zhoujian cao et al- turbulence control with local pacing and its implication in cardiac...

8/3/2019 Zhoujian Cao et al- Turbulence control with local pacing and its implication in cardiac defibrillation

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Turbulence control with local pacing and its implicationin cardiac defibrillation

Zhoujian Cao and Pengfei LiDepartment of Physics, Beijing Normal University, Beijing 100875, China

Hong ZhangZhejiang Institute of Modern Physics and Department of Physics, Zhejiang University,Hangzhou 310027, China

Fagen Xie Research Department, Kaiser Permanente, Pasadena, California 91101

Gang Hua

Department of Physics, Beijing Normal University, Beijing 100875, Chinaand BeijingHong KongSingapore Joint Center of Nonlinear and Complex Systems,Beijing Normal University Branch, Beijing, China

Received 19 October 2006; accepted 9 February 2007; published online 30 March 2007

In this review article, we describe turbulence control in excitable systems by using a local periodic

pacing method. The controllability conditions of turbulence suppression and the mechanisms un-

derlying these conditions are analyzed. The local pacing method is applied to control Winfree

turbulence WT and defect turbulence DT induced by spiral-wave breakup. It is shown that WTcan always be suppressed by local pacing if the pacing amplitude and frequency are properly

chosen. On the other hand, the pacing method can achieve suppression of DT induced by instabili-

ties associated with the motions of spiral tips while failing to suppress DT induced by the insta-

bilities of wave propagation far from tips. In the latter case, an auxiliary method of applying

gradient field is suggested to improve the control effects. The implication of this local pacing

method to realistic cardiac defibrillation is addressed. 2007 American Institute of Physics.

DOI: 10.1063/1.2713688

The only clinical method currently accepted in perform-

ing cardiac defibrillation is to apply large shocks to body

surface or directly to cardiac muscle.1,2

These large-

amplitude shocks may damage the cardiac tissue and

cause serious pains.3

Therefore, in both fields of nonlin-

ear science and cardiac physiology there are growing ex-

perimental and theoretical efforts to develop low-

amplitude defibrillation methods.49

In Ref. 7, the authors

performed suppression of Winfree turbulence by apply-

ing global periodic low-amplitude forcing.10

The difficulty

of injecting an external signal to whole cardiac tissue re-

stricts the practical utilization of this global method.

Therefore, so-called local and low-amplitude pacing

control,9

which applies low-amplitude signals to the local

boundary area of turbulent systems for global turbulence

suppression, may become a promising method for

defibrillating cardiac tissue. On one hand, boundary re-

gion control is very convenient, and on the other hand,

simple periodic pacing does not require instant measure-

ments and feedbacks of the system variables, and can be

easily realized in practical situations. In this article, we

review turbulence control in excitable systems by using a

local periodic pacing method, and both Winfree turbu-

lence (WT) and defect turbulence (DT) are

considered.11,12

It is shown that WT can always be sup-

pressed by local pacing as the pacing amplitude and fre-

quency are properly chosen. On the other hand, the pac-

ing method can achieve suppression of DT induced by

instabilities associated with the motions of spiral tips

while failing to suppress DT induced by the instability of

wave propagation far from tips. In the latter case, an

auxiliary method of applying gradient coupling is sug-

gested to improve the control effects. The implication of

this local pacing method to realistic cardiac defibrillation

is addressed.

I. INTRODUCTION

More and more experiments indicate that ventricular fi-

brillations VF are induced by transitions of cardiac electric

propagating waves from spiral waves to turbulent waves.13,14

Different mechanisms of turbulence formation in cardiac

systems are suggested, among which two cases will be dis-

cussed in this paper. One case of turbulence formation is due

to wave breakup through instabilities of two-dimensional

2D spiral waves. This spiral wave breakup turbulence is

associated with formation of topological defects, and thus is

called defect turbulence DT.6,15

In the other case, turbu-

lence is caused by Winfree instability, called Winfree turbu-

lence WT. Winfree turbulence, an unstable state of scroll

waves, is a special kind of spatiotemporal chaos that exists

exclusively in 3D excitable media, whereas its correspon-

dence in 2D system is stable spiral waves. WT results from

aAuthor to whom all correspondence should be addressed. Electronic mail:

