Georg Gottwald, Alain Pumir and Valentin Krinsky- Spiral wave drift induced by stimulating wave...

8/3/2019 Georg Gottwald, Alain Pumir and Valentin Krinsky- Spiral wave drift induced by stimulating wave trains

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nario and is robust in the sense that it does not depend on the

particular0

values of the control parameters, but is instead a

generic) situation.

For

meandering spirals, again we are faced with nonzero

drift5

velocities for stimulation periods larger than Tg

s seeA

Figs. 2 a1

2 cH

,h case C . IfTs is taken to be the time between

two

consecutive points of equal phases of the freely mean-

dering5

spiral, i.e., Tg

s is

associated with the smaller radius

R

S ,h and R

is

taken to be the smaller of the two radii R

RS ,h formulas 9

and1

10 are1

in good agreement with the

numerical simulation Fig. 6 .Free

meandering itself is a strongly nonstationary pro-

cessH where the spiral wave tip moves periodically into its

own2

refractory tail. On the smaller circle the excitability is

high

and the spiral curls. It will meet its own refractory tail

and1 moves into an area with low excitability where it con-

tinues

to move on a large circle with RL until4 an inhibitor-

free hole opens and the wave tip can freely curl again with its

growing) velocity. Hence, meandering is basically due to the

fact

that the spiral wave periodically changes the excitability

of2 the medium it moves through by its own inhibitor. Peri-

odic2

stimulations such as the emission of wave trains drasti-

callyH change the excitability of the active medium and dis-

turb

this inherent periodicity of the spiral wave and impose

their

own periodicity. This suppresses and transforms the

meandering and a steady drift will be established. Barkley20T

has

identified the Euclidean symmetry group as being essen-

tial

for the onset of meandering. The invariance under the

action1 of the Euclidean group, rotation, reflection, and trans-

lation|

leads to a reduction of the original system to a set of

five7

ordinary differential equations. In this system, the onset

of2

meandering is described by a Hopf bifurcation. The peri-odic2 stimulation by wave trains does break the symmetry and

destroys5

the meandering.

VII.k

DISCUSSION

W

e have investigated drift of spiral waves induced by a

periodic0 wave train which is launched close to the core. The

surprisingA result of nonzero drift velocities for stimulating

periods0

larger than the period of the freely rotating spiral has

beenD

observed. We note that for stimulations far from the

coreH the spiral wave arms shield the core and will prevent a

drift5

of the core, but once a drift has been induced by a

stimulationA

close to the core, this drift will be stationary. This

seemsA to contradict the conclusions of former work.4,5E

We

found

that a transitory phase caused by an interaction of the

wave3 train with the refractory tail of the spiral wave is re-

sponsibleA

for this new phenomenon. Essentially this transi-

tory

phase introduces a larger spiral wave period and hence

allows1 for stimulating periods larger than the original spiral

wave3 period Ts . A phenomenological model was established

which3 quantitatively describes the drift velocities for moder-

ately1 sparse spirals. Initially meandering spirals were also

stimulatedA and we observed a steady drift along a straight

line as in the nonmeandering cases. The stimulation by wave

trains

does dominate the inherent periodic nature of mean-dering5

spirals. We could again describe the drift with our

phenomenological0

formula.

The new result of nonzero drift for stimulation periods

lar|

ger than Tg

s was3 mainly due to two separate factors: ( i

l

),X

we3 leave the parameter region of extreme densities and ( i il

),X

we3 stimulate close to the core. In the remainder we comment

on2 these two issues and put them into a perspective from a

clinicalH point of view.

Considering

stimulations close to the core is relevant

from a cardiological point of view. Here typical wave veloci-

ties

are of the order 10 cm/s and typical time scale is of the

order2 of 0.2 s, which implies a typical wavelength of 2 cm,

which3

is not too small if compared with the heart size. In thiscase

H

obvious general collision arguments, as employed in the

aforementioned1 classical theories, do not apply. Neverthe-

less,|

we saw that for the case of extremely sparse spirals

there

is no drift for Tg

fv T

g

s also1 in the case where the source

of2

stimulation is close to the core caseH

A in Figs. 2 a1

2 cH

.

After

one initial collision and the resulting displacement of

the

core, the spiral wave arm develops and starts shielding

the

core. For moderately sparse spirals, though, we do ob-

serveA

nonzero steady drift velocities for Tfv Ts . This brings

us4 to the problem of the density of spirals.

TheF

density

of2 a spiral can be defined as the ratio of

the

width of the spiral wave arm and the wavelength of the

FIG. 7. Trajectories of the tip of a meandering spiral. a Freely meandering

spiral with two clear defined radii.b

Same under the influence of a stimu-

lating periodic wave train coming from the lower boundary. Parameters are

t

hoseo fc aseCi nF ig.2 .

493Chaos, Vol. 11, No. 3, 2001 Spiral wave drift

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Georg Gottwald, Alain Pumir and Valentin Krinsky- Spiral wave drift induced by stimulating wave...

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