the Normal Heartbeat Chaotic or Homeostatic? Ary L. Goldberger

I I Limits to the usefulness of homeostasis as a guiding physiological principle are revealed by new mechanisms derived from study of nonlinear systems that generate a type of variability called chaos. Loss of complex physiological variability may occur in certain pathological conditions including heart rate dynamics before sudden death and with aging.

A dom inant tenet in physio logy for m .ore t han half a century h as been the principle of homeostasis. The term was coined by W. B. Cannon (2) to describe the observation that physiological systems normally op- erate to reduce variability and to maintain a constancy of internal function. Accordingly, one might ex- pect most any physiological variable, including heart rate, to return to its normal value after it has been per- turbed and to remain steadily at that value until for some reason it is per- turbed again.

This notion of a physiological steady state, however, appears to be complicated by recent observations that indicate a good deal of intrinsic variability in many aspects of healthy function (9). For example, the normal heartbeat is not predict- ably regular. The normal electroen- cephalogram shows erratic brain wave activity. Hormone levels, as- sayed in the serum of healthy indi- viduals, fluctuate in a manner that also seems to violate the constancy required of unperturbed homeo- static systems. Is there a mechanism for this apparently nonhomeostatic variability that characterizes the healthy structure and function of the heart and many other physiological systems? Can one quantitate this complex variability and the way that it changes during disease? Answers

A. L. Goldberger is Associate Professor of Med- icine, Harvard Medical School, and Director of Electrocardiography, Beth Israel Hospital, 330 Brookline Avenue, Boston, MA 0.2215, USA.

0886- 17 14/9 1 $1.50 0 199 1 Int, Union Physiol. Sci./Am. Physic

to these questions may come from recent investigations in fractal math- ematics and chaotic dynamics.

Fractals, chaos, and strange attractors

The interrelated topics of fractals and chaos are central concepts in a relatively new branch of science called nonlinear dynamics (4, 9, 13). A major finding of nonlinear dynam- ics is that chaotic behavior may oc- cur in systems whose equations of motion, parameters, and initial con- ditions are known and are not ran- dom. Such systems are referred to as deterministic (4, 13).

The term fractal is a geometric concept related to chaos (10,lZ). The smooth and regular forms in classi- cal geometry (lines, circles, spheres) have integer dimensions (1, 2, and 3, respectively). Fractals are highly ir- regular objects and, as a result, have noninteger, or fractional, dimen- sions. Consider a fractal line. Unlike a smooth Euclidean line, a fractal line, which has a dimension be- tween 1 and 2, is wrinkly and irreg- ular. Furthermore, if one examines these wrinkles with the low-power lens of a microscope, it becomes ap- parent that there are smaller wrin- kles on the larger ones, and so on.

This internal look-alike property of fractals is referred to as self-simi- larity. The more closely you inspect a fractal, the more detail you see, and the small scale structure is sim- ilar to, though not necessarily iden- tical to, the larger scale form, As a consequence, fractal objects do not have a well-defined length. The

measured length of a fractal line (e.g., a coastline) will vary depending on the size of the ruler used. The smaller the measuring stick, the longer the apparent length, since the smaller ruler will “pick up” the mi- croscopic bends and turns of the small scale structure.

Fractal geometry is widespread in nature: coastlines, clouds, lightning flashes, and winding rivers, to name but a few. Examples of fractal-like anatomies include the vascular sys- tem, the His-Purkinje network (Fig. l), the tracheobronchial tree, as well as the folds of the small bowel and brain (5, 10, 16). A surprising discov- ery in nonlinear dynamics is that these fractal architectures are also exhibited by chaotic processes and describe the geometric structure of so-called strange attractors.

One way to analyze the dynamic patterns of a complex nonlinear sys- tem geometrically is to track the lo- cation of its variables using a so- called phase space or delay map rep- resentation, The state of such a sys- tem is defined by a point in this phase space, the axes of which cor- respond to each of the independent variables of the system (or to the value of any single variable plotted against the value of that same vari- able after some fixed time delay). As time proceeds, this point traces out a curve, called an orbit or trajectory, that describes the system’s evolu- tion. If there is a limiting set to which neighboring trajectories converge after a sufficiently long time, then the system dynamics are described by an attractor. The simplest case is that of the fixed-point attractor, where all the trajectories converge to a single point, a type of homeo- static equilibrium. Another pattern is that of a periodic attractor (limit cycle) in which the trajectories fol- low a regular path corresponding to a process that is cyclic, The name strange attractors was given to those attractors on which the system dy- namics do not converge to a fixed point and or a limit cycle, but instead are chaotic.