[email protected]

CHAOS 17, 015107 2007

1054-1500/2007/171 /015107/9/$23.00 2007 American Institute of Physics17, 015107-1
http://dx.doi.org/10.1063/1.2713688http://dx.doi.org/10.1063/1.2713688http://dx.doi.org/10.1063/1.2713688http://dx.doi.org/10.1063/1.2713688http://dx.doi.org/10.1063/1.2713688http://dx.doi.org/10.1063/1.2713688http://dx.doi.org/10.1063/1.2713688


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the negative-tension instability of scroll waves, and is cur-

rently considered as one of the principal mechanisms of car-

diac fibrillation.7,16

Cardiac tissues are typical excitable me-

dia, and cardiac fibrillations are produced by various

dynamic transitions to turbulence. Therefore, defibrillation is

essentially a task of turbulence suppression in excitable me-

dia.

So far, a number of feedback and nonfeedback control

methods1723 have been suggested for suppressing spatiotem-poral chaos, among which nonfeedback approaches have

some advantages of flexibility and convenience in realistic

applications.9,22

In this paper, we are interested in one of the

nonfeedback methods, called the local pacing method, for

turbulence suppression in excitable media. For controlling

turbulence, we inject weak periodic signals just to a small

space region around the system surface. This local pacing

method is one of the low-amplitude defibrillation methods

developed recently in cardiac physiology. For instance, some

experiments explored the use of boundary periodic pacing

for cardiac defibrillation.2429

They found that the pacing had

only a local effect, and could not achieve global control offibrillation. Once the pacing was suspended, the local region

of capture was reinvaded by surrounding turbulent electrical

activity, and the tissue remains in a state of fibrillation. In

order to fully understand these failures and effectively im-

prove the technique of low-amplitude defibrillation, it is nec-

essary to study systematically local pacing methods for tur-

bulence suppression. In this review paper, we make a survey

on turbulence suppression with local pacing in excitable me-

dia. The control mechanism, efficiency, and the controllabil-

ity conditions are discussed in detail, and possible improve-

ment of the pacing method is discussed as well. Moreover,

the implication of this turbulence suppression to the problem

of cardiac defibrillation is also addressed. In the next section,

we introduce the models of excitable media used in this pa-

per for the analysis of turbulence suppression with local pac-

ing. In Sec. III and IV, we investigate WT and DT suppres-

sions, respectively. The influences of various parameters on

the control efficiency of the local pacing method are dis-

cussed in Sec. V. In the last section, we address some impli-

cation of this local pacing method to cardiac defibrillation.

II. MODELS, LOCAL PACING METHOD,AND GENERAL DISCUSSIONON CONTROLLABILITY

There are a number of mathematical models describing

the activities of the cardiac tissue, among which the Luo-

Rudy model

V

t=

1.0

CmINa + Isi + IK+ IK1 + IKp + Ib + D

2V 1

has been most extensively used. In Eq. 1, Vx , t is the

transmembrane potential of the cardiac cell at the space point

x, Is represent various iron currents through cell iron gates,

and Is are controlled by iron gate variables, which are con-

trolled in turn by the potential voltage V.9,30

Neglecting the complexity of various iron currents andgate variables, we can greatly simplify the dynamics by us-

ing only two variables,3133

the difference of electric poten-

tial and the total iron current. In the following, we use Bar-

kley and Br models,31,32

which have been extensively

investigated as typical excitable media for describing this

simplified dynamics. These two models together can be ex-

pressed by

u

t

=1

u1 u

u

v + b

a + 2u , 2a

v

t= fu v, 2b

where is the ratio of the time scale of the activator u over

that of the inhibitor v. For the Barkley model,31

function f

reads

fu = u 3

and for the Br model,32

the function f is specified as

fu =

0 0 u 1/3

1 6.75uu 12 1/3 u 1

1 1 u .

4

These two models have been used as the simplest homoge-

neous monodomain models representing the excitability of

cardiac electric activities. Although the two models are akin

to each other, there is an important difference between them.

