Spatio-temporal patterns in two-dimensional excitable media subject to Robin boundary conditions

Transcription

1 Applied Mathematics and Computation 146 (2003) Spatio-temporal patterns in two-dimensional excitable media subject to Robin boundary conditions J.I. Ramos E.T.S. Ingenieros Industriales, Universidad de Malaga, Room I-325-D, Plaza El Ejido, s/n Malaga, Spain Abstract A numerical study of spiral wave propagation in two-dimensional excitable media subject to homogeneous Robin boundary conditions is presented, and it is shown that, depending on the magnitude of the transfer coefficient, periodic, almost periodic and nonperiodic dynamics may be observed. For transfer coefficients equal to or smaller than one, spiral waves almost identical to those obtained with homogeneous Neumann boundary conditions are observed, whereas arms of spiral waves, layers along the boundaries of the domain, break-up, attachment to and detachment from boundaries, islands and complex spatio-temporal patterns result when the transfer coefficients for both the activator and inhibitor are larger than one along either the top and bottom boundaries or the left and right boundaries. When the transfer coefficients are larger than one along all the boundaries of the domain, it has been found that there is a layer of high activator s concentration along the boundaries and an arm of spiral waves may appear and interact with the boundary layers, and the dynamics of the excitable medium is complex and nonperiodic. Ó 2002 Elsevier Inc. All rights reserved. Keywords: Spiral waves; Wave breakup; RobinÕs boundary conditions; Wave attachment; Boundary layers 1. Introduction Although spiral waves have been observed in many chemical and biological systems [1 4] as well as in numerical solutions of the reaction diffusion address: jirs@lcc.uma.es (J.I. Ramos) /$ - see front matter Ó 2002 Elsevier Inc. All rights reserved. doi: /s (02)

2 56 J.I. Ramos / Appl. Math. Comput. 146 (2003) equations that model such systems, most of the two-dimensional studies on excitable oscillatory media performed to date have been concerned with the spiral tip dynamics, the effects of convective flow fields on wave distortion and break-up [5 7], and forcing by means of, for example, electrical fields [8] and illumination [9], anisotropy [10], the presence of holes [11] and time-periodic modulation [12] due to their importance in biological systems, catalysis, cardiology, pattern formation, etc. Spiral waves have been found to be robust under domain truncation [13]; however, they have been found to undergo several bifurcations and transitions when they are forced by illumination, electric fields, spatial and/or temporal modulations, convective flow fields, etc. For example, Schebesch and Engel [9] used a modified Oregonator model for the light-sensitive Belousov Zhabotinskii (BