. Wallace and J.T. Gallagher, Circulation 63 (1981) 1371.
[27] E.T. Angelakos and G.M. Shepherd, Circ. Res. 5 (1957) 657. (281 E.J. Battersby, Circ. Res. 17 (1965) 296. [29] J.N. Herbschleb, R.M. Heethaar, I. van der Tweel, A.N.E. Zimmerman and F.L. Meijler, Computers in Cardiology (IEEE Computer Society, Long Beach, 1979) p. 49. [30] M. Kobayashi and T. Musha, IEEE Trans. Biomed. Res. 29 (1982) 456. [31] A.J. Mandell, P.V. Russo and S. Knapp, in: Evolution of Order and Chaos in Physics, Chemistry, and Biology, H. Haken, ed. (Springer, New York, 1982) pp. 270-286.
NONLINEAR DYNAMICS OF THE HEARTBEAT II. SUBHARMONIC BIFURCATIONS OF THE CARDIAC INTERBEAT INTERVAL IN SINUS NODE DISEASE
Ary L. GOLDBERGER, School of Medicine,
University of California, San Diego, La Jolla, California, USA
Valmik BHARGAVA, schoolof Medicine, University
of California, San Diego, La Jolla, Catifornia, USA
Bruce J. WEST Center for Studies of Nonlinear L.a Jolla, California, USA
Dynamics,
La Jolla
Institute
(a@liated
with the University
of Caii/ornia,
San
Diego),
and
Arnold J. MANDELL? School of Medicine,
University of California,
San Diego, Lu Jolla, California, USA
Received 30 November 1984 Revised 16 April 1985
Changing the coupling of electronic relaxation oscillators may be associated with the emergence of complex periodic behavior. The electrocardiographic record of a patient with the “sick sinus syndrome” demonstrated periodic behavior including subharmonic bifurcations in an attractor of his interbeat interval. Such nonlinear dynamics which may emerge from alterations in the coupling of oscillating pacemakers are not predicted by traditional models in cardiac electrophysiology. An understanding of the nonlinear behavior of physical and mathematical systems may generalize to pathophysiological processes.
1. Introduction Under physiologic conditions, the sino-atrial (SA) node, the normal cardiac pacemaker, and the atrio-ventricular (AV) junction, a subsidiary pacemaker, can be modeled as a pair of coupled relaxation oscillators [l-3]. The faster intrinsic firing frequency of the SA node appears to entrain the slower subsidiary pacemaker which results in $Address for correspondence and reprints: Ary L. Goldberger, M.D., Harvard Medical School, Cardiovascular Division, Beth Israel Hospital, 330 Brookline Ave, Boston, MA 02215. USA.
1 : 1 phase-locking of these oscillators [3]. Studies of model systems have indicated that perturbation of such a network of entrained nonlinear oscillators may alter the coupling (interactions) such that a new cooperative, dynamic state emerges. For example, Gollub et al. [4, 5) observed a variety of complex periodic states in a system of variously coupled tunnel diode relaxation oscillators. Analogous alterations in the interaction of relaxation oscillators might occur in the cardiac electrophysiologic system, possibly induced by pathology in SA, atrial, and AV junctional pacemaker activity. One example of this kind of global alteration in
0167-2789/85/$03.30 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
208
A. L. Goldberger et al./ Nonlinear cardiac oscillators II
the cardiac conduction system is the so-called “sick sinus syndrome” [6, 71 which may reflect a wide variety of structural and metabolic disturbances. We hypothesize that “pathologic” coupling of pacemaker relaxation oscillators in patients with the sick sinus syndrome will be associated with periodic behavior similar to that observed in a variety of physical systems. As early as 1928 van der Pol and van der Mark [l] suggested the use of coupled relaxation oscillators as a model for the heart and built a working analog circuit to demonstrate their ideas. In other studies involving relaxation oscillators [8], they established experimentally that if a small amplitude electromotive force (emf) of frequency f is applied to an electrical relaxation oscillator of free-running frequency f,,, then if f is not too different from fO, i.e. If-&] ef, the oscillator will respond at the frequency f. As they varied the intrinsic frequency of the oscillator ( fa) they found that the system continued to vibrate with the period of the impressed emf (f). This is quite similar to the response of the AV junction to the “applied emf” of the SA node. In addition, they observed that it was also relatively easy to entrain the system to a subharmonic of the impressed frequency f, i.e. the fundamental frequency of the “entrained” oscillation in the system was observed to be f/n, n being an integer up to 100 or 200. They referred to this phenomenon as frequency demultiplication, a term that has been supplanted by subharmonic bifurcation [9]. Mathematically, this phenomenon is understood in terms of an originally stable attractor (orbit) becoming unstable and bifurcating due to the variation in the intrinsic frequency (control parameter). The first bifurcation leads from an orbit of period 2?r/f to one of twice that period, i.e. 2(2a/f) or of frequency f/2. Further variation of f0 may induce a second bifurcation to a period 4(27r/f) or frequency f/4, etc. There has been a resurgence of interest in processes associated with this type of bifurcational behavior since their richness in structure has been shown to emerge from simple discrete mappings [9, lo] having universal properties. In addition to
these mathematical models, period-doubling behavior has been uncovered in a variety of physical systems [ll, 121. A central question, bearing on the problem of sudden cardiac death, is whether these physical-dynamical models can apply to physiological oscillators [13, 141 as originally proposed by van der Pol and van der Mark. Their nonlinear concept of the cardiac conduction system lay fallow for many years in part because it was not apparent how this model would lead to practical consequences not anticipated by traditional electrophysiology. A stringent test of the relevance of the nonlinear viewpoint is how well it explains spontaneous pathologic variations in the normal beating of the heart. It is in response to such perturbations that the distinguishing characteristics between linear and nonlinear systems become particularly evident. Derangements in cardiac electrophysiology seem well suited to examination of these issues since the output of the system can be readily assayed with surface electrocardiography. This report of a patient with conduction system disease is the first quantitative, clinical description of emergent periodicity and subharmonic bifurcations in an attractor of the cardiac interbeat interval.
2. Case study A 60 year old man without cardiac symptoms or evidence of myocardial dysfunction underwent 24-hour ambulatory electrocardiographic monitoring because of a very slow heart rate. Analysis of the recording confirmed the clinical impression of severe conduction system disease (“sick sinus syndrome”) marked by periods of inappropriate sinus bradycardia, prolonged (> 4 s) sinus pauses, and atria1 or AV junctional extrasystoles.
3. Data analysis A continuous section of the recording encompassing 2300 consecutive beats which suggested
A.L. Goldberger et al./ Nonlinear cardiac oscillators II
209
2.8 r\ 0
2.4
a u ”
2.0
_I < >
1 .8
vz W t-
1.2
z n -0.8 w
I fY
0.4
C
4
D
A’
C’
, 50
100
150
HEART
200
250
300
BEATS
Fig. 1. Representative moving-frame heartbeat series from patient with “sick sinus syndrome,” encompassing 342 consecutive data points ( = 500 s). Periodic state (A) is followed by subharmonic bifurcations (B, C, D) and return to relatively aperiodic steady state (S). Similar pattern is seen throughout the record with reappearance of periodic bursts (A’, C’). Note reverse bifurcation sequence in right-hand section (arrow) in which period-2 (A’) follows longer cycle length oscillation.
the greatest variability in heart rate with the patient in bed were selected for more detailed analysis. Interbeat intervals were measured with an x, y digitizer interfaced to a Tektronix 4051 computer. Because of the relatively brief duration (instability) of the periodic episodes, analysis of the data at 8-beat intervals was performed. First, as shown in fig. 1, the data were displayed as a moving-frame heartbeat series obtained by plotting the interbeat (R-R) interval versus consecutive heartbeat number. Second, graphical representation of the periodic trajectory of the interbeat interval was provided by phase plane mapping (AR-R vs. R-R) for selected periodic sequences.
4. Results Fig. 1, a moving-frame heartbeat series from a representative portion of the data, demonstrates spontaneous shifts in the dynamics of the interbeat interval (R-R) through quasi-stable steady states (S, corresponding to either an AV junctional escape rhythm or sinus rhythm), bifurcation to periodicity (A), subharmonic bifurcation to longer wavelengths (B, C, D), and return to relatively aperiodic steady state. Fig. 2 shows the actual electrocardiographic records, corresponding to data sets A-D and S. Fig. 3, a graph of the phase plane, maps the interbeat interval attractor at a steady state (S) and after bifurcation to period-2 (sequence A), the
A. L. Goldherger et ul./ Nonlineur curdiuc oscillators II
210 PERIOD 2 latrial bigeminVl
PERIOD 4
PEmm4 I
C’J-
c,
PERIDD 8
D-J-J-J-
I
I
1.
I^
I
A-
STEADY STATE
S-I
Fig. 2. Electrocardiographic records corresponding to portions of data identified in fig. 1. Period-2 corresponds to atrial bigeminy in which two supraventricular (i.e. sinus, atrial, or AV junctional) beats are clearly coupled together. Period-4 and period-8 sequences correspond to more complex patterns of supraventricular beats. Bottom panel (S), the steady state rhythm corresponds either to AV junctional escape rhythm (with no apparent P waves) or to sinus rhythm. Record has been retouched for clarity.