Spiral waves in the Barkley model are always stable, i.e., the

spirals there never break up in 2D systems. This model is

suitable for the study of WT, which occurs in the 3D Barkley

model in certain parameter regions due to filament expan-

sion of the corresponding 3D scroll waves.7,34

As to the Br

model, its spirals have wealthy behaviors including 2D DTinduced by the Doppler effect,

31backfire instability,

32and so

on. This is therefore an ideal model studying DT and its

suppression. With local pacing, Eq. 2a becomes

u

t=

1

u1 uu v + b

a + 2u +xFcost,

x = 1 x0 x,

5

where is the boundary region to which we apply pacing.

We fix at the left site x =0 for 1D systems and left stripe

x =0 for 2D systems. The local pacing of Eq. 5 models the

low-amplitude control of cardiac systems through electric

poles placed on the surface of heart. Throughout the paper,

we apply no-flux boundary condition for different spacial

dimensions.

With local pacing control, we excite some local space

area periodically to generate regular propagating waves or

called controlling waves, and to use these waves to suppress

turbulence. The local control method functions in three key

processes of wave generation, wave propagation, and wave

competition. More specifically, the following three condi-

tions are of crucial importance for the success of the local

pacing method. a The periodic pacing must be able to gen-erate ordered controlling waves; b the controlling waves

015107-2 Cao et al. Chaos 17, 015107 2007


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must be able to propagate regularly in the medium; c the

propagating controlling waves must out-compete with turbu-

lent waves.

Now we analyze how these three conditions play roles in

turbulence suppression with local pacing. As to condition a,

we find that pacing frequency higher than certain threshold

cannot produce regular waves having the pacing frequency.

This condition can be investigated with a 1D chain. In Fig. 1,

we present a response curve of input-output frequencies ofthe Barkley model with a =1.1, b =0.19, =0.02, F=2.0.

When the pacing frequency increases too high 1.49, we

find that the output waves can no longer follow each excita-

tion pulse of the pacing, namely, the system output cannot

keep 1:1 frequency relation with the input, rather it can gen-

erate only 1: n n larger than or equal to 2 frequency re-

sponse. In Fig. 1 we observe 1:2 response for 2.88

1.61 and 1:3 response for 3.643.0, and so on. Note,

the highest output frequency is observed at the highest input

frequency of the 1:1 response region where the input fre-

quency is relatively low. And when the pacing frequency is

too high, it can only result in waves with low frequency see

1.26 in Fig. 1. Therefore, when the pacing frequency istoo high or too low, the output frequency cannot be high. The

former generates low output frequency due to1: n blocks

out=

nn2, while the latter generates low output fre-

quency out= due to low pacing frequency. In the discus-

sion of condition c, we will see that sufficiently high fre-

quency of controlling waves is a necessary condition for

turbulence suppression. In Figs. 2b and 2c, we show that

neither too high nor too low pacing frequencies can suppress

turbulence while suitable intermediate pacing frequency suc-

ceeds in doing so Fig. 2d. This is consistent with the

above analysis.

Condition b is self-evident, because in order to com-pete turbulent waves, the regular controlling waves must be

stable themselves. The problem for this condition is to illus-

trate the cases in which regular waves cannot propagate sta-

bly in excitable media. For instance, two mechanisms can

certainly make wave propagation unstable: one is the back-

firing effect and the other is alternans instability. In Fig. 3,

we compare essentially different behaviors of wave propaga-tions without a-c and with backfiring d-f. When

backfiring occurs, the forward-propagating waves produce

daughter waves propagating backward Fig. 3e, which

produce their daughters again. Finally, the whole space is

filled with these randomly distributed waves Fig. 3f. With

this backfiring instability, the controlling waves generated by

the periodic pacing are unstable, they produce turbulence by

themselves, and one can thus certainly not expect to use

these waves to suppress turbulence. The alternans phenom-

enon can be observed in Luo-Rudy35

and Aliev-Panfilov

models,33

which serve as typical models describing cardiac

dynamics. Originally, alternans means that a single cardiac

cell responds to external stimuli with long-short-long-short

period-doubling temporal behavior Fig. 4a.20,36

These

temporal alternans can produce also period-doubling in space

with long-short-long-short wavelengths shown in Fig. 4b.

