16640433

Wave propagation and suppression in excitable media with

holes and external forcing

J.I. Ramos

Room I-320-D, E.T.S. Ingenieros Industriales, Universidad de Malaga, Plaza El Ejido, s/n, 29013 Malaga, Spain Accepted 5 March 2001

Abstract

The propagation and suppression of spiral waves in inhomogeneous excitable media is studied numerically. The inhomogeneities correspond to either holes whose size and location are varied or constant external forcing on the activator's reaction rate. For the case of a single hole and localized external forcing, it is found that the spiral wave is robust, although it may break up into two branches which reconnect with each other after the wave overcomes either the hole or the region where the external forcing is applied. When ®ve holes are presented but there is no external forcing or the external forcing is localized, the spiral wave is robust and exhibits high concentration of the activator between holes and corners when it approaches and interacts with the holes. In the presence of several holes and external forcing across the excitable medium, the spiral wave is suppressed and the activator exhibits a breathing behaviour characterized by fronts that propagate towards the boundaries of the excitable medium, and complex patterns when the front is located near the holes. Ó 2002 Elsevier Science Ltd. All rights reserved.

1. Introduction

Spiral waves are ubiquitous in two-dimensional excitable media [1] and are amongst the most prominent examples of spontaneous formation of spatiotemporal patterns in macroscopic systems driven far from thermodynamic equilibrium. They occur as rotating waves of chemical activity in the Belousov±Zhabotinsky (BZ) reaction, circulating waves of neuromuscular activity in the cardiac muscle tissue, etc. For example, it is now believed that some forms of ventricular tachycardia arise when a spiral wave of action potential is generated and drives the ventricle at a much faster rate than the normal sinus rhythm, and that ventricular ®brillation, i.e., a more spatially and temporally disorganized state, arises from the subsequent breakdown of this spiral wave into multiple drifting and meandering spiral waves [2].

The simplest transition in the spiral wave family is from a stationary, periodic and stable spiral wave to a quasi-periodic meander, and this transition corresponds to a secondary Hopf bifurcation; however, more complicated phe-nomena such as hypermeandering [3] and breakup are not yet well understood, and most studies to date have dealt with the breakup regime. For example, Qu et al. [4] used the Luo±Rudy action potential model, carried out a two-dimen-sional simulation of a cardiac tissue model, and found that a transition from quasiperiodic meandering to sustained chaotic meander, and then to persistent breakup occurs. In the case of chaotic meander, chaos was found to be localized in the spiral wave core area.

The control and suppression of spiral waves is of paramount importance for the treatment of potentially fatal cardiac arrhythmias. Conventional methods for the elimination of spiral waves in cardiac tissue include de®brillation which is achieved by the direct activation of most of the cardiac tissue by a single electric shock but may cause tissue damage, overdrive local pacing of an arrhythmia, parametric resonance drift of spiral waves whereby the spiral wave is www.elsevier.com/locate/chaos

*_{Tel.: +34-95-2131402; fax: +34-95-2132816.} E-mail address:[email protected] (J.I. Ramos).

0960-0779/02/$ - see front matterÓ 2002 Elsevier Science Ltd. All rights reserved. PII: S 09 6 0- 07 7 9 ( 01 ) 0007 3 - X

drifted and can be eliminated at the boundary of the medium provided that certain properties of the medium are varied with the period of rotation of the spiral wave, etc. [5]. However, since electrical forcing can modify the period of the spiral waves, the determination of the resonant frequency may require a feedback loop control mechanism.

A variant of the BZ reaction that uses a photosensitive complex has been used in the past to study the e€ect of external stimulation on spiral wave dynamics and control, for, under illumination, the catalytic complex becomes photochemically excited and releases the inhibitor of the reaction; therefore, the local excitation of the medium depends on the applied light intensity. Braune and Engel [6] externally forced a light-sensitive BZ medium at a ®xed frequency and employed feedback-controlled forcing to control the spiral wave dynamics.

