Chaos and chaos control in biology.

J N Weiss, … , M L Spano, W L Ditto

J Clin Invest. 1994;93(4):1355-1360. https://doi.org/10.1172/JCI117111.

Research Article

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Perspectives

Chaos and Chaos Control in BiologyJames N. Weiss, * Alan Garfinkel, * Mark L. Spano,t and William L. DittoI*Cardiovascular Research Laboratory and the Department of Medicine, UCLASchool of Medicine, Los Angeles, California 90024;tNaval Surface Warfare Center, Silver Springs, Maryland 20903; and Department of Physics, Georgia Institute of Technology,Atlanta, Georgia 30332

IntroductionChaos, in its mathematical sense, refers to irregular behaviorthat appears to be random, but is not. The recognition that anirregular behavior is chaotic, rather than random, signifies thata set of precise rules, rather than chance, governs the irregularbehavior of the system. Therefore, if the system is sufficientlywell understood, the irregular behavior can be predicted, elimi-nated, or controlled. Chaos theory has been applied very suc-cessfully to a wide variety of phenomena exhibiting irregularbehavior, ranging from astrophysics to quantum mechanics,chemistry, biology and medicine, social sciences, and even thestock market (1). In the biomedical field, chaos theory hasbeen used successfully to explain observed phenomena such asthe response of cardiac and neural tissue to pacing stimuli (2-8), fluctuations in leukocyte counts in patients with chronicmyelogenous leukemia (4), variations in renal blood flow inhypertensive versus normal rats (9), and the epidemiology ofmeasles in an urban environment ( 10). A particularly excitingnew area has been the development of a method for controllingchaotic behavior that does not depend on a detailed under-standing of the mechanisms producing chaos in the system( 11). In this Perspective, we illustrate how this new techniquehas been applied to a biological problem by describing ourresults applying chaos control to a chaotic form of ventriculartachycardia induced by the drug ouabain in isolated rabbit ven-tricular muscle ( 12). Webegin with a general description ofchaos, without assuming that the reader has any mathematicalbackground in chaos theory.

Somegeneral background in chaos theoryAs noted above, chaotic behavior, although irregular, is actu-ally generated by an underlying deterministic, i.e., nonprobabi-listic, process. As an illustration, consider the data set in Fig. 1,which appears to be completely irregular and has no repeatingpattern. Statistical autocorrelation tests applied to these dataare negative. One possibility is that the fluctuations are ran-dom; but it is also possible that an underlying deterministicprocess is governing the irregular fluctuations. By determinis-tic, we mean that the value of the system at the previous point

Address correspondence to Dr. James N. Weiss, Division of Cardiol-ogy, Rm. 3645, MRLBuilding, UCLASchool of Medicine, Los An-geles, CA90024.

Receivedfor publication 12 November 1993.

(x,_- ) has determined the value of the current point (x") ac-cording to some set of rules (i.e., mathematical equations). Anobvious way, then, to look for a deterministic relationship be-tween one point and the next is simply to plot xs vs. x,,- for allthe data points, i.e., point 2 against point 1, point 3 againstpoint 2, etc. Whenthe data in Fig. 1 are plotted in this manner,known as a Poincare plot, a very simple relationship betweenxn and x,,- becomes apparent (Fig. 2). In this example, therelationship is simply the curve described by the equation for aparabola:

Xn = aXnI (1 -Xn-l)-As the successive data points appear in Fig. 1, they do not fillout the parabola in a particular sequence, but instead bounceback and forth between different regions, producing the com-plex irregular behavior. But despite the irregularity in the val-ues of successive points, the relationship between one pointand the next is always very simple, though nonlinear.

The ability of this simple nonlinear equation to producesuch complex behavior, however, is critically dependent on thevalue of the parameter a in eq. 1. This is illustrated in Fig. 3. Ifa has a low value, such as 2, no matter what initial value xO youbegin with, the system comes to an equilibrium after severaliterations of eq. 1 and remains there (Fig. 3 A). If a has asomewhat higher value of 3.2, then regardless of the initialvalue (x0), successive values eventually oscillate between twovalues in an ABABAB. . . pattern (Fig. 3 B). A mathemati-cian calls this period-2 behavior; a cardiologist calls it bige-miny. If a has a higher value still, 3.5, then more complex butstill periodic behavior occurs, such as the ABCDABCD. . .

period-4 (or quadrigeminal) pattern shown in Fig. 3 C. At acritical value of a, slightly greater than 3.8, however, the systembecomes completely irregular and aperiodic, and has no re-peating patterns whatsoever (Fig. 3 D). The value of a, there-fore, determines whether the system behaves in a stable peri-odic fashion, or in an irregular chaotic fashion.

The chaotic behavior produced by eq. 1 can be attributed toa defining feature of all chaotic systems, their extreme sensitiv-ity to initial conditions. A famous example is the "butterflyeffect," described by meteorologist Edward Lorenz ( 13 ). Usinga relatively simple set of nonlinear differential equations tosimulate atmospheric weather changes, he assigned a set of ini-tial conditions and computed the subsequent behavior of thesystem. The system evolved over time in an irregular chaoticpattern. If he used a slightly different value for the initial condi-tions of the same equations, by changing one part in a million,the system initially behaved in a similar manner; however,after time, it started to diverge and evolved a completely differ-ent pattern. Fig. 4 illustrates this phenomenon using the data inFig. 1, obtained from eq. 1 with a = 4.0. Whenthe initial value

Chaos in Biology 1355

J. Clin. Invest.©The American Society for Clinical Investigation, Inc.0021-9738/94/04/1355/06 $2.00Volume 93, April 1994, 1355-1360

casts falls off dramatically as the forecast period increases, andit applies equally well to any physical or biological system thatexhibits chaotic behavior.

The appreciation that a simple equation, such as eq. 1, canproduce extremely irregular behavior has profound implica-tions: it suggests that despite the very complex behavior that istypical of most physical and biological phenomena, they mayyet be governed by a simple set of rules. Whereas previouslymany of these phenomena seemed too hopelessly complex to

l lsolve by traditional analytic methods, chaos theory offers a0 20 40 60 80 100 fresh approach. First, identify the underlying equations de-

n scribing a system; next, locate a critical parameter equivalent toa in eq. 1; then, modulate the value of the critical parameter by

A sequential data set (x~) showing a very irregular (non- some intervention to eliminate chaos and restore periodic be-havior to the system. Of course, this approach assumes that thesystem is simple enough to develop an accurate mathe-

as changed from 0.2000000 (solid line) to 0.2000001 matical model, and that the model contains a critical parame-line), the curves remained virtually superimposed for ter that can be manipulated in a practicalfashion. In reality, a20 iterations of eq. 1, but then rapidly diverged and physical or biological system may be too complex to develop acompletely different behavior. This extreme sensitivity sufficiently accurate model, or the critical parameters deter-

Ll conditions, in which small differences diverge expo- mining its behavior may not be accessible or easily manipu-y, is a feature that characterizes all chaotic systems. (A lated. Recently, however, a new technique has been developedin which initial differences diverge exponentially rather that focuses on controlling, rather than eliminating, chaosearly is a nonlinear system, and for this reason, chaos ( 11 ). This method does not require a detailed knowledge of theis also called nonlinear dynamics.) The "butterfly ef- system, but only the ability to observe the chaotic behavior inso-named because one can imagine that even a butterfly real time and to apply brief small perturbations to the system.g its wings could create enough of a disturbance in the The method has now been applied successfully to several physi-onditions of a chaotic weather system to cause a com- cal and chemical systems ( 14-17). It also has been speculateddifferent weather pattern to evolve over time, perhaps that chaos control may be a physiologic mechanism of infor-ing a hurricane that would not have occurred otherwise. mation processing by the brain ( 18 ). Weadapted chaos control-orollary to this property is that the accuracy to which theory for its first biological application, the chaotic cardiacial conditions are defined determines how accurately arrhythmia described below.

the future behavior of a chaotic system can be predicted, evenwhen the exact equations governing the system are known.Since computers can perform calculations only to a finite num-ber of decimal places, the ability to predict future behavior of achaotic system is always limited from a practical standpoint.This is an important reason why the accuracy of weather fore-

0.5

Xn-1

Figure 2. A Poincar6 plot of the data from Fig. 1, in which the currentvalue of x (x") is plotted against the previous value (x.- I), revealinga simple underlying relationship.