If the alternans effect is too strong, successive waves can

collide with each other, leading to wave breakup and turbu-

lence Fig. 4c. If the controlling waves fall into this strong

alternans range, the above controllability condition b is not

met, and turbulence suppression with local pacing cannot be

achieved.

When the local pacing can generate regular controlling

waves, which can effectively propagate in the excitable me-

dium, there is still one more condition c that should befulfilled for successful turbulence suppression. Suppose two

FIG. 1. Output frequency versus input frequency for a driven 1D Barkley

model 5. The periodic pacing is applied at x = 0. a =1.1, b =0.19, =0.02,F=2.0. 1:1 frequency response is observed for 1.26. For 1.26, the

system cannot respond to each excitation of the pacing, and thus the con-

trolling waves generated by the pacing have frequency m : nnm to theinput frequency. The highest output frequency of the controlling waves,

which is most important for turbulence suppression, is determined by thelargest input frequency in the 1:1 region.

FIG. 2. Contour patterns of the 2D Br system a =1.1, b =0.19, =0.09, F

=2.0. a F=0. Turbulent state freely evolved from a spiral initial state.

b-d The boundary stripe control Eq. 5 is applied to the state a. b=1.34. Defect turbulence control with frequency that is too high no sup-pression. t=2000. c = 1.05. Defect turbulence control with frequencythat is too low no suppression. t=2000. d = 1.25. Successful defectturbulence suppression with suitable frequency. t=1000. Figure fromRef. 12.

015107-3 Local pacing control Chaos 17, 015107 2007


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waves wave A and wave B interact and compete with each

other. The phase-matching condition governing the motion ofthe interface between A and B reads

37

k1x + 1t+ = k2x + 2t+ , 6

where k1,2 are the wave numbers of waves A and B, respec-

tively, and x is the distance between the interface point and

some reference point e.g., wave source. 1,2 are the fre-

quencies of two wave chains and is the phase angle of the

interface point. Thus the speed of the interface VIdx1

dt=

dx2

dtis given by

VI =1 2

k1 + k2. 7

Positive negative velocity VI represents that wave A wave

B suppresses wave B wave A. This formula is consistent

with the finding that in excitable media, propagating waves

with the highest excitation frequency can always overtake all

other waves with lower frequencies.38

In conclusion, in order

for the controlling waves generated by the local pacing tosuppress turbulent waves in excitable media, the frequency

of the former must be higher than that of the latter.

III. WINFREE TURBULENCE SUPPRESSIONWITH LOCAL PACING

From the analysis of Sec. II, one can conclude that all

stable spiral waves can be suppressed by local pacing with

amplitude and frequency properly chosen. This strong con-

clusion can be understood heuristically. A spiral wave tip

serves as a pacing source generating wave. In the presence of

stable spiral waves, local pacing with the spiral frequencycan certainly satisfy the above conditions a and b since

the spiral wave tip can successfully generate and propagate

stable spiral waves. Generically, the frequency of the spiral

denoted by s is not the highest frequency say, h sup-

ported by the medium. We can then easily find a pacing

frequency sh with which the local pacing can still

satisfy conditions a and b on one hand, and the frequency

of the controlling waves generated by this pacing is higher

and that of the spiral and then can win the competition with

the stable spiral on the other hand since s, and condi-

tion c is satisfied. WT is induced by the 3D effect of stable

2D spiral waves.7

Due to the negative tension of the filament

of the corresponding 3D scroll waves, all the above reason-

ing on the controllability of stable spirals exists in the pro-

cess of WT suppression note, filaments that cause instability

of scroll waves to WT do not exist in controlling waves.

This makes us propose a conjecture that all WT can be sup-

pressed by local pacing with proper pacing amplitude and

frequency. We use the Barkley model Eqs. 1 and 2 to

discuss the problem of WT suppression. It is well known that

all spirals in the 2D Barkley model are stable, and in certain

parameter regions scroll waves of the 3D Barkley model can

be destabilized via filament expansion instability,7

develop-

ing WT as shown in Fig. 5. Now we apply periodic pacing to

a small cube nnn sites region on a surface11

and adoptthe pacing protocol suggested in Ref. 7,

FIG. 3. Comparison of wave propagation with and

without backfiring. ac Wave propagation without

backfiring. df Wave propagation with backfiring.a and d, b and e, c and f are at the same times,respectively.