The control and suppression of spiral waves can also be achieved by the introduction of small spatial inhomo-geneities in the medium [7], external forcing [6], the interaction of the wave with the boundaries of the domain [8], convective ¯ow ®elds [9,10], anisotropy [11], modulation, etc. The presence of parameter gradients can lead to drift and subsequent annihilation of the spiral wave at the boundaries of the domain, while the geometry and size of the inhomogeneities can signi®cantly in¯uence the wave dynamics in excitable media. In the presence of external forcing, it has been observed that the spiral centre drifts primarily in the direction of the applied electric ®eld, but there is a smaller component of the drift that is perpendicular to the ®eld and whose sign depends on the chirality of the spiral wave.

Convective ¯ow ®elds in excitable media can yield a wide variety of patterns depending on the boundary conditions that the velocity satis®es at the boundaries, e.g., slip or no-slip, irrotationality and/or solenoidality. If the ¯ow is not solenoidal, the divergence of the velocity ®eld introduces an additional source term in the equations for the activator and the inhibitor which corresponds to volumetric expansion. Moreover, the ¯ow ®eld strains and stretches the spiral wave, and may result in local suppression or extinction depending on the magnitude and spatial frequency of the imposed velocity ®eld [9,10]. Anisotropic di€usion tensors for the activator and inhibitor may result in elongated spiral waves or stripes [11].

In this paper, a numerical study of the e€ects of inhomogeneities on spiral wave propagation and suppression in two-dimensional excitable media is presented. The study is based on the BZ reaction and considers media with holes and with or without inhomogeneities in the reaction or source term for the activator.

2. Governing equations

The numerical study presented here is based on the BZ reaction which is often modelled by the Oregonator equations [12,13] and may be written as

ou

otˆ r …Duru† ‡Fu; …1†

ov

otˆ r …Dvrv† ‡Fv; …2†

wheretis time,uandvdenote the concentrations of the activator and the inhibitor, respectively,DuandDvare the

di€usivity tensors foruandv, respectively, and the source terms in Eqs. (1) and (2) can be written as

Fuˆ1 u

u2 _fv‡/†u q

u‡q

; Fvˆu v; …3†

where / represents the light-induced ¯ow of Br [14],ˆ0:01,f ˆ1:4 andqˆ0:002, and are the same as those employed in the BZ model. When light, i.e., external forcing, is not applied to the excitable medium,/ˆ0.

In this paper, it is assumed thatDuˆIandDuˆ0:6I, whereIis the unit tensor of second rank, and it is known that,

for these di€usivity tensors, the Oregonator model has spiral wave solutions if homogeneous Neumann boundary conditions are applied at the boundaries, and there are no holes and/ˆ0.

Eqs. (1) and (2) were solved in the spatial domainXˆXT nSXi, whereXT ˆ ‰ Lx=2;Lx=2Š ‰ Ly=2;Ly=2Šwith LxˆLyˆ7:5, andXidenotes theith (rectangular) hole, subject to homogeneous Neumann boundary conditions on all

the boundaries and to the following initial condition inX:

uˆ0for 0<h<0:5; uˆq…f‡1†=…f 1†elsewhere; …4†

wherehis the angle with respect to the origin of coordinates measured counterclockwise from the positivex-axis. In the absence of holes and light-induced ¯ow of Br , this initial condition results in the formation of a spiral wave which rotates counter-clockwise. It should be pointed out that some authors, e.g., Mu~nuzuri et al. [14] and references therein, have previously used Eq. (3) with /6ˆ0to simulate inhomogeneities/obstacles in excitable media; in this paper, however,/6ˆ0indicates that a light-induced ¯ow of Br is employed, whereas the term obstacle is used to denote the holes.

When there is a single hole, this is speci®ed by the coordinates of its lower left corner, i.e.,…xi;yi† 2XT, and its

lengthslxandlyin thex- andy-directions, respectively, wherelx<Lxandly<Ly. When several holes are considered,

they are identi®ed by the locations of their lower left and upper right corners.