I 4 4 8.8
first subharmonic bifurcation to period-4 and an apparent second subharmonic bifurcation to period-g. The bifurcation to period-2, is represented by beat-to-beat oscillation of the R-R interval between low (L) and high (H) values (see sequences A and A’ in fig. 1) and corresponds to the well known clinical arrhythmia referred to as atria1 bigeminy (fig. 2, Panel A). Thus, over an g-beat cycle, this periodic pattern will yield the sequence L-H-L-H-L-H-L-H. For example, in fig. 1 (sections A and A’), the R-R interval oscillates between relatively low values (c 1.0 s) and relatively high values (> 2 s) on a beat-to-beat basis. The period-4 cycles repeat twice in an g-beat interval. Sequences B and C in figs. l-3 show two different period-4 cycles. Sequence B corresponds to the interbeat interval pattern: L-L-H-H-L-L-H-H, while sequence C corresponds to H-H-H-L-H-H-H-L. Over the entire 2300-beat data set, the period-4 sequence in
S
,.2
Lpi 2.0
2.4
2.8
3.2
3.6
4.4
A. L. Goldberger et al./ Nonlinear cardiac oscillators II
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Fig. 3. Phase-plane graph (AR-R/R-R) with A, B, C, D, and S indicating the identical portions of data shown in figs. 1 and 2. Relatively aperiodic steady states (AV junctional rhythm or sinus rhythm) were interrupted by bifurcations to period-2 (A), a subharmonic bifurcation to period-4 (B and C), and an apparent second subharmonic bifurcation to period-8 (D). Period-4 in C includes two points close together near the abscissa.
panel B was noted 23 times ( p < 0.001, &i-square test) and the period-4 sequence in panel C was noted three times ( p < 0.001). A period-8 pattern (sequence D of figs. l-3) appeared to be present as well. However, this cycle did not ever repeat itself in 16 consecutive beats. Therefore, the identification of period-8 remains tentative due to the relative instability of these apparent cardiac bifurcation patterns. Furthermore, it was noted that period-2 was not invariably followed by period-4, and that period-4 sometimes appeared without a preceding burst of period-2. Finally, as shown in the right side of fig. 1, reverse bifurcation sequences also occurred. In this case, period-2 (sequence A’) emerged immediately following some longer cycle oscillations in R-R interval.
5. Discussion These data are consistent with the hypothesis that under pathologic conditions the cardiac electrophysiologic system behaves as if there were nonlinear coupling of multiple oscillatory pacemakers. Such nonlinear interaction between relaxation oscillators as seen in coupled tunnel diodes [3-51 can lead to the emergence of longer periods than those intrinsic to any participant and
represents the kind of subharmonic solutions obtained mathematically with the forced van der Pol equation [15-181, the earliest model of the heartbeat [l]. In a model of the cardiac conduction system consisting of two coupled anharmonic oscillators (corresponding to the SA node and the AV junction) we demonstrated a bifurcation from 1 : 1 phase locking (corresponding to sinus rhythm) to a 2 : 1 pattern which occurred when the SA oscillator was driven over a critical range of frequencies [3]. This type of phenomenon mimics the periodicity which may appear when the right atrium is driven at increasing rates with an external pacemaker. In a general mathematical model of two coupled van der Pol oscillators, Keith and Rand [19] showed that the transition from 1: 1 to 2 : 1 phase entrainment can also result from a change in the magnitude of the coupling or from a change in the ratio of the natural frequencies 01/w2. In a possibly analogous way, period-doubling behavior has been observed in the spontaneous rhythmic activity of aggregates of chick embryo heart cells perturbed by injection of current pulses [13] and in the intact canine heart stressed with norepinephrine [14]. Cardiac tamponade, a pathologic condition in which fluid accumulates in the pericardial space and compresses the heart, is associated with a subharmonic bifurcation in the
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frequency of cardiac mechanical oscillation as the heart swings back and forth in the pericardial fluid
A
;
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Our data indicate that perturbation of the intact human electrophysiologic system may be also associated with the emergence of periodic behavior of sinus node and ectopic (non-sinus) pacemakers, manifested by atria1 bigeminy (period-2) or more complex, quasi-periodic rhythms (period-4, period8). The clinical term “sick sinus syndrome” is a misnomer since it suggests impairment of a single physiologic element. Phase plane mapping of the electrocardiogram in this case (fig. 3) is more suggestive of a new pathophysiologic mechanism emerging from the nonlinear interactions of multiple perturbed pacemakers. The graph of the restricted time series (fig. 1) suggests that the overall heartbeat time trace is that of an attractor undergoing bifurcations (and reverse bifurcations) due to spontaneous changes in physiologic parameters modulating the interaction of the SA node and subsidiary