An appreciation for the fractal na- ture of the processes that are de- scribed by strange attractors can be obtained most simply by inspecting

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FIGURE 1. His-Purkinje system distributes electrical impulses from atria to ventricles via a fractal-like network with self-similar branchings. [Adapted from Goldberger et al, (5))

the time series of a chaotic process. At first glance, the ragged irregular time series plot of such a process appears random. However, if you now inspect a shorter section of the same time series in greater detail, you find that there is a layer of fluc- tuations visible on this finer time scale. You can dissect this process even further by examining these shorter time scales at greater reso- lution Therefore, just as a fractal line has no characteristic length scale, a chaotic process has no char- acteristic time scale. Remarkably, in chaos the fluctuations (temporal wrinkles) on different time scales are self-similar (fractal).

The concept of noninteger dimen- sionality is an important link be- tween these notions of a geometrical fractal and a chaotic process. Several techniques have been developed for computing a dimension from the output of complex processes. The di- mension of a process can be used to estimate the minimum number of independent variables needed to fully characterize the process. Ap- plication of these dimensional cal- culations to biological\ and physio- logical data sets, in which there may be random noise in addition to de-

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terministic chaos, is currently an area of active research (14).

Nonlinear dynamics and sudden cardiac death

Attempts to apply chaos theory to physiology in general and to the heart in particular began about a decade ago. At first it was widely assumed that chaotic time series were produced by pathological sys- tems and that this new nonlinear theory would be most useful as a way of modeling cardiac arrhyth- mias and especially in understand- ing the dynamics of atria1 or ventric- ular fibrillation. Contrary to initial theories, the weight of current evi- dence indicates that the dynamics of these erratic arrhythmias are not chaotic in the technical sense de- scribed here (7). We have proposed that the most compelling clinical ex- ample of cardiac chaos is paradoxi- cally found in the dynamics of nor- mal sinus rhythm (9, 15).

The heart rate in healthy individ- uals, even those at bedrest, is neither constant nor strictly periodic. In- stead, plots of beat-to-beat heart rate variability demonstrate an erratic pattern of fluctuations in healthy subjects. The time series of the nor-

mal heartbeat, in fact, is quite remi- niscent of the wrinkly type of fractal line described earlier (Fig. 2). Fur- thermore, the wrinkliness of the normal heartbeat time series is seen over many different orders of tem- poral magnitude (hours, minutes, and seconds); there is no character- istic time scale. Spectral analysis of these normal heart rate time series supports their fractal nature, since it demonstrates a type of broad-band spectrum (Fig. 3).

The Fourier spectrum decomposes a complex waveform (such as a time series) into its constituent frequency components. Highly regular proc- esses (those characterized by peri- odic attractors) are represented by a narrow spectrum, one having only one or a few closely spaced fre- quency peaks with relatively little background. In contrast, chaotic processes, which have no character- istic frequency, generate a broad or “noisy” spectrum with a continuum of frequency components. Spectral analysis of normal heart rate varia- bility reveals a type of broad spec- trum suggestive of chaos. Further, phase space mapping of the normal heart rate reveals a strangelike at- tractor rather than the periodic type of attractor that would be seen if a healthy heartbeat were truly regular (Fig. 3).

What is the mechanism of this un- expected healthy chaos of the nor- mal heartbeat? The answer appears to lie in the chaos of the nervous system, since heart rate variability is directly modulated by autonomic in- puts, Parasympathetic stimulation decreases the firing rate of pace- maker cells in the heart’s sinus node. Sympathetic stimulation has an ac- celerative effect. The competing in- fluences of these two branches of the nervous system result in a constant tug of war on the sinus node. The result of this continuous neural buf- feting is the type of chaotic heart rate variability that is recorded in healthy subjects.

An intriguing model for chaotic feedback loops that could generate this type of complex variability is suggested by recent simulations of neural network interactions. A neural network is an interactive group of nerve cells or nerve cell collections. The differential equa- tions governing the cross talk be-

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FIGURE 2. Normal sinus rhythm time series, Heart rate in healthy subjects, even at rest, is not strictly regular but fluctuates in a complex way (bpm, beats/min). Furthermore, there are self- similar fluctuations on multiple different orders of temporal magnitude, a fractal feature of chaotic dynamics (9, 10).

tween these neural cells are nonlin- ear and include time delays. Studies by Freeman (3) have shown how chaos can arise out of a deterministic feedback system in a model of the olfactory system. An analogous in- teraction of the nuclei of the auto- nomic nervous system most likely accounts for chaotic fluctuations of heart rate. The importance of time delays in generating chaotic dynam- ics has also been investigated by Glass and Mackey (4) in nonlinear difference equations.

What are the functional advan- tages of chaotic dynamics? Chaotic systems are by nature variable, and this variability serves as an impor- tant mechanism for adaptability and flexibility. Such plasticity is essen- tial for coping with the exigencies of an unpredictable and changing en- vironment. The absence of a char- acteristic time scale for heart rate fluctuations, for example, broadens the frequency response of the cardi- ovascular system and prevents the system from being locked into one frequency response (mode locking).

Recent evidence from a number of laboratories suggests that the normal dynamics of other physiological sys- tems are also chaotic. Chaos in the nervous system function was men- tioned earlier. Evidence for the existence of chaos in the central nervous system is also supported by dimensional analysis of electroen- cephalographic (EEG) waveforms of healthy individuals (1). Like the nor- mal electrocardiogram, the normal EEG also shows a broad type of spec- trum and a phase space attractor that is quite unlike a limit cycle. Chaos in neuroendocrine function, a con- cept advanced by Rossler and Gotz (15) and others, is suggested by time- series analysis of serum hormone levels in healthy human subjects. Such plots, as noted above, do not display the regularity expected by a classical homeostatic system.

Nonlinear diagnostic indexes

Not surprisingly, a number of pa- thologies are characterized by in- creasingly periodic behavior and a

loss of complex variability. For ex- ample, we and others have analyzed the heart rate patterns that may pre- cede sudden cardiac death (8). The major dynamic finding that emerges from this analysis is the loss of phys- iological heart rate variability seen in patients minutes to months before sudden cardiac death (Fig. 3). In some cases, the loss of normal dy- namics is represented by an overall reduction in beat-to-beat variability and in other cases by highly periodic, relatively low-frequency (0.0~0.04 Hz), sometimes-sustained oscillations that usually start and stop abruptly. In the parlance of non- linear dynamics, these abrupt changes are referred to as bifurca- tions.

Similar heart rate patterns involv- ing a loss of variability and the ap- pearance of low-frequency oscilla- tions have been previously reported in survivors of cardiac arrest, in high-risk patients following heart at- tacks, and in fetal distress syndrome. Current investigations are aimed at establishing the practical prognostic and diagnostic importance of nonlin- ear heart rate patterns in clinical medicine.

In the nervous system, an analo- gous loss of complex variability and emergence of pathological periodic- ities is seen in several maladies in- cluding epilepsy, tremors, and manic-depressive oscillations, This type of nonlinear trait may also gen- eralize to other complex systems in physiology. For example, under nor- mal conditions white blood cell counts in healthy subjects have been reported to fluctuate chaotically from day to day. Of interest, in cer- tain cases of chronic granulocytic leukemia, periodic oscillations in white blood cell counts occur (6). Similarly, epilepsy has been shown to cause a reduction in the dimen- sionality of EEG time series related to that seen in normal individuals (1). Heart rate dynamics with aging are also characterized by a loss of complex variability (11).

Chaos vs. homeostasis

Chaos theory is not completely in- compatible with the classical con- cept of homeostasis. However, there are several fundamental differences between these approaches to under- standing physiological variability

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FIGURE 3. Heart rate dynamics. Normal sinus rhythm in healthy subjects (left) shows complex variability with a broad spectrum and a phase space plot consistent with a strange (chaotic) attractor, Patients with heart disease may show altered dynamics, sometimes with oscillatory sinus rhythm heart rate dynamics (middle) or an overall loss of sinus variability (ri@t). With the oscillatory pattern, spectrum shows a sharp peak, and phase space plot shows a more periodic attractor, with trajectories rotating about a central hub, With the flat pattern, spectrum shows an overall loss of power, and phase space plot is more reminiscent of a fixed-point attractor, [Adapted from Goldberger et al. (9))

(Table 1). Chaotic systems generate variable behavior even in the ab- sence of external stimuli, whereas homeostatic systems should settle down to a constant steady state in the absence of perturbation. There- fore, the type of nonlinear mathe- matical modeling required to simu-

late chaotic systems is quite differ- ent from the linear equations that suffice for systems with one steady state. Furthermore, homeostatic sys- tems might be expected to become more variable as they are destabi- lized.