Eqs. (1) and (2) were solved numerically by means of an implicit, time-linearized, second-order accurate (in both space and time) ®nite di€erence method [15]. This method factorizes the elliptic equations that result upon discreti-zation of time at each time level into two one-dimensional boundary value problems and employs an iterative tech-nique to account for the approximate factorization errors. Computations were performed on a 10001000 point equally spaced mesh and a time step of 10 4_{. Computations were also performed with equally spaced meshes of} 200200 and 500500 points and di€erent time steps in order to insure that the results were independent of both the number of grid points and the time step. In the next section, some sample results obtained with 10001000 point equally spaced meshes and a time step equal to 10 4_{are presented; however, only 100100 points are illustrated in the} graphs.

3. Presentation of results

In this section, some sample results corresponding to one and ®ve holes and with and without light-induced ¯ow of Br are presented. For /ˆ0and a single hole characterized by …x1;y1† ˆ …47d;47d†, and lxˆlyˆ7d where

dˆ15=101, i.e., a square hole located almost in the centre ofXT, which corresponds to Case 1 of Table 1, it has been

observed that the tip of the spiral wave is ¯attened when the spiral wave approaches and comes in contact with the hole boundary and may exhibit a corner. A similar ®nding was also observed for/ˆ0and a single hole characterized by …x1;y1† ˆ …37d;47d†,lxˆ37dandlyˆ7d, i.e., a rectangular hole with the largest side in the x-direction, which

cor-responds to Case 2 of Table 1, but the spiral wave was thicker and shorter than that for Case 1; moreover, Case 2 also showed the corners in the spiral wave tip when the wave turned around the corners of the hole. In both Case 1 and Case 2 with/ˆ0, the spiral wave rotates round the hole and the values ofuandvat…25d;20d†and…20d;20d†are periodic functions of time characterized by spikes ofu; the peak valuesuare not constant and the separation between two successive spikes ofuis shown as a function of the location and length of the hole in Table 1.

Table 1 indicates that the separation between two successive pulses or spikes ofuincreases slightly aslxis increased

for square holes; the separation between pulses also increases aslxorlyis increased. Since the centre of the domainXTis

located at…50:5d;50:5d†, the results presented in Table 1 also indicate that the separation between pulses ofuis not very sensitive to the eccentricity of the hole provided that its size is small. For Cases 9 and 11 with/ˆ0, i.e., for holes o€ the centre ofXT, it has been found that the hole may break the spiral wave into two branches which reconnect to each other

as soon as the spiral wave overcomes the hole.

Calculations have also been performed with a single hole and with light-induced ¯ow of Br , i.e., /ˆ1 in ‰47d;47dŠ ‰54d;54dŠand/ˆ0elsewhere, for the same locations and sizes of the holes as those shown in Table 1. Since the region of light-induced ¯ow coincides with or is included in the hole for the cases 1, 5, 6, 7, 10, 12, 13 and 14 shown in Table 1, this ¯ow does not a€ect the results for these cases and this has been veri®ed numerically. Moreover, in cases 2, 3 and 4, the region where/6ˆ0is larger than the hole, whereas, in cases 9 and 11, the hole is o€ the centre ofXT

and the e€ects of the light-induced ¯ow are not felt unless the spiral wave is suciently close to the region where/6ˆ1. For Case 11 with/ˆ1 in‰47d;47dŠ ‰54d;54dŠand/ˆ0elsewhere, it was found that the tip of the spiral wave was ¯attened when the wave approached the region where/ˆ1 and exhibited corners analogous to those discussed

Table 1

Separation between the pulses in the activator's concentration u as a function of …x1;y1† ˆd…m;n†, lxˆdM, lyˆdN at …x;y† ˆd…25;20†for a single hole and/ˆ0

Case 1 2 3 4 5 6 7 8 9 10 11 12 13 14

(m,n,

M,N) (47,47,7,7) (49,49,3,3) (49,47,3,7) (47,49,7,3) (47,37,7,37) (37,47,37,7) 37,37)(37,37, (37,47,7,7) (76,47,7,7) (47,37,7,7) (47,76,7,7) (42,42,7,17) (42,47,17,7) (47,42,7,17) Period 1.6216 1.6757 1.6216 1.6216 1.9459 1.9459 2.4324 1.7838 1.6216 1.7838 1.6216 1.6757 1.5676 1.5135

above for Case 1 with/ˆ0, when the spiral wave rotated around the light-induced ¯ow region. Moreover, the spiral wave was anchored to the light-induced region and was distorted by and broken by the hole; however, the two broken branches reconnected with each other as soon as the inner or outer edge of the spiral wave overcame the hole.

