The role of nonlinear dynamics in cardiac arrhythmia control

David J. Christini, Ph.D., Kenneth M. Stein, M.D., Steven M.Markowitz, M.D., Suneet Mittal,M.D., David J. Slotwiner, M.D., and Bruce B. Lerman, M.D.

Division of Cardiology, Department of Medicine,Cornell University Medical College,

New York, NY 10021, USA

July 28, 1999

This work was supported in part by a grant from the National Institutes of Health (R01 HL56139).

Address correspondence to:David J. Christini, Ph.D.Division of Cardiology, Starr 463Cornell University Medical College520 E. 70th St.New York, NY 10021phone: (212)746-2240FAX: (212)746-8451email: dchristi@med.cornell.edu

Keywords: Cardiac dynamics, cardiac electrophysiology, arrhythmiacontrol, nonlinear dynamics,chaos control, spiral waves, scroll waves.

Christini et al. 2

Abstract

The field of nonlinear dynamics has made important contributions towards a mechanistic under-

standing of cardiac arrhythmias. In recent years, many of these advancements have been in the

area of arrhythmia control. In this paper, we will review theanalytical, modeling, and experimen-

tal nonlinear dynamical arrhythmia control literature. Wewill focus on stimulation and pharma-

cological techniques that have been developed, and in some cases used in experiments, to control

reentrant rhythms (including spiral and scroll waves) and fibrillation. Although such approaches

currently have practical limitations, they offer hope thatnonlinear dynamical control techniques

will be clinically useful in the coming years.

Christini et al. 3

1 Introduction

In recent years, cardiac electrophysiology has evolved from a discipline that was of interest primar-

ily to physicians and physiologists to one that has caught the attention of physicists, mathemati-

cians, and engineers. Such scientists have come to realize that cardiac dynamics are characterized

by many of the same principles that underlie the physical systems with which they are intimately

familiar. The corresponding influx of new (to cardiology) analyses and techniques has led to many

important contributions. In particular, techniques from the field of nonlinear dynamics have led to

important advancements in the understanding of cardiac electrophysiological dynamics [1–3] and

to computational and experimental success in the control ofarrhythmias [4–15].

In this paper, we will review the advancements in the area of nonlinear dynamical arrhythmia

control. First, we will introduce nonlinear dynamics and chaos. Next, we will discuss chaos

control (also known as model-independent control), a classof control techniques which exploit

the underlying dynamics of nonlinear systems to achieve a desired motion or rhythm. We will

then review recent analytical, computational, and experimental applications of nonlinear dynamics

and chaos control to arrhythmia termination. Specifically,we will discuss investigations which

have analyzed and controlled a wide range of cardiac dynamics (from stable periodic rhythms to

aperiodic fibrillatory-like behavior to spiral waves) and the relevance of these studies to tachycardia

and fibrillation control. We will conclude with an evaluation of the prospects of clinical nonlinear

dynamical arrhythmia control.

2 Nonlinear dynamics and chaos

A dynamical system can be defined as a system that changes withtime [16]. Like most physiolog-

ical systems, the heart is dynamical in nature. Because of the dynamical nature of fibrillation and

tachycardia, the field of nonlinear dynamics has made important contributions towards a mecha-

nistic understanding of cardiac arrhythmias.

Christini et al. 4

For certain nonlinear systems, known as chaotic systems (ormore formally as deterministic

chaos), behavior is aperiodic and long-term prediction is impossible even though the dynamics

are entirely deterministic (i.e, the dynamics of the systemare completely determined from known

inputs and the system’s previous state, with no influence from random inputs). Such seemingly

random behavior occurs indefinitely (i.e., the system neversettles down).

An example of a simple chaotic system is the “logistic map”, which was suggested as a means

of representing the year-to-year population dynamics of a species [17]. The logistic map is given

by xn+1 = rxn(1 � xn), wherexn (xn 2 [0; 1℄) is the current population at yearn, xn+1 is the

predicted population for the next yearn + 1, andr (r 2 [0; 4℄) is the population growth rate. Forr > 3:57, x is chaotic, as shown in Fig. 1 forr = 4:0. Figure 1a showsxn iterated from an

arbitrary initial value (x0 = 0:7). When viewed in this manner, the fluctuations resemble those of