FIG. 4. Schematic figures of alternans and alternans DT in the Luo-Rudy

model. a and b Alternans effect of long-short-long-short period-doublingmotion in the 1D Luo-Rudy model. a An alternans evolution of the actionpotential V at an arbitrary space point. b An evolution contour pattern in

the x-t plane. c A snapshot contour pattern of alternans DT in the 2DLuo-Rudy model at an arbitrary moment.

015107-4 Cao et al. Chaos 17, 015107 2007


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F=1

au1 ubf. 8

Figure 6 demonstrates the process of WT suppression by

local pacing. The local excitation on a small cube stimulates

spherical target waves, which propagate in the excitable me-

dium, invading the turbulent surrounding, and suppress tur-

bulence by pushing filaments of scroll waves out of the ex-

citable medium during their propagation Figs. 6a6d. In

Figs. 6e and 6f, we show how the ordered waves wipe out

the remaining filaments and drive the system to the desired

rest state, after lifting the periodic forcing at the turbulent

state of Fig. 6c.

In Fig. 7, we study a quantity important for the control

efficiency, the control speed that is defined as R = 1 /T with Tbeing the time needed for turbulence suppression. R depends

on forcing frequency, injection area, and other pacing param-

eters. Figure 7 shows the parameter range where the local

pacing can successfully suppress the existing turbulent scroll

waves. There are several characteristic boundaries to this

range corresponding to the controllability conditions ac

emphasized in Sec. II. First, this range is restricted within a

frequency zone Fig. 7a, and frequencies above or below

this zone do not provide successful turbulence suppression.

The forcing frequency needs to be higher than a threshold 00 1.19 is the rotation frequency in unforced 2D system

due to condition c. This is because the target waves gener-ated by the local excitation can suppress existing scroll

waves only if its frequency is higher than that of the latter.38

There is an upper frequency bound for effective control. The

target waves are generated by the periodic signals in the

presence of turbulence and scroll waves. In order to effec-

tively stimulate target waves from the existing scroll wave

background, the forcing frequency should not be too far from

0 for achieving 1:1 resonant excitation due to condition

a. This gives rise to the upper bound of the zone of Fig.

7a. Secondly, the injection area should be sufficiently large.

Our WT suppression with cubic stimulation fails for n4

Fig. 7

b

. The reason for this condition is clear. In order toproduce regular waves, the injection area needs to be larger

than a threshold so that the wave front generated has curva-

ture smaller than a critical value for allowing target wave

propagation otherwise, no spherical wave front can be

formed, condition a.39 Nevertheless, when n is above the

threshold, the speed of turbulence annihilation is no longer

sensitive to n. In other words, further increase of n does not

further reduce the suppression time. This characteristic al-

lows us to take some optimal control area for performing WT

suppression. In Fig. 8, we test our conjecture of WT control-

lability with local pacing by examining the 3D Barkley

model for different parameter values. With =0.02, b =0.19

fixed, we vary a in the interval a = 0.98,1.16 where WT

occurs. Note, WT does not exist for smaller and larger a. For

a0.98, the system cannot support any waves due to too

low excitability,7

and the system has only a single stable

state, the stationary homogeneous rest state. With a1.16,

scroll waves become stable because of positive tension of

filaments.7

Figure 8 satisfactorily confirms at least, for the

Barkley model the conjecture that WT can be always sup-

pressed by local pacing.

As a possible mechanism of cardiac fibrillation,7,16

WT

suppression is one of the crucial tasks in cardiac defibrilla-

tion. Since local pacing methods are effective in WT sup-

pression in excitable media, they have some interesting im-plication in defibrillation. As a possible defibrillation

FIG. 5. Evolution of WT in the 3D Barkley model. a =1.1, b =0.19, and

=0.02. a Initial state with a regular filament circle located at the center ofsimulation domain. b Filament expansion to an irregular while still con-tinuous curve at t=50. c The expanded filament touches space boundaries,and the boundaries cut the filament into many segments. t=150. d A suf-ficiently developed WT state at t=250.