The separation between the pulses inuwhen there is a single obstacle and a light-induced ¯ow region is presented in Table 2, and a comparison between Tables 1 and 2 indicates that the presence of light-induced ¯ow reduces the sep-aration between successive pulses of the activator concentration.

Some sample results illustrating the interaction of spiral waves in excitable media with ®ve holes but without light-induced ¯ow of ions are presented in Figs. 1 and 2. Fig. 1 corresponds to holes X1ˆ ‰…9d;9d†;…24d;24d†Š, X2ˆ ‰…76d;9d†;…91d;24d†Š, X3ˆ ‰…9d;76d†;…24d;91d†Š, X4ˆ ‰…76d;76d†;…91d;91d†Š and X5ˆ ‰…43d;43d†;…58d;58d†Š, indicates that the tip of the spiral wave is ¯attened when it approaches the holeX5which is located near the centre of XT, and presents indentations when approaching the other four obstacles which are located near the corners ofXT; in

some cases, the holes break the wave into two branches that reconnect with each other after the wave overcomes the hole. The remarkable feature of the results presented in Fig. 1 is that the spiral wave is robust in the sense that it is not destroyed by the presence of the holes. The separation between two successive spikes ofufor the parameters of Fig. 1 is 1.7341.

Table 2

Separation between the pulses in the activator's concentration u as a function of …x1;y1† ˆd…m;n†, lxˆdM, lyˆdN at …x;y† ˆd…25;20†for a single hole and/ˆ1 in…47d;47d† …54d;54d†

…m;n;M;N† …49;49;3;3† …49;47;3;7† …47;49;7;3† …76;47;7;7† …47;76;7;7†

Period 1.6185 1.6185 1.6185 1.5607 1.5607

Fig. 1. Concentration of the activatoruat (from left to right, from top to bottom)tˆ100.4, 100.6, 100.8, 101, 101.2, 101.4, 101.6, 101.8 and 102 for domain with ®ve holes X1ˆ ‰…9d;9d†;…24d;24d†Š, X2ˆ ‰…76d;9d†;…91d;24d†Š, X3ˆ ‰…9d;76d†;…24d;91d†Š, X4ˆ ‰…76d;76d†;…91d;91d†ŠandX5ˆ ‰…43d;43d†;…58d;58d†Š, and/ˆ0.

Fig. 2 corresponds to holes X1ˆ ‰…9d;34d†;…24d;49d†Š, X2ˆ ‰…76d;34d†;…91d;49d†Š, X3ˆ ‰…9d;51d†;…24d;66d†Š, X4ˆ ‰…76d;51d†;…91d;66d†ŠandX5ˆ ‰…43d;43d†;…58d;58d†Š, and shows very complex patterns when the spiral wave approaches the holes which are concentrated in a relatively narrow horizontal band. Of particular interest are the second, third and seventh frames of Fig. 2 which show a high concentration of the activator between two holes; this high-concentration region is not connected with the main arm of the spiral wave. Fig. 2 also shows that the wave is displaced outwardly when it approaches or interacts with a hole. The separation between two successive spikes of the activator concentration is 1.7341 for Fig. 2.