a stochastic (i.e., random) system. However, whenxn is plotted on a first-return map (in whichxn+1 is plotted versusxn; Fig. 1c), all of the points fall on a parabolic curve. When viewed as a

first-return map, a stochastic system will appear as a space-filling scattering of points (a “shotgun”

pattern), a periodic system will appear as a finite number of points (e.g., a system that alternates

between two values will appear as two discrete points on a first-return map), and a chaotic system

will appear as an infinite number of points in a structure (such as a parabola) that depicts the

dependence of the current system value on past system values. The iteration of the logistic map

can be visualized by looking at the annotated segments1; 2; :::; 6 of Fig. 1b in their corresponding

locations as single points on the first-return map of Fig. 1c. Each point in Fig. 1c (known as state

points) represents a particularxn andxn+1, which characterize the state of the system at a givenn.

Iteration of the system can be visualized from the first-return map using the cobweb method [16].

With the cobweb method, a horizontal line is drawn from the present state point to the line of

identity (the diagonal line for whichxn+1 = xn). Next, a vertical line is drawn from the line of

identity to the parabola. The intersection of the parabola and the vertical line marks the location of

the next state point. This method is shown in Fig. 1c for the state points 1 to 6.

Christini et al. 5

The intersection of the parabola with the line of identity marks the location of a special state

point known as the unstable period-1 fixed point��. If the state point equals��, it will stay there

indefinitely becausexn+1 = xn, xn+2 = xn+1, xn+3 = xn+2, etc. Such repetition from one

iteration to the next is termed “period-1”. Similarly, if a system cycles back and forth between

2, 3, or 4 values, it is referred to as period-2, -3, or -4, respectively. �� is referred to as unstable

because any small deviation away from it will cause the statepoint to march away from��. This

can be verified graphically by using the cobweb method to iterate from a state point very close to��.In real-world systems, because of noise and natural unstable dynamics, a system’s state point

can never remain exactly equal to an unstable periodic fixed point. Therefore a real-world chaotic

system cannot settle down to a periodic rhythm without external intervention. Importantly, recent

developments (techniques known as chaos control or model-independent control) have made such

interventions possible.

3 Model-independent control

In recent years, advancements have been made in the understanding of the mechanisms which

underlie cardiac electrophysiological dynamics [18–24].Importantly, together with increased un-

derstanding of underlying dynamical mechanisms comes the possibility of exploiting those mech-

anisms to alter (i.e., control) system behavior. Well-established model-based feedback (i.e., closed

loop) control techniques from the field of control engineering utilize a system’s governing equa-

tions (i.e., an analytical system model) to control the dynamics of a system [25]. Unfortunately,

although many physiological mechanisms are well-understood qualitatively, quantitative relation-

ships between physiological system components are usuallyincomplete. Thus, because accurate

analytical system models cannot be developed for such systems, model-based control techniques

are typically not applicable to most physiological systems. However, for nonlinear dynamical

Christini et al. 6

systems, there is a class of model-independent (i.e., no explicit knowledge of a system’s under-

lying equations is required) feedback control techniques for which a qualitative understanding of

the underlying mechanisms is sufficient. (For a review of model-independent control, see [26].)

Model-independent techniques extract necessary quantitative information (about the functional de-

pendence of the variable to be controlled on a system parameter) from system observations and then

use this information to exploit the system’s inherent dynamics to achieve a desired control result.

Thus, because these techniques are applicable to systems that are essentially “black boxes”, they

are inherently well-suited for the control of physiological systems, for which analytical models are

typically unavailable or incomplete.

In the seminal work in the area of model-independent feedback control, Ott, Grebogi, and

Yorke [27] developed a control technique for chaotic dynamical systems. The goal of model-

independent control of a dynamical system is to eliminate anunwanted higher-order rhythm by

stabilizing a system variable about a desired lower-order unstable periodic fixed point. For non-

chaotic systems the higher-order rhythm is periodic [5, 28], while for chaotic systems it is aperi-

odic [26, 29]. Such system dynamics can be seen on a bifurcation diagram, such as that for the

“Henon map” shown in the bottom panel of Fig. 2. The Henon map is a system that was developed

to study the mathematical dynamics of chaos and is given byxn+1 = 1:0 � px2n + 0:3xn�1. The

bifurcation diagram shows all possible values of the systemvariablex (the dots in the bifurcation

diagram) for a particular value of a system bifurcation parameterp. A bifurcation parameter is a

system parameter that dictates the particular rhythm of a system. (For the logistic map discussed

earlier,r is the bifurcation parameter.) For example, ifp = 0:3, x will always be (after an initial

transient period) the single valuex = 1:00 (also shown in the discrete-time domain in Fig. 2a).