Applying chaos theory to a cardiac arrhythmiaWhentwo or more oscillators are coupled together so that theirfeedback influences each other, chaotic behavior often occurs.In heart, intracellular Ca is closely regulated by a number ofcoupled processes that cyclically augment and decrease intra-cellular Ca, analogous to a system of coupled oscillators. Fur-thermore, cyclical fluctuations in intracellular Ca are a cause ofafterdepolarizations and triggered activity in heart, a well-known arrhythmogenic mechanism. We therefore reasonedthat intracellular Ca overload in heart might produce irregularbeating patterns due to chaos. To test this idea, we exposed anintact cardiac muscle preparation, consisting of the interven-tricular septum of a rabbit heart, isolated and arterially per-fused through the left coronary artery (Fig. 5), to a toxic con-centration of the cardiac glycoside ouabain (± epinephrine) toinduce intracellular Ca overload. A monophasic action poten-tial was recorded continuously, digitized by a computer, andanalyzed on-line to calculate the interbeat intervals (I") be-tween successive beats. As the ouabain took effect, the heartfirst started beating on its own at a fast but regular rate (Fig. 6A). Whenwe plotted the current interbeat interval (I") againstthe previous interbeat interval (I"_) we obtained a singlepoint on the line of identity (where In = I since all theintervals were the same (Fig. 6 B). Subsequently, the heartbegan beating in a bigeminal or period-2 pattern, with a longinterval followed by a short interval in a repeating ABABAB

. . . pattern. The Poincare plot now showed two groups ofpoints on either side of the line of identity, corresponding to thelong interval followed by a short interval, and the short intervalfollowed by a long interval, respectively. Typically, the arrhyth-

1356 J. N. Weiss, A. Garfinkel, M. L. Spano, and WL. Ditto

1.0

X 0.5

0.c

Figure 1.periodic)

of xO w;(dashedthe firstevolvedto initianentiallsystem ithan lintheory ifect" isIflappinginitial cpletely (

generatiA key cthe init

1.0

cx 0.5

0.00.0

A a=2.0

0 10 20 30 40 50n

3 a=3.2r x

C1.0

x 0.5

0.0

I:

1 .C

x 0.5

0.C

0 10 20 30 40 50n

a=3.5

0 10 2 3 I

0 1 0 20 30n

a=4.0

0 10 20 30n

40 50

Figure 3. The behavior of eq. I with differ-_ ent values of a. (A) a = 2.0; (B) 3.2; (C)

40 50 3.5; and (D) 4.0 (the same as in Fig. 1).See text for details.

mia spontaneously developed higher order periods, such as

quadrigeminy (period-4), the ABCDABCD. . . patternshown in Fig. 6 E, with the corresponding Poincare plot illus-trated in Fig. 6 F. In 85% of hearts, the arrhythmia eventu-

ally became completely irregular, with no repeating patternwhatsoever (Fig. 6 G). The Poincare plot no longer showed a

discrete set of points, but instead a cloud of points (Fig. 6 H).However, the cloud of points was not a diffuse cloud, as wouldexpected if the interbeat intervals during the irregular arrhyth-mia were occurring in a random pattern. Instead, there wereareas that were much more highly populated than other areas.This is the classic hallmark of chaos. The rough structure out-lined by the densely populated regions is known as a strangeattractor, since the points are preferentially attracted to theseregions.

The observation that the ouabain-induced ventriculartachycardia was chaotic is generally consistent with the ideathat intracellular Ca level is a critical parameter, analogous tothe parameter a in eq. 1, that can push the heart from periodicinto chaotic beating. However, our understanding of the de-tailed mechanisms by which chaos developed in this setting is

1.0

X 0.5

limited by the lack of an exact mathematical model that canaccount for all of the complexities of intracellular Ca regulationin the biological preparation. Aside from avoiding ouabain tox-icity, we have no real insight into how to prevent or eliminatethe chaotic arrhythmia by manipulating a critical parameter of

7o -TPPC

o ok 1UJ IifI W'Vl'J U Hf 1 Figure S. Schematic diagram of the isolated arterially perfused rabbit, interventricular septal preparation. The triangular-shaped septum is

0 20 40 60 80 100 mounted in a chamber and arterially perfused through a perfusionn cannula (PC) at 370C in a nitrogen-filled (N2) atmosphere. Temper-

ature is continuously monitored with a probe (TP), and the prepara-Figure 4. Illustration of extreme sensitivity to initial conditions. Eq. 1, tion is stimulated via stimulating electrodes (SE) connected to an

with a = 4.0, was iterated 100 times, starting with an initial value electronic pacemaker. A monophasic action potential catheter (MAP)(xO) of 0.2000000 (solid line) or 0.2000001 (dashed line). The curves connected to a preamplifier is used to monitor electrical activity, andbegin to diverge markedly after 20 iterations. contraction is recorded with a tension transducer( TT) tied to the apex.

Chaos in Biology 1357

1.0

cx 0.5

0.0

kE

1.0

X 0.5

0.0

I

I

iII

1t (a)

1.0c-I0.I10.6

0103.4 - --

To (s)

0.30 0.35I.1 (a)

Figure 6. Recordings of monophasic action potentials (MAPs) (A, C,E, and G) and the respective Poincar6 plots of interbeat intervals (B,D, F, and H) at various stages of ventricular tachycardia induced byouabain (± epinephrine) in typical rabbit septal preparations. Seetext for details. Note that in the Poincar6 map of the final stage (H),the points form an extended structure that is not space filling (i.e.,not random), consistent with chaos. (Reprinted from reference 12,with permission.)

the system. This is where the concept of controlling, as opposedto eliminating, chaos is useful.

To understand how to control chaos, it is important to ex-amine the Poincare plot more closely (Fig. 7 A). Note theregion where the attractor crosses the line of identity (nearcoordinates 0.30, 0.30). A point on this line means that thecurrent interbeat interval is the same as the previous interbeatinterval, which defines an equilibrium point. The equilibriumpoint is unstable, because when a point falls very close to thisequilibrium point, it quickly wanders away. However, if oneclosely observes the sequence by which points approach anddepart from this unstable equilibrium point, one finds that theapproach consistently occurs along a characteristic direction(called the stable manifold, illustrated approximately by points

Iu., ( )

Figure 7. (A) The position of the unstable equilibrium point, stable,and unstable (linearized) manifolds for the Poincar6 plot in Fig. 6H. The numbers refer to successive pairs of interbeat intervals, illus-trating the characteristic approach and departure from the unstableequilibrium point, as described in the text. The heavy arrows are lin-ear approximations of the directions of the stable and unstable mani-folds in the vicinity of the unstable equilibrium point. (B) Schematicof our method of chaos control, using the features of the unstableequilibrium point (intersection of diagonal lines) and its associatedstable and unstable manifolds. See text for details. (Reprinted fromreference 12, with permission.)

163 and 164). Similarly, departure from the unstable equilib-rium point also consistently occurs along a different direction(called the unstable manifold, illustrated by points 164, 165,166, and 167, which alternately flip across the line of identity,each time landing further away from the unstable equilibriumpoint). This structure of an unstable equilibrium point withassociated stable and unstable manifolds is known mathemati-cally as a saddle point, and the analogy is intuitive. Imaginebalancing a ball on a saddle. If placed exactly at the center ofthe saddle, the ball will remain there (in unstable equilibrium)until any small perturbation, such as a whiff of air, displaces theball to one side or the other. The ball will then accelerate awayfrom the center, but always in the transverse direction (towardsone of the stirrups) and never towards the front or the backalong the centerline of the saddle. The transverse direction istherefore equivalent to an unstable manifold. Conversely, if theball is placed on the centerline towards the front or back of thesaddle, it will roll towards the center of the saddle. The center-line of the saddle is therefore equivalent to a stable manifold.Mathematically, all chaotic systems exhibit saddle points or,more generally, periodic saddle cycles.

To control chaos, one takes advantage of this saddle pointstructure. Continuing with the ball and saddle analogy, if onewishes to keep the ball in the center of the saddle (i.e., near theunstable equilibrium point), one could either move the saddleto compensate every time the ball shifted off-center, or give theball a nudge to move it back towards the centerline of thesaddle. With either perturbation, it is not necessary to keep theball near the center of saddle, but only near the centerline of thesaddle, since the ball will always roll closer to the center of thesaddle from any position on the centerline (i.e., the stable man-ifold).

Fig. 7 B illustrates schematically how we used this methodto control the chaotic ouabain-induced arrhythmia in the rab-bit heart. The computer program computed on-line each inter-beat interval from the monophasic action potential recordings.By keeping track of the sequence of points as they were plottedon a Poincare plot, the program estimated the location of anunstable equilibrium point. By watching how successive points

1358 J. N. Weiss, A. Garfinkel, M. L. Spano, and W. L. Ditto

moved towards and away from this point, the program thenestimated positions of the stable and unstable manifolds by alinear regression method. After this learning phase was com-pleted, the computer waited for a pair of interbeat intervals tofall near the unstable equilibrium point, such as the point a inFig. 7 B. The point a represents a long interbeat interval (thecoordinate on the I.-, axis) followed by a shorter interbeatinterval (the coordinate on the I,, axis). By construction of thePoincare plot, the short interbeat interval becomes the I,,- axiscoordinate for the next point (if), so ,3 must fall somewhere onthe line defined by the vertical arrow in Fig. 7 B. Since a waslocated very close to the unstable manifold, (3 will tend to bepropelled further away from the unstable equilibrium pointnear the intersection of the vertical arrow and the unstablemanifold. That is, the saddle property of the unstable equilib-rium point predicts that the point # will consist of a short inter-beat interval followed by a long interbeat interval. Wedesignedthe computer program to introduce an electrically stimulatedpremature beat (delivered by an electronic pacemaker) toshorten the predicted long interbeat interval, so that the posi-tion of the next point after a was deflected to the position (3'instead of f3. The point 3' was chosen since it falls near thestable manifold instead of the unstable manifold. Thus, thenatural dynamics of the system will cause the next point y tofall even closer to the unstable equilibrium point. As long as thesubsequent points remain near the stable manifold, they con-tinue to approach closer to the unstable equilibrium point.Every time a point falls near the unstable manifold, anotherpaced beat is introduced to reposition the subsequent pointback close to the stable manifold. In principle, all of the pointson the Poincare plot could be confined within a small region ofthe unstable equilibrium point using this pacing algorithm.That is, all of the interbeat intervals would be nearly identicaland the heart rhythm would be regularized.

The results of applying this pacing algorithm based onchaos control theory to the rabbit ventricular preparation areshown in Fig. 8. As illustrated by the three examples, when thechaos control pacing algorithm was activated during the aperi-odic phase of the ouabain-induced ventricular tachycardia, theirregular chaotic beating pattern was controlled and replacedby periodic beating, typically with a period-3 or -4 pattern.When chaos control pacing was terminated, the arrhythmiareverted to its irregular pattern. During chaos control, onlyevery third or fourth beat was an electrically paced beat, indi-cating that we were not simply overdrive-pacing the heart. Fur-thermore, periodic or random pacing of the preparation at asimilar average rate, as during chaos control pacing, did noteliminate the irregularity of the arrhythmia, and generallyseemed to make the irregularity more marked (Fig. 8 C). Over-all, we were successful in converting aperiodic beating to peri-odic beating in 8 of 11 preparations using this chaos controlpacing algorithm.