pacemakers. The global nature of this perturbation is supported by electrophysiologic studies in patients with the sick sinus syndrome documenting a high prevalence of conduction abnormalities involving the AV junction as well as the His-Purkinje system [21-231, in addition to the SA node. Cardiac perturbations, therefore, may be associated with the appearance of bifurcations which are not predicted by traditional models. On the basis of conventional analysis, the electrocardiographic records in fig. 2 would be interpreted as showing depressed sinus node automaticity or conduction with frequent ectopic beat activity due to AV junctional or atria1 automaticity and possibly reentry. The pattern changes seen in the sequential panels of fig. 2 can be explained by implicating a number of- such different mechanisms occurring at different times and at different local sites. This level of analysis, however, does not account for the period-doubling sequences apparent in the long time trace in fig. 1 and the phase plane maps in fig. 3. Bifurcation behavior of this kind is most consistent with perturbation of a system of coupled
2.6 2
24
2
2.8
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Fig. 4. Moving-frame heartbeat series from another patient (an 83 year old man) during cardiopulmonary resuscitation (CPR) after cardiac arrest. Periodicity and visually obvious bifurcations (corresponding to multiple ectopic supraventrkular and ventricular beats) are noted at the beginning of the record, which was interrupted during resuscitation, after which a return to a relatively aperiodic state (sinus rhythm) occurred.
nonlinear oscillators. The precise factor(s) (e.g., neural, metabolic) modulating the global interaction of these nonlinear pacemakers cannot be identified in the present case. From a clinical viewpoint, patients with sick sinus syndrome commonly develop atria1 fibrillation [7] which may represent a new temporal and spatial bifurcation in the electrodynamics of the heartbeat. The suggestion, however, that fibrillation represents cardi,ac chaos [24] emerging at the end-point of a series of subharmonic bifurcations is not supported in the present case. In this patient with sick sinus syndrome, atria1 fibrillation did not appear after any of the bifurcation’ sequences. Furthermore, recent [25] as well as previous [26-291 data based on epicardial mapping studies and spectral analysis support the contention that fibrillation may be a relatively periodic, not a chaotic process, associated with some type(s) of organized nonlinear traveling waves. The transition to fibrillatory activity of the atria or ventricles, therefore, appears to represent a further bifurcation to pathologic periodicity rather than to chaos [25]. The relevance of bifurcational behavior to the problem of sudden cardiac death is suggested in
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band stability [31] associated with l/f-like scaling, to periodic, pathologic states with the capacity for further bifurcations to subharmonic, quasi-periodic, and perhaps in the case of flutter [l] and fibrillation [25] to travelling wave solutions.
Im Fig. 5. Representative electrocardiographic record from data set in fig. 4. During cardiopulmonary resuscitation (CPR) multiple ectopic (non-sinus) beats were seen (top panel) followed by resumption of sinus rhythm. Deflections marked with x in top panel are artifacts caused by external cardiac compressions.
Acknowledgements
This work was supported in part grants from the W.M. Keck Foundation, the Academic Senate of the University of California, San Diego, and the La Jolla Institute.
fig. 4, a moving-frame heartbeat series from another
patient, in this case during the course of a cardiopulmonary arrest. Evidence for emergent periodicity and subharmonic bifurcations is present at the beginning of the record at which time the electrocardiogram (fig. 5) showed a complex combination of ventricular and supraventricular extrasystoles. A return to a relatively aperiodic state (sinus rhythm) was observed after resuscitation. Although the brevity of this record (reflecting the patient’s clinical instability) precludes detailed analysis, the interbeat interval graph in fig. 4 is reminiscent of the dynamics in fig. 1. In both cases, the appearance of frequent ectopic activity is associated with oscillation in R-R intervals as well as the suggestion of additional frequency bifurcations. These periodicities might not be discerned by simple inspection of the electrocardiographic records in figs. 2 and 5 which suggest a highly erratic sequence of depolarization pulses. Finally, the marked periodicity of the heartbeat under a variety of pathologic conditions contrasts with its physiological dynamics. Fourier analysis of interbeat interval variability in healthy subjects has demonstrated a broadband l/f-like spectral pattern with an additional spike noted at the breathing frequency [30]. We have proposed, therefore, that the emergence of certain arrhythmias may be viewed as a bifurcation from normal sinus rhythm, a steady state with broad-
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