If p = 0:6, x alternates between� 1:39 and��0:22 (Fig. 2b). If p = 1:0, x cycles periodically

through� 1:27! ��0:66! � 0:95! ��0:10! � 1:27::: (Fig. 2c). If p = 1:2, x is chaotic

and can take many values (Fig. 2d). The bifurcation diagram demonstrates that asp is increased,

the dynamical system undergoes a period-doubling route from period-1 to chaos. At each period-

Christini et al. 7

doubling bifurcation, a period-k cycle (k = 1; 2; 4; 8; :::) loses stability while a stable period-2kcycle emerges. For example, a period-1 cycle is stable whenp is less than� 0:36, but becomes

unstable asp is increased beyond�0:36 (at which point a stable period-2 cycle emerges simultane-

ously). Model-independent control attempts to move a system from a particular stable higher-order

(periodic or aperiodic) rhythm to one of the unstable lower-order rhythms (without changing the

nominalp value). For example, for the Henon map withp = 1:0, model-independent control might

attempt to move the system from the stable period-4 rhythm tothe underlying unstable period-2

rhythm depicted by the open diamonds in the bifurcation diagram.

The Ott-Grebogi-Yorke control technique attempts to achieve such a goal by making time-

dependent bifurcation-parameter perturbations to alter the motion of the state point. The initial

perturbations attempt to force the state point onto the unstable periodic fixed point; subsequent

perturbations then hold the state point near that unstable periodic fixed point. While it is beyond

the scope of this review to explain the specific mathematics of model-independent control, such

control is analogous to balancing a ball on a saddle (Fig. 3).The ball represents the state point,

while the center of the seat represents the unstable periodic fixed point. If the ball is exactly on

the center of the seat, it will remain there indefinitely. If,however, it moves a tiny bit, it will roll

down one of the saddle’s sides. The sides of the saddle are termed unstable, while the front and

back of the saddle, which tend to force the ball back towards the seat, are termed stable (similar

dynamical regions typically exist near unstable periodic fixed points). If one can shift the saddle

(i.e., perturb the bifurcation parameter) and properly utilize the stable direction while avoiding the

unstable direction, the ball (state point) can be balanced on the seat (unstable periodic fixed point).

Importantly, all of the dynamical control parameters are estimated from observations of the system.

Thus, model-independent control is practical from an experimental standpoint because it requires

no analytical model of the system.

Christini et al. 8

4 Previous applications of model-independent control

Model-independent chaos control techniques, have been applied to a wide range of systems, in-

cluding magneto-elastic ribbons [29], electronic circuits [30–33], lasers [34–36], chemical reac-

tions [37, 38], and driven pendulums [39–42]. The success ofmodel-independent control in stabi-

lizing physical systems has fostered interest in applying these techniques to biological dynamical

systems, which are often well-understood qualitatively, but for which quantitative relationships

between system components are usually insufficient for the development of an analytical model.

In the first such application, Garfinkelet al. [4] stabilized drug-induced irregular cardiac activ-

ity in a section of tissue from the interventricular septum of a rabbit heart. They determined that the

aperiodic arrhythmias of thein vitro rabbit heart tissue were suitable for chaos control by plotting

one interbeat intervalIn�1 versus the previous interbeat intervalIn (similar to the first-return map

of Fig. 1). As shown in Fig. 4 (top panel), the rabbit heart interbeat intervals appeared to form a

finite geometric pattern. Recurrent patterns of points which approached an unstable periodic fixed

point along a characteristic direction (stable direction)and departed that same fixed point along

a different characteristic direction (unstable direction) were detected. Because of these charac-

teristics, the aperiodic activity of the rabbit heart tissue were deemed to be appropriate for chaos

control.

Garfinkelet al. used a technique similar to OGY control to constrain the aperiodic rabbit heart

oscillations to a low-integer (3 or 4) periodic oscillation(Fig. 4, bottom panel). In part because

Garfinkelet al. were unable to obtain the targeted period-1 rhythm, it has been suggested that the

control-induced rhythms could have been the result of simple pacing dynamics [43]. However,

Christini and Collins [44] used computational analysis to show that the low-integer periodic oscil-

lations stabilized by Garfinkelet al. are to be expected. Indeed, the period-3 and -4 dynamics may

simply have been the result of small misestimations of the system dynamics [44].

Because of the complications of such misestimations, model-independent control techniques

Christini et al. 9

that can adaptively estimate system dynamics are a must for biological control. Several adap-

tive control techniques have been developed [44–49]. Unfortunately, even with adaptive capa-

bilities, there is no theoretical basis to expect that low-dimensional temporal control techniques

(such as those of Refs. [4, 44, 48, 49]) will be able to controlfibrillation in the human heart; fib-

rillation is high-dimensional and spatiotemporal, meaning that multiple variables are required for

full-characterization and that its dynamics occur in both space and time. Such dynamics are not

accounted for by low-dimensional temporal control techniques [50, 51]. However, there currently

appear to be at least three possible means by which nonlineardynamics could have an impact in

arrhythmia control. One is controlling spatially stationary low-dimensional arrhythmias (such as

stable reentrant tachycardia) using temporal control algorithms. Another is using high-dimensional

techniques capable of controlling spatiotemporal arrhythmias such as fibrillation. Yet another is us-

ing pharmacological therapies that could prevent fibrillation by exploiting the nonlinear dynamics

of its onset. These approaches are discussed in the following sections.

5 Nonlinear dynamical control of low-dimensional temporal ar-rhythmias

Many arrhythmias are characterized primarily by temporal dynamics (e.g., ectopic tachycardia and

sinus-node dysfunction). Although reentrant rhythms are spatially-extended by nature, some types

of reentry can be viewed temporally from a single spatial location. Because of this, such rhythms

may be susceptible to nonlinear dynamical control at a single spatial location with low-dimensional

temporal techniques.

To investigate model-independent control of regular cardiac arrhythmias, a series of studies [5–

7, 9] has focused on controlling conduction dynamics of the atrioventricular (AV) node. (Although

AV-nodal conduction arrhythmias are not of great clinical relevance, they serve as an accessi-

ble system to enhance the understanding of how the temporal dynamics of an arrhythmia can be

exploited to alter the arrhythmia.) Specifically, these studies investigated the complex tempo-

Christini et al. 10

ral atrioventricular-nodal conduction dynamics that can be associated with orthodromic reentrant

tachycardia — an arrhythmia that occurs when a cardiac impulse passes anterogradely through the

atria, AV node, and His-Purkinje system, and then conducts back into the atrium via an accessory

conduction pathway (Fig. 5, top panel). In such a situation,a bifurcation from constant AV con-

duction to atrioventricular nodal conduction alternans (abeat-to-beat alternation in AV-nodal con-

duction time: fast, slow, fast, etc.) can occur [52–55]. As mentioned in Sec. 3, such a bifurcation

suggests the existence of an underlying unstable period-1 rhythm (i.e., constant conduction inter-

vals). Importantly, such an unstable rhythm is the fundamental requirement of model-independent

control.

With this in mind, Christini and Collins [5] used a mathematical model [56] of AV-nodal con-

duction to demonstrate that model-independent control could suppress alternans. They simulated

orthodromic reentry via a protocol, called fixed-delay stimulation, in which the model right atrium

was stimulated at a fixed time interval (HA, which simulates the reentrant pathway conduction

time) following detection of the cardiac impulse at the bundle of His (Fig. 5, bottom panel). When

the delay is reduced (simulating faster reentry), the period-1 rhythm destabilizes and the con-

duction time through the AV node bifurcates into alternans.They then stabilized the underlying

unstable period-1 AH fixed point [5] by making perturbationsvia simulated stimulation shorten-

ing of the bifurcation parameter HA. (The AH interval, whichis the time between a given atrial

activation and the corresponding His-bundle activation, represents the AV-nodal conduction time.)

They were able to adaptively estimate and stabilize the underlying unstable period-1 rhythm.

Motivated by the computational control results of Ref. [5],Hall et al. used a similar control

technique to suppress alternans in a series ofin vitro rabbit heart experiments [9]. In five rabbit

heart preparations alternans was induced using fixed interval pacing and then suppressed with

model-independent control. Figure 6 shows the control results of one preparation. The top trace

shows that the underlying unstable period-1 AH fixed point was stabilized while control was active.

The bottom trace of Fig. 6 shows the control perturbations made to the bifurcation parameter HA.

Christini et al. 11

When control was terminated, alternans quickly resumed, verifying that alternans did not simply

subside naturally during control; rather, control was required for alternans suppression. It is worth

noting that the alternans of Fig. 6 were nonstationary, i.e., they were characterized by an increasing

difference between the alternating conduction times and a slow average upward drift of the average

conduction time. Yet, because the adaptive component of thecontrol algorithm repeatedly re-

estimated the unstable period-1 fixed point, the nonstationarity was successfully tracked over the

course of the control trial.

These studies suggest that model-independent algorithms could serve as the foundations for

clinically useful techniques that terminate reentrant arrhythmias by exploiting their underlying

nonlinear dynamics. Modeling and experimental work in nonlinear dynamics have already pro-

duced considerable insight into the dynamics and control ofreentrant circuits [57–60]. Specif-

ically, the nonlinear dynamics of impulse propagation around a one-dimensional ring of cardiac

tissue (a model of reentrant excitation) have been studied in great detail [57–59]. These analyses

have provided mathematical underpinnings for how to determine the amplitude, phase, and spatial

location of a single stimulus that will extinguish a reentrant impulse. Additionally, the dynamics

of periodic stimulation have been analytically extrapolated from the dynamics of a single stimu-

lus [59]. These studies have shed considerable light on the nonlinear dynamics of antitachycardia

stimulation when the topology of the reentrant circuit and the relative location of the stimulating

electrodes are well-characterized. For the most part however, because such geometries cannot be

determined precisely in the clinical setting with present technologies, arrhythmia pacing dynamics

are difficult to characterize. However, it is possible that an adaptive technique may be able to char-

acterize such dynamics during control, thereby making model-independent reentrant arrhythmia

termination clinically feasible.

Christini et al. 12

6 Nonlinear dynamical control of spatiotemporal arrhythmias

Unlike the spatially-constrained dynamics of the arrhythmias discussed in the previous section,

fibrillation and some forms of tachycardia are characterized by spatiotemporal dynamics. It is be-

lieved that fibrillation is the result of multiple excitation wave fronts wandering about the cardiac

tissue in highly irregular patterns of local excitation [61]. Although the dynamical nature of fib-

rillation in the human heart is not fully understood, recentnonlinear dynamical evidence supports

the hypothesis that fibrillation is comprised of both deterministic and stochastic components [62].

Thus, fibrillation is not entirely random.

One hypothesis for the nature of fibrillation (and some complex tachycardias) is that it is the re-

sult of spiral waves. In cardiac tissue, a spiral wave is a reentrant wave that rotates around a center

core in a spiral shape. Because the core is comprised of excitable tissue (in contrast to the anatom-

ical barrier typical of the types of reentry discussed in Sec. 5), it is essentially an electrical vortex,

topologically analogous to the hydrodynamic vortices present in a turbulent fluid [63]. Because

spiral waves can occur in many types of excitable media in which local suprathreshold excitation

initiates propagating waves (e.g., convection [64] and autocatalytic chemical reactions [65]), much

of spiral wave theory was developed in areas outside of cardiology.

Cardiac electrophysiological vortex dynamics were suggested in the 1970’s in the “leading cir-

cle” explanation of functional reentry [66–68]. Such reentry occurs, in the absence of an anatomi-

cal core, as a result of dispersion of refractoriness. A reentrant wave develops and then perpetuates

about a functional obstacle that remains refractory as a result of the collision of the circuit’s cen-

tripetal impulses (Fig. 7). Spiral waves are another form offunctional reentry that develop when

a propagating wave is broken, creating a discontinuity or “wavebreak”. Such a discontinuity can

occur when a propagating wave passes around an obstacle [62], passes through inhomogeneous

tissue [62], or when one propagating wave collides with another propagating wave [69] as might

occur in the presence of an ectopic beat (Fig. 8). At the discontinuity, the wave front merges with

Christini et al. 13

the wave tail, causing a decrease in the wave front velocity.Depending on the conduction proper-

ties of the tissue, the corresponding gradient of conduction can create a curvature because of the

very slow wave front propagation close to the wavebreak. In such a situation, the wavebreak can

form the pivoting point (core) of a spiral. This core can meander about the tissue indefinitely while

the spiral wave propagates around it. Unlike with the centripetal wavelet collision of leading-circle

reentry, the core of a spiral wave remains excitable due to the high curvature of the leading edge

of the rotating wave front [70].