FIG. 6. WT control with local pacing on a 555 cube in the center of the

back surface of the space. bf=0.75,=1.6 and all other parameters are the

same as Fig. 5. The pacing is applied to Fig. 5d at t=250. ad The localpacing generates spherical target waves near the injected area, which then

propagate in the space and push turbulence and all scroll wave filaments out

of the excitable medium. e,f The evolution of the medium after the injection signals are lifted at t=331 i.e., from the state c. The systemapproaches the desired rest state without the external forcing, and complete

black indicating the rest state is observed after t360. Figure fromRef. 11.



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method, the present local stimulation shows a number of

advantages in comparison with some other low-amplitude

defibrillation methods, such as global controlling methods.

First, the global method7

needs to inject external signals to

all space sites, and this is inconvenient in many practical

situations e.g., it is difficult to apply electrical impulses to a

large area of interior cardiac body. On the contrary, it is

much more convenient for local pacing methods to injectsignals to partial areas around the surface of the system.

Second, the turbulence suppression speed of our local stimu-

lations is faster11

than that of the global ones in Ref. 7 if

similar powers of pacing signals are applied in both cases. It

is worthwhile to remark that in Fig. 6 we suppress WT by

applying signals to 53 sites only among the total 1503 sites

with signal amplitude being of the same order as the system

variables. This is desired for developing a low-amplitude

defibrillation method.

IV. DEFECT TURBULENCE SUPPRESSION

WITH LOCAL PACING

Considering another possible mechanism of cardiac fi-

brillation, we study DT suppression in excitable media. Ap-

parently, the previous arguments on the general controllabil-

ity of stable spirals and WT are no longer valid for DT

control because spirals are now not stable. In excitable me-

dia, spirals can break up through different mechanisms. They

may break up in the regions near the spiral tips, such as

through the Doppler effect,31

or may break up through the

instabilities in arm wave propagation far from the tips, such

as through backfiring32

and alternans instabilities.20,36

We

will observe in this section that the former DT can be easily

suppressed by local pacing while the latter cannot be doneso, and the reasons for these observations will be explained

later. Let us consider the 2D Br model with a =1.1, b

=0.19, and we apply periodic pacing of Eq. 5 with F=2.0

to analyze the problem of controllability of DT by external

pacing, which is applied to the left boundary of the space,

x 0 ,x ,y0 ,L. In Fig. 9, the controllable region on

the - plane is presented. Without control, the entire region

0.069 corresponds to breakup spiral to DT in this model,

and Doppler-effect-induced DT is justified in the interval

0.0690.103. It is interesting to find that, with the

boundary pacing, the controllable region covers all the Dop-

pler DT domain, and this region is ended at =0.103, whichis just the threshold of backfiring.

32The coincidence of the

controllable parameter region boundary with the backfiring

boundary confirms our analysis of controllability condition

FIG. 7. Turbulence suppression speed

R = 1 /T of the cubic nnn injec-tion plotted for different forcing pa-

rameters where T is the time needed

for turbulence suppression. a n = 6,bf=0.75. R plotted vs . b bf=0.75,

=1.6. R vs n. Figure from Ref. 11.

FIG. 8. Controllable region of WT suppression in the Barkley model. With

=0.02, b =0.19, WT exists only in the parameter region 0.98a1.16.

Thus WT in all this range can be suppressed by boundary local pacing withthe pacing amplitude and frequency properly chosen.

FIG. 9. Controllability region, where all spiral waves and DT wavelets can

be suppressed, in the 2D Br model. The system state is DT for 0.069.

=0.103 is the critical parameter above which backfiring takes place. The

upper line and the lower line are the highest and the lowest frequencies of

controllability. Pacing with frequency between these two curves can sup-press spirals and DT successfully.