Implications for cardiac arrhythmias.These observations support previous studies demonstratingthat the heart is capable of chaotic behavior (2-6). More gener-ally, they demonstrate that chaos control techniques are adapt-able to the biological setting. It is still controversial whetherchaos plays a role in clinically important cardiac arrhythmias.It is likely that the irregular conduction patterns sometimesseen during second degree atrioventricular block and modu-lated parasystole are examples of chaos (6). Of the tachyar-rhythmias, the most likely candidates for chaos are the irregu-

Figure 8. Chaos control of theX0Ao ^ I: .::. - . . ouabain-induced ventricular

tachycardia in the isolated rab-bit heart. Interbeat interval In

0.2 ~is plotted vs. the beat numbern during the chaotic phase ofthe arrhythmia. The chaos

25!l 300 350 400 450 500 550 NOO control pacing algorithm was

applied during the period indi-0.6 , cated, and successfully con-Cl;'c ' verted the aperiodic arrhyth-ONg . mia to a period-3 pattern (AB-0.6 . CABC.. . ) in A, to a period-4

pattern (ABCDABCD.. . ) inw. > . . : ~~~B. and to a period-2 pattern-,o.^*;.. (ABAB.. . ) in CA-C are

___ - ais.,.,fromdifferent hearts. In C, pe-

0.2 *it 'riodic pacing delivered at thesame average rate failed to sta-

cPon . bilize the arrhythmia. See text. PREAC for details. (Reprinted from

200 300 400 500. ..260-sean400 500

reference 12, with permission.)

lar arrhythmias such as multifocal atrial tachycardia, polymor-phic ventricular tachycardia, Torsade de pointes, and atrialand ventricular fibrillation. From a clinical standpoint, atrialand ventricular fibrillation remain the most commonand vex-ing of these arrhythmias. Evidence both for and against thepresence of low-dimensional chaos during fibrillation has beenreported ( 19-24). Weare currently investigating whether hu-man atrial fibrillation shows hallmarks of chaos in patientsundergoing cardiac electrophysiology studies. Our analysis isstill preliminary, but results such as those illustrated in Fig. 9are encouraging. Not only does the Poincare plot of humanatrial fibrillation show evidence of nonrandom structure, butfeatures such as saddle points (unstable equilibrium pointswith associated unstable and stable manifolds) are detectable.If further studies indicate that fibrillation is chaos, and thatstructures such as saddle points needed to implement chaoscontrol are consistently present, then this raises an interestingissue. Can a pacing algorithm based on chaos control theory,perhaps implemented by a "smart pacemaker," be developedas a new therapeutic strategy? Although intriguing, whetherthis possibility can be realized will depend on surmounting anumber of critical obstacles, such as dealing with the spatio-temporal complexity of fibrillation and determining how toterminate the arrhythmia once chaos is controlled.

Chaos in Biology 1359

MAP

500 ms400

200

200

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0 200 400

In-i (Ms)Figure 9. A Poincare plot of interbeat interhuman atrial fibrillation in a patient undergelectrophysiology study. An intracardiac catmonophasic action potentials (MAP) from tiing spontaneous atrial fibrillation (top tracinAplot of interbeat intervals (I._- vs. I") illustrathat is not space filling (i.e., not random), c(Right) The presence of a saddle point (un,with associated stable and unstable manifold7. The numbers refer to successive pairs of i71), demonstrating the approach towards alunstable equilibrium point (at the intersecticharacteristic and consistently observed direare the linear approximations of the directionstable manifolds in the vicinity of the unstat

Implications for biology.Biomedical research has traditionally febehavior of an organ system by breakinual components and studying these inremarkable success of this reductionist E

limitations when one tries to reassemtindividual components to predict behavplex as a biological organ. This is especigactions between the components are nc

tial for chaos exists. Whether chaotic bference between health and disease stldebate in most instances. However, thecal chaos control strategies strengthenstecting chaos in biological systems, sincegy for therapeutic intervention. Anykey physiological parameter, e.g., bloodels, immunological responses, etc., ispotentially amenable to control by intr(timed perturbations based on this tech

ACKNOWLEDGMENTS

Wethank Donald Walter, Ph.D., for his th(manuscript.

This work was funded in part by National Institutes of Healthgrants ROl HL-36729 and ROI HL-44880 (J. N. Weiss), the LaubischFund (J. N. Weiss), the Chizuko Kawata Endowment (J. N. Weiss),the Naval Surface Warfare Center Independent Research Program(M. L. Spano), and the Office of Naval Research (M. L. Spano and W.L. Ditto).

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1360 J. N. Weiss, A. Garfinkel, M. L. Spano, and W. L. Ditto

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