Spiral waves have been studied extensively in computational models of cardiac tissue [63, 71–

80], in in vitro animal preparations [69, 81–88], and recently inin vivoanimal experiments [70, 89–

92]. While such studies have provided a tremendous amount ofinsight into the dynamics of cardiac

spiral waves, it is unclear to what degree (if any) these dynamics translate to clinical arrhythmias

in humans. Nevertheless, one proposal [93] is that: (i) a stationary spiral wave might cause some

types of monomorphic ventricular tachycardia, (ii) a meandering spiral wave might underlie poly-

morphic tachycardia and/or torsade de pointes, and (iii) the breakup of spiral waves into many

small, asynchronous wavelets may characterize fibrillation [80, 94–97]. If these hypotheses are

correct, then there is good reason to pursue the eliminationof spiral waves and/or the prevention of

spiral wave breakup. Such therapies could be effective in both ventricular tachycardia termination

and fibrillation prevention.

Several methods for controlling or suppressing spiral waves have been proposed [98]. One

scheme utilizes the application of an external forcing signal to the excitable medium [10–15,

99–101] in an attempt to move the spiral wave in a desired direction.This approach is shown in

Fig. 9, in which spiral waves with regular behavior�Figs. 9a andc

�are caused to drift

�Fig. 9b

�or meander in complex patterns

�Fig. 9d

�. For cardiac tissue, such an algorithm might employ

electric stimulation from one or more electrodes to direct the spiral wave into the interventricular

septum, or some other non-conducting tissue, where it wouldbe annihilated. Another scheme is

global feedback via modulation of a parameter that affects the system as a whole (as opposed to

Christini et al. 14

stimulation at one or more local sites) [102, 103]. For cardiac tissue, such an algorithm might

employ electric field modulation across the entire heart. Osipov et al. have developed an approach

that directs the spiral wave towards a boundary by introducing localized spatial inhomogeneities

between the boundary and the core of the spiral wave [98]. Osipov and Collins have proposed

another approach which utilizes weak impulses to perturb and destabilize the wave front and wave

back velocities such that the wave self annihilates [104]. Rappelet al. demonstrated in simulations

that spiral-wave breakup into fibrillatory-like activity can be prevented by using feedback current

perturbations applied via multiple electrodes in an electrode array [105]. They speculated that

stabilization of spirals might be achievable in intact hearts with a 1 cm inter-electrode spacing.

The aforementioned spiral wave control techniques utilizefeedback perturbations of various

types to annihilate a spiral wave. Clinical realizations ofsuch algorithms, if possible, would con-

ceivably utilize implantable devices similar to implantable cardiac defibrillators. Alternatively,

pharmacological approaches to spiral wave control have been suggested recently [106–109]. This

idea is based on evidence that ventricular fibrillation results from the breakup of the spiral waves of

stable ventricular tachycardia [80, 94–97]. Simulations have shown that such spiral wave breakup

is dependent on specific electrophysiological properties of the cardiac tissue. Specifically, the resti-

tution properties of the cardiac action potential durationand conduction velocity have been shown

to be important determinants of the stability of spiral waves. Restitution is the relationship between

action potential duration and conduction velocity and the diastolic interval of premature beats.

Typically, as diastolic interval decreases, action potential duration and conduction velocity also

decrease. Spiral wave breakup is expected when the slope of the restitution curve relating action

potential duration and preceding diastolic interval is greater than 1. Recent simulations [106, 107]

and animal experiments [107, 109] have provided evidence that when the slope of the restitution

curve is held less than 1, spirals do not break up into fibrillatory-like activity. These studies suggest

that a drug that can hold the slope of the restitution curve below 1 may prevent the progression of

ventricular tachycardia to ventricular fibrillation. Thus, development and testing of such a drug

Christini et al. 15

seems warranted.

One limitation to the spiral wave theory of reentrant arrhythmias is its failure to account for the

three-dimensional nature of the ventricular wall. Thus, because spiral waves are two-dimensional,

they are merely a rough approximation ofin situ electrophysiological dynamics. In order to prop-

erly model such dynamics, investigators are now studying scroll waves [62, 110–113], which es-

sentially are stacks of two-dimensional spiral waves. Because of the increased complexity involved

in analysis and simulations and the inability to perform three-dimensional imaging in experiments,

scroll waves are not as well understood as spiral waves. Nevertheless, it is anticipated that as

technologies advance, it will be possible to extend many of the dynamical analyses and control

techniques developed for spiral waves to scroll waves, and therefore one step closer to possible

clinical utility.

7 Prospects

Although numerous practical limitations must be overcome,recent advancements, as described in

this review, offer hope that nonlinear dynamical control techniques will be clinically useful. For

tachycardia control, model-independent feedback-based techniques may be advantageous over an-

titachycardia or defibrillation algorithms employed by current implantable cardiac defibrillators.

Such implantable defibrillator algorithms are dynamicallysimplistic (brute force for defibrillation;

simple ramp or burst patterns for antitachycardia) and utilize little, if any, feedback information

regarding their beat-to-beat effects on the arrhythmia — the rates, intervals, thresholds, and other

algorithmic parameters do not adapt automatically on a beat-to-beat basis. Furthermore, pacing ef-

ficacy decreases as tachycardia rate increases [114]. Nonlinear dynamical stimulation techniques

that can control reentrant tachycardia or fibrillation by exploiting inherent dynamics to terminate

the arrhythmia, may improve efficacy. Alternatively, nonlinear dynamics might contribute to new

pharmacological therapies that prevent the breakdown of ventricular tachycardia to ventricular fib-

Christini et al. 16

rillation, thereby eliminating the need for implantable cardiac defibrillators in some patients. In

short, the nonlinear dynamical nature of cardiac electrophysiology, combined with the innovative

research being carried out on many fronts, could lead to exciting developments in cardiac arrhyth-

mia control in the coming years.

Acknowledgements

This work was supported in part by a grant from the National Institutes of Health (R01 HL56139).

We thank Alan Garfinkel, Ph.D. for valuable suggestions.

Christini et al. 17

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Figure Captions

Figure 1: Discrete-time domain (a andb, whereb shows a magnified segment ofa) and first-returnmap (c) of the chaotic logistic map given byxn+1 = rxn(1�xn), with r = 4:0. Note that the solidlines ina andb are not meant to represent continuous time, but rather, are used to show progressionfrom eachxn to xn+1.Figure 2: Discrete-time domain (a, b, c, andd) and bifurcation diagram (bottom panel) for theHenon map given byxn+1 = 1:0� px2n + 0:3xn�1. The dots in the bifurcation diagram representall possible values ofx for a given value ofp. The open circles and diamonds in the bifurcationdiagram are the analytically-derived unstable period-1 (p ' 0:36) and period-2 (p ' 0:91) fixedpoints, respectively.

Figure 3: A schematic representation of model-independentcontrol. The ball represents the statepoint, while the center of the seat represents the unstable periodic fixed point. The sides of thesaddle represent the system’s unstable directions, while the front and back of the saddle representthe system’s stable direction. By perturbing the bifurcation parameter (shifting the saddle) in amanner that utilizes the stable directions and avoids the unstable directions, the ball can be balancedon the seat.

Figure 4: The top panel shows one interbeat intervalIn�1 versus the previous interbeat intervalIn during drug-induced irregular cardiac activity in a section of tissue from the interventricularseptum of a rabbit heart. The points form a structure that is not space-filling, evidence of thepossible presence of deterministic chaos. The bottom panelshows interbeat intervalsIn versusbeat numbern for a chaos control trial of such irregular activity. Control stabilized the rhythmin a period-4 rhythm (between the “Control On” and “Control Off” annotations). Modified andreprinted with permission from Ref. [4].

Figure 5: Two schematics showing normal conduction from thesinoatrial (SA) node, throughthe right and left atria (RA,LA), the atrioventricular (AV)node, and the right and left ventricles(RV,LV). The top panel shows an abnormal retrograde pathwaybetween the right ventricle andatrium that produces an orthodromic reentrant tachycardia. The bottom panel shows how, in theabsence of such an abnormal retrograde pathway, orthodromic reentrant tachycardia can be sim-ulated (as depicted by the loop containing the computer) by fixed-delay stimulation of the rightatrium (at time A) at an interval HA following detection of his-bundle activation (at time H).