015107-6 Cao et al. Chaos 17, 015107 2007


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b in Sec. II. In the Doppler DT region, DT is induced by

collisions between spiral tips topological defects and spiral

wave arms.32

Since the controlling waves do not have topo-

logical defect see Fig. 2d and they do not suffer from the

Doppler instability, the controlling waves can surely suppress

Doppler DT when the local pacing frequency is chosen larger

than that of the spiral wavelets. On the other hand, backfiring

makes propagation of any regular waves fail, and in the pres-

ence of such instability the controlling waves generated bythe local pacing must fail in suppressing DT no matter how

one adjusts the pacing amplitude and frequency, because DT

can be induced by the controlling waves themselves during

their propagation. The parameter regions with these kinds of

instabilities are thus called an absolutely uncontrollable re-

gion for local pacing. In Fig. 9 the backfiring region,

0.103, is a kind of absolutely uncontrollable region. The

existence of an absolutely uncontrollable region may be an

interpretation of failures of some experiments for cardiac

defibrillation by applying local periodic pacing.2429

It is em-

phasized that backfiring is a sufficient but not a necessary

condition for an absolutely uncontrollable parameter region.Other instabilities occurring in wave propagation, e.g., wave-

breakup induced by alternans, can also determine the abso-

lutely uncontrollable parameter domain.

It is possible that some cardiac fibrillation may appear

right in the parameter domain of absolute uncontrollability; it

is interesting to seek realistic means to remedy the condition

b for improving the local pacing method. In the following,

we describe a convenient method to aid local pacing to con-

trol backfiring DT. Our method is to introduce an additional

gradient field term into Eq. 5,4043

u

t =

1

u1 uu v + b

a + 2u +xFcos t E

u

x. 9

It is argued that in some physical situations, the gradient

term in Eq. 9 can be easily realized. For instance, if the

variables of the system represent some densities of irons, the

gradient term appears when we apply a certain electric po-

tential difference on the left and right boundaries. In this

case, the gradient term is realized also conveniently by

boundary actions. In Fig. 10a, we compare the DT control-

lable regions of Eq. 9 in the - plane with different gra-

dient intensities. It is shown that the gradient term improves

the control effect surprisingly well. The physical mechanism

underlying this striking improvement can be clearly under-

stood. The essential influence of the gradient force to the

wave propagation in excitable media is to make defects and

waves move along the gradient direction Fig. 10b. This

influence has two effects simultaneously. On one hand, it

reduces the frequency of spiral and turbulent waves of the

system due to the Doppler effect Fig. 10c. On the other

hand, it makes the pacing-generated ordered waves leave the

pacing boundary faster along the gradient direction, and this

allows the system to keep 1 : 1 frequency response against

faster pacing and to generate controlling waves with largerfrequency. The former effectively moves down the low con-

trollable frequency threshold while the latter considerably

moves up the high-frequency boundary of the controllable

region. These two tendencies together effectively push back-

firing threshold B to larger Fig. 10d, and makes the

large absolutely uncontrollable parameter area controllable.From Fig. 10 it is obvious that the gradient coupling can

greatly improve DT control with local pacing. Nevertheless,

the applicability of this gradient term to realistic cardiac

defibrillation is still an open problem. First, it is not clear

how to apply the gradient coupling in cardiac systems. More-

over, it is commonly believed that cardiac fibrillation is, to a

large extent, related to DT induced by alternans

instability,20,36

and the effect of gradient force on suppression

of alternans DT is another open problem.

V. OTHER FACTORS INFLUENCING

THE EFFECTS OF LOCAL PACING

Besides the pacing frequency and pacing area discussed

in previous sections, there are some other factors influencing

the effects of the local pacing method. First, the pacing am-

plitude must be a relevant factor. It is trivial to conclude that

a pacing cannot stimulate any controlling waves and thus

cannot achieve any turbulence suppression if the pacing am-

plitude is too small. There exists a threshold for the ampli-

tude, and the periodic local pacing can generate controlling

waves only if the pacing amplitude is larger than this critical

value. Another less trivial effect of pacing amplitude is that

changing amplitude can change the 1:1 response region of

Fig. 1, and can indirectly change the controllable parameterregion. We again use the Br model to illustrate this effect. In

FIG. 10. Numerical results of Eq. 9 with F= 2. a The controllable zonewithout circles, E= 0 and with disks, E=0.35; stars, E=0.6 gradient field.The backfiring thresholds B for different Es are shown in the figure with

vertical lines. b The velocity of pacing-generated waves plotted vs E for=0.13. c The principal frequency of turbulent waves plotted vs E for =0.13. d B vs E for Eq. 9. Figure from Ref. 12.