Similar results to those presented in Figs. 1 and 2 have also been observed for X1ˆ ‰…34d;9d†;…49d;24d†Š, X2ˆ ‰…34d;76d†;…49d;91d†Š,X3ˆ ‰…51d;9d†;…66d;24d†Š, X4ˆ ‰…51d;76d†;…66d;91d†Š andX5ˆ ‰…43d;43d†;…58d;58d†Š which roughly corresponds to the domainXof Fig. 2 rotated 90°; these holes resulted in a separation between successive spikes ofuequal to 1.7341. ForX1ˆ ‰…34d;34d†;…49d;49d†Š,X2ˆ ‰…34d;51d†;…49d;66d†Š,X3ˆ ‰…51d;34d†;…66d;49d†Š, X4ˆ ‰…51d;51d†;…66d;66d†ŠandX5ˆ ‰…43d;43d†;…58d;58d†Š, the ®ve holes are so closed to the centre ofXT that the

spiral wave preserves its identity, exhibits sharp corners when it comes in contact with a hole, and exhibits almost rectangular tongues between two holes, and these tongues are connected to the main wave; moreover, in this case, the spiral wave does not break up into two smaller ones and the separation between two successive spikes of the activator concentration is 2.2543 which is much longer than those corresponding to Figs. 1 and 2.

For X1ˆ ‰…19d;19d†;…24d;24d†Š, X2ˆ ‰…76d;19d†;…81d;24d†Š, X3ˆ ‰…19d;76d†;…24d;81d†Š, X4ˆ ‰…76d;76d†;

…81d;81d†ŠandX5ˆ ‰…43d;43d†;…58d;58d†Š, the thickness of the spiral wave is larger than the sizes of the holes Xi, iˆ1;2;3;4, but smaller than those ofX5; as a consequence, the tip of the spiral wave ¯attens as it approachesX5, and experiences some outward displacement when approaching the other four holes, and the separation between two successive spikes of the activator concentration is 1.7341.

The e€ects of ®ve holes on the propagation of spiral waves in excitable media with light-induced ¯ow of ions, i.e., /6ˆ0, have also been investigated and some sample results are presented in Figs. 3±5. The domain of Fig. 3 coincides Fig. 2. Concentration of the activatoruat (from left to right, from top to bottom)tˆ100.4, 100.6, 100.8, 101, 101.2, 101.4, 101.6, 101.8 and 102 for X1ˆ ‰…9d;34d†;…24d;49d†Š, X2ˆ ‰…76d;34d†;…91d;49d†Š, X3ˆ ‰…9d;51d†;…24d;66d†Š, X4ˆ ‰…76d;51d†;…91d;66d†Š andX5ˆ ‰…43d;43d†;…58d;58d†Š, and/ˆ0.

with that of Fig. 1 and/was set equal to 0.5 in the rectangular region‰46d;54dŠ ‰ Ly=2;Ly=2Š, i.e., across the domain

in the y-direction, and to zero elsewhere. This ®gure clearly indicates that the spiral wave has been annihilated or suppressed by the light-induced ¯ow, exhibits clearly the location of the ®ve holes and symmetry with respect to thex -andy-axes, and indicates that the excitable medium presents a breathing behaviour along thex-axis characterized by a high concentration ofuthat propagates fromxˆ0towards the boundaries located atxˆ Lx=2 andxˆLx=2; once the

activator front reaches these boundaries, there is little chemical activity in the domain until the phenomenon just de-scribed is repeated in a periodic fashion with a period equal to about 4.0462 which is larger than that when/ˆ0 . A similar behaviour to the one just described was also observed in the same domain but with/ˆ1 in the rectangular region‰46d;54dŠ ‰ Ly=2;Ly=2Š.

For the same domain as in Fig. 1 and/ˆ0:5 or/ˆ1 in‰ Lx=2;Lx=2Š ‰46d;54dŠ, the results correspond to those

of Fig. 3 rotated 90°, and thus con®rm the accuracy of the numerical solutions reported in this paper.