Christini et al. 30

Figure 6: AV nodal conduction times AH (filled circles, top trace; left y-axis) and delay times HA(open diamonds, bottom trace; right y-axis) versus beat numbern during a control trial of alternansin an isolated rabbit heart [9]. Control was implemented from beat 266 to 787 (2 minutes).

Figure 7:A) In leading circle reentry, an electrical impulse rotates around a refractory core. Such afunctional obstacle is produced by centripetal impulses that collide and render that tissue refractory.B) Reentry can be elliptical or assymetric when the tissue’s conducting properties are anisotropic.C) Two reentrant circuits sharing a common pathway can produce “Figure-8” reentry. Reprintedwith permission from Ref. [62].

Figure 8: Simulations (96x96 FitzHugh-Nagumo excitable cells) showing two methods of initi-ating spiral wave activity. The top panel (consecutive snap-shots shown ina throughj) showsthe effects of the collision of two planar waves S1 and S2. The arrow in each snap-shot indicatesthe direction of the wave front. The bottom panel (consecutive snap-shots shown ina throughe)shows the effects of two point stimuli S1 and S2. S1 was delivered at the site of the asterisk, whileS2 was delivered 14 pixels to the left of the S1 stimulation site (b). Reprinted with permission fromRef. [69].

Figure 9: Computer simulations showing the effects of periodic forcing on spiral-wave dynamics.a) Periodically rotating solution in a reaction-diffusion model of homogeneous excitable media.The white curve shows the path of the spiral tip.b) The same system asa, but with spatiallyhomogeneous periodic forcing that caused the spiral to drift from the lower right to the upperleft. c) The quasiperiodic motion of the spiral tip of an ordinary differential equation model [100].d) The spiral tip movement of the system shown inc under the influence of periodic forcing.Reprinted with permission from Ref. [115].

Christini et al. 31

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

ξ*

6

5

4

3

2

1

xn+1

= xn

xn+1

xn

0 100 200 300 400 5000.00.20.40.60.81.0

n

n

244 246 248 2500.0

0.5

1.0

c

b

a

65

43

21x

n

xn

Fig. 1Christiniet al.

Christini et al. 32

0 25 50 75 100n

0.9

0.95

1

1.05

1.1

xn

p = 0.3a

0 25 50 75 100n

−1

0

1

2

xn

p = 0.6b

0 25 50 75 100n

−1

0

1

2

xn

p = 1.0c

0 25 50 75 100n

−2

−1

0

1

2

xn

p = 1.2d

0.2 0.4 0.6 0.8 1.0 1.2

-1.0

-0.5

0.0

0.5

1.0

1.5

x

p

Fig. 2; Christiniet al.

Christini et al. 33

STABLESTABLE

UNSTABLE

Fig. 3Christiniet al.

Christini et al. 34

Fig. 4Christiniet al.

Christini et al. 35

ORT

RA LA

AV

SA

RV LV

−delay HA

−computer

RA LA

AV

SA

measurement

RV LV

A H

stimulator

Fig. 5Christiniet al.

Christini et al. 36

0 200 400 600 800 1000

beat number

50

75

100

125

150

AH

90

100

110

120

130

HA

(n)

(ms)(ms)

AH

HA

Fig. 6Christiniet al.

Christini et al. 37

FIG. 2. Functional re-entry requires no anatomical obstacle but is based on

Fig. 7Christiniet al.

Christini et al. 38

Fig. 8Christiniet al.

Christini et al. 39

Figure 2.1: Illustration of the e�ects of periodic forcing on the spiral-wavedynamics. (a) Periodically (rigidly) rotating solution in a reaction-di�usionmodel of homogeneous excitable media [Barkley, 1991]. Grey-scale showsvalues of the slow species, and the white curve shows the path of the spiraltip. Model parameters are: a = 0:8, b = 0:05, and " = 0:02. (b) The sameconditions as (a), except with spatially homogeneous forcing at the rotationfrequency of the unforced spiral. The induced drift in the spiral location isfrom bottom right to top left in this case. Here b = 0:05+0:0035 sin(1:806t).(c) Quasiperiodic state from the ordinary di�erential equation model de-scribed in the text. The state corresponds to a \meandering" spiral wave.Parameters are: �1 = 10=3, �2 = �7:5, and 0 = 3:8. (d) The e�ect ofperiodically forcing this wave: �2 = �7:5 + 0:6 sin(0:3222t).23

Fig. 9Christiniet al.