8/9

Fig. 11a, we do the same as Fig. 1 with different pacing

amplitudes applied. It is interesting to see that increasing the

pacing amplitude can considerably increase the highest fre-

quency of the controlling waves. In Fig. 11b, the highest

response frequency is plotted against pacing amplitude. The

highest response frequency increases with amplitude for

small F, and saturates to a certain value for large F. With this

picture we can understand why increasing the pacing ampli-

tude can effectively improve the pacing results for relativelysmall F, while as the amplitude is sufficiently large, a further

increase of F no longer changes the control effect anymore.

In all the previous discussions, we used only a cosine

signal for the local pacing. It has been found that a different

form of pacing signal can also considerably change the pac-

ing results. This is because that different pacing form can

make the highest frequency of the controlling waves differ-

ent. And this again influences the effect of turbulence sup-

pression indirectly. This problem has been discussed in Ref.

9 in detail.

One more relevant factor of turbulence suppression with

local pacing is the system size. Because some instabilities,

such as alternans and backfiring instabilities, emerge only for

sufficiently large systems, changing the system size may also

essentially change the control results. As an example, we

compute the input-output frequency response curves for the

1D Luo-Rudy model and adopt the local pacing strategy of

Ref. 9. It is shown that the response behavior for small sys-

tem size Fig. 12a, and also Fig. 1 in Ref. 9 is considerably

different from that for large system size Fig. 12b. From

these response behaviors, one concludes that turbulence sup-

pression with local pacing in small systems is easier than in

large systems, a conclusion familiar to experimentalists in

this field.

There are certainly some other factors influencing the

effects of local pacing. To our understanding, the factor of

input-output frequency relation is the most important. The

effects of many other factors can often influence the controlresults by modifying the input-output frequency relations,

similar to what is shown in Figs. 11 and 12.

VI. CONCLUSION

In this paper, we present a survey of local pacing control

in suppressing turbulence in excitable media. We show that

the local pacing method can effectively suppress WT and DT

induced by near-field instabilities associated with spiral

tips. The local pacing method fails in DT suppression when

the turbulence occurs via some instabilities in wave propa-

gation, such as backfiring and alternans instabilities. In the

latter case, some auxiliary method of gradient coupling is

suggested to aid the local pacing to perform DT suppression.

These understandings are expected to be useful in develop-

ing practical methods for cardiac defibrillation. It was argued

that WT may be one of the major causes of fibrillation.16,44

The above controllability prediction of WT may be of help in

searching low-amplitude local pacing methods to perform

FIG. 11. a The same as Fig. 1 with different pacingamplitudes applied. a =1.1, b =0.19,=0.03. It is ob-

served that different pacing amplitudes may result in

different input-output relations. b The highest outputfrequency vs pacing amplitude. When F2.2, the high-

est output frequency saturates to a certain constant.

FIG. 12. The same as Fig. 1 with the 1D Luo-Rudy model computed. The system dynamics and parameters are given in Ref. 9 and the response curves areobtained for different system sizes. a L =0.84 cm. Figure from Ref. 9. b L =8.4 cm.

015107-8 Cao et al. Chaos 17, 015107 2007


9/9

defibrillation in this case. On the other hand, propagating

waves breaking up to DT often occur for steeply sloped res-

titution curves,45,46

where low-amplitude pacing may not

succeed in cardiac defibrillation. We suggest that in these

cases one has to search some effective auxiliary methods

cooperating with the low-amplitude pacing to perform car-

diac defibrillation. The methods reviewed here focus on WT

and DT suppression only in some simple models of excitable

media. In the future, we will investigate these methods inmodels representing more realistic electric and chemical ac-

tivities of heart systems, and develop applications of these

methods in direct cardiac defibrillation experiments.

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