Chaos theory predicts just the op-

TABLE 1. Homeostasis vs. chaos in physiology

Homeostasis System will settle down to a steady state [constancy) if unperturbed Fluctuations result from external influences Destabilizing factors such as disease or aging are anticipated to decrease order [increase

chaos) Chaos

System does not settle down to constant steady state Fluctuations arise from internal feedback and do not require external perturbation Destabilizing factors such as disease or aging usually decrease the degree of complex

variability (reduce chaos)

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posite: namely, that factors such as disease and aging may decrease the dimensionality or degree of chaos. On the other hand, chaotic systems are not random and, like homeo- static systems, the range of chaotic fluctuations is constrained. Chaotic systems may also be perturbed by external stimuli that further compli- cate their dynamic patterns.

Nonlinear investigation of phys- iological systems is just at its begin- ning. Indeed, physiology may prove one of the richest of laboratories for the study of fractals and chaos, as well as other types of nonlinear dy- namics (9). Nonlinear mathematics could soon become the most power- ful tool for quantitatively describing the apparently nonhomeostatic var- iability of normal physiological dy- namics and the changes that ac- company a variety of diseases. The jargon of nonlinear dynamics, how- ever, may have contributed to some of the initial misunderstanding about the applications of chaos to physiology. Strange attractors, far from unusual, may be dynamic maps of healthy fluctuations in the heart and nervous system observed under most ordinary circumstances. Broad spectra of time series appear to be markers of physiological informa- tion, not “noise.” Finally, determin- istic chaos, contrary to its vernacular connotations, is not a completely random state. Instead, chaos gives rise to fractal structures and com- plex variability that bring an elegant and essential order to physiological self-organization.

This work was supported in part by grants from the National Aeronautics and Space Ad- ministration (NAGZ-514) the National Heart, Lung, and Blood Institute (ROl HL-42172), and the G. Harold and Leila Y. Mathers Char- itable Foundation.

References

1. Babloyantz, A., and A. Destexhe. Low- dimensional chaos in an instance of epi- lepsy. Proc. NatI. Acad. Sci. USA 83: 3513-

3517, 1986.

2. Cannon, W. B. Organization for physiolog- ical homeostasis. Physiol. Rev. 9: 399-431, 1929.

3. Freeman, W. J. The physiology of percep- tion. Sci. Am. 264: 78-85, 1991.

4. Glass, L., and M. C. Mackey. From Clocks to Chaos: The Rhythms of Life. Princeton, NJ: Princeton Univ. Press, 1988.

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5. Goldberger, A. L., V. Bhargava, B. J. West, and A. J. Mandell. On a mechanism of cardiac electrical stability: the fractal hy- pothesis. Biophys. J. 48: 525-528, 1985.

6. Goldberger, A. L., K. Kobalter, and V. Bhargava. l/f-like scaling in normal neu- trophil dynamics: implications for hema- tologic monitoring. IEEE Trans. Biomed. Eng. 33: 874-876,1986.

7. Goldberger, A. L., and D. R. Rigney. Non- linear dynamics at the bedside. In: Theory of Heart, edited by L. Glass and P. Hunter. New York: Springer. In press.

8. Goldberger, A. L., D. R. Rigney, J. Mietus, E. M. Antman, and S. Greenwald. Nonlin- ear dynamics in sudden cardiac death

syndrome: heartrate oscillations and bi- furcations. Experientia Base1 44: 983-987, 1988.

9. Goldberger, A. L., D. R. Rigney, and B. J, West. Chaos and fractals in human phys- iology. Sci. Am. 262: 42-49, 1990.

10. Goldberger, A. L., and B. J. West. Fractals in physiology and medicine. Yale J, Biol. Med. 60: 421-435,1987.

11. Lipsitz, L. A., J. Mietus, G. B. Moody, and A. L. Goldberger. Spectral characteristics of heart rate variability before and during postural tilt. Relations to aging and risk of syncope. Circulation 81: 1803-1810,199O.

12. Mandelbrot, B. B. The Fractal Geometry ofNature. New York: Freeman, 1982.

13. Moon, F. C. Chaotic Vibrations: An Intro- duction for Applied Scientists and Engi- neers. New York: Wiley, 1987.

14. Rigney, D. R., J. E. Mietus, and A. L. Gold- berger. Is sinus rhythm “chaotic”? Meas- urement of Lyapunov exponents (Ab- stract). Circulation 82, Suppl. III: 236, 1990.

15. Rossler, 0. E., and F. Gotz. Chaos in en- docrinology (Abstract). Biophys. J. 25: 216A, 1979.

16. Van Beek, J. H., S. A. Roger, and J. B. Bassingthwaighte. Regional myocardial flow heterogeneity explained with fractal networks. Am. J. Physiol. 257 (Heart Circ. Physiol. 26): H1670-H1680, 1989.

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