For the same domain as in Fig. 1 and/ˆ0:5 or/ˆ1 in…46d;54d† …34d;86d†, a thick spiral wave was observed. The tip of this wave was ¯attened on account of the holeX5located near the centre ofXand the light-induced ¯ow, the wave presented indentations when it approached the other four holes or the light-induced ¯ow region, and the sepa-ration between two successive spikes ofuwas about 2.2543. The wave broke up into two branches that reconnected to each other after overcoming the hole, and was very robust. A similar behaviour was also observed for the same domain as in Fig. 1 and/ˆ0:5 or/ˆ1 in…34d;86d† …46d;54d†. It was also found that the regions where/P0:5 behaved as if they were obstacles or holes, and, therefore, these regions acted as suppressors or annihilators of chemical activity. Since spiral wave suppression was only observed when/ˆ0:5 or/ˆ1 along either a horizontal or a vertical band that extends across the whole domainX, calculations were performed with/ˆ1 in‰ Lx=2;Lx=2Š ‰46d;54dŠin order

to determine the e€ects of the location and size of the ®ve holes on wave propagation in excitable media, and some sample results are presented in Figs. 4 and 5. Fig. 4 corresponds to X1ˆ ‰…9d;34d†;…24d;49d†Š, X2ˆ ‰…76d;34d†;

…91d;49d†Š, X3ˆ ‰…9d;51d†;…24d;66d†Š, X4ˆ ‰…76d;51d†;…91d;66d†Š and X5ˆ ‰…43d;43d†;…58d;58d†Š, i.e., the same Fig. 3. Concentration of the activatoruat (from left to right, from top to bottom)tˆ100.4, 100.6, 100.8, 101, 101.2, 101.4, 101.6, 101.8 and 102 for domain with ®ve holes X1ˆ ‰…9d;9d†;…24d;24d†Š, X2ˆ ‰…76d;9d†;…91d;24d†Š, X3ˆ ‰…9d;76d†;…24d;91d†Š, X4ˆ ‰…76d;76d†;…91d;91d†ŠandX5ˆ ‰…43d;43d†;…58d;58d†Š, and/ˆ0:5 in‰46d;54dŠ ‰ Ly=2;Ly=2Šand/ˆ0elsewhere.

domain as in Fig. 2, and shows a high value ofubetween the holes located on the right. This value propagates in thex -andy-directions and around ®rst the holes on the right, then the hole in the middle and, ®nally, around the holes on the left, and such a propagation occurs in a symmetric manner aboutyˆ0but not aboutxˆ0. Once a high concentration of the activator has surrounded the ®ve holes, the wave front propagates in the y-direction until it reaches the boundaries located atyˆ Ly=2 andyˆLy=2, and this is followed by a period of relative quietness as indicated by the

last four frames of Fig. 4. The separation between two successive spikes ofuwas found to be 4.0462 for Fig. 4. For X1ˆ ‰…34d;9d†;…49d;24d†Š, X2ˆ ‰…34d;76d†;…49d;91d†Š, X3ˆ ‰…51d;9d†;…66d;24d†Š, X4ˆ ‰…51d;76d†;

…66d;91d†ŠandX5ˆ ‰…43d;43d†;…58d;58d†Š, i.e, when the ®ve holes are located in a narrow vertical line alongxˆ0, high values ofupropagate from near… Lx=2;0†and…Lx=2;0†towardsxˆ0until they reach the holeX5; later on, the

front propagates in they-direction but becomes distorted by the presence of the other four holes and creates a wake between the holes and the upper or lower boundaries of the domain.

For X1ˆ ‰…34d;34d†;…49d;49d†Š, X2ˆ ‰…34d;51d†;…49d;66d†Š, X3ˆ ‰…51d;34d†;…66d;49d†Š, X4ˆ ‰…51d;51d†;

…66d;66d†ŠandX5ˆ ‰…43d;43d†;…58d;58d†Š, the ®ve holes are very close to each other and the high values ofuoccur along vertical lines on the periphery of the leftmost and rightmost holes, and this high concentration propagates to-wardsxˆ Lx=2 andxˆLx=2. This front eventually surrounds the ®ve holes, shows rectangular tongues in the spacing

between the uppermost and lowermost holes, and propagates in the y-direction until it reaches the boundaries

yˆ Ly=2 and yˆLy=2. Before reaching these boundaries, some chemical activity can be observed in the spacing

between the uppermost and lowermost holes, and no chemical activity is observed between these tongues and they -propagating front.

For X1ˆ ‰…19d;19d†;…24d;24d†Š, X2ˆ ‰…76d;19d†;…81d;24d†Š, X3ˆ ‰…19d;76d†;…24d;81d†Š, X4ˆ ‰…76d;76d†;

…81d;81d†ŠandX5ˆ ‰…43d;43d†;…58d;58d†Š, the results presented in Fig. 5 exhibit similar trends to the ones just de-scribed except that the small holes near the corners ofXdo not a€ect much they-propagating front, and the separation between two successive spikes ofuwas found to be 4.0462.

Fig. 4. Concentration of the activatoruat (from left to right, from top to bottom)tˆ100.4, 100.6, 100.8, 101, 101.2, 101.4, 101.6, 101.8 and 102 for domain with ®ve holes X1ˆ ‰…9d;34d†;…24d;49d†Š, X2ˆ ‰…76d;34d†;…91d;49†Š, X3ˆ ‰…9d;51d†;…24d;66d†Š, X4ˆ ‰…76d;51d†;…91d;66d†ŠandX5ˆ ‰…43d;43d†;…58d;58d†Š, and/ˆ1 in‰46d;54dŠ ‰ Ly=2;Ly=2Šand/ˆ0elsewhere.

It must be noted that/in Eq. (3) reduces the reaction rate foruand that the largest value offvwhen/ˆ0is about 0.308 which is smaller than the values of/, i.e., 0.5 and 1, employed in the calculations presented here. These values of /were selected so as to suppress or annihilate the spiral wave that has been reported in the absence of light-induced ¯ow regions; when such a suppression is local, the main e€ects were to ¯atten the wave's tip or distort the spiral wave. However, when the light-induced region extended across the domain, the results presented in Figs. 3±5 indicate that a global suppression of the spiral wave results.

4. Conclusions

A two-dimensional numerical study of the Oregonator equations in inhomogeneous excitable media has been performed in order to determine the e€ects of holes and external forcing on wave propagation and, in particular, the conditions under which spiral waves can be suppressed. In the absence of external forcing, it has been found that a robust spiral wave propagates through the medium, is distorted by the holes and may even break up into smaller branches which reconnect to each other after the wave overcomes the holes; the thickness of the wave increases as the size or the number of holes increases. When several holes are present, the spiral wave preserves its integrity, i.e., it is robust, and a high concentration of the activator may be observed between holes.

It has also been found that the e€ects of external forcing, which have been accounted for by means of a parameter in the activator's reaction rate, are almost identical to those of holes provided that the external forcing is localized, and the tip of the spiral wave ¯attens as it approaches and interacts with either the holes or the regions were the external forcing is applied.

It has also been found that complete wave suppression can be achieved when the external forcing is applied along a narrow band across the excitable medium. When this is done, very complex patterns, which propagate ®rst from two Fig. 5. Concentration of the activatoruat (from left to right, from top to bottom)tˆ100.4, 100.6, 100.8, 101, 101.2, 101.4, 101.6, 101.8 and 102 forX1ˆ ‰…19d;19d†;…24d;24d†Š,X2ˆ ‰…76d;19d†;…81d;24d†Š,X3ˆ ‰…19d;76d†;…24d;81d†Š,X4ˆ ‰…76d;76d†;…81d;81d†Š andX5ˆ ‰…43d;43d†;…58d;58d†Šand/ˆ1 in‰ Lx=2;Lx=2Š ‰46d;54dŠand/ˆ0elsewhere.

boundaries towards the holes, interact in a complex manner with the regions between holes, and then propagate to-wards the other two boundaries of the excitable medium, are observed. In some cases, islands of high concentrations are observed in the spacings between holes. Although the spiral wave may be suppressed by external forcing in either the presence or absence of holes, the concentrations of both the activator and inhibitor exhibit a periodic breathing be-haviour which has a longer period than in the absence of external forcing and a still longer one than that corresponding to spiral waves propagating in homogeneous excitable media.

Acknowledgements

The research reported in this paper was supported by Project PB97-1086 from the D.G.E.S. of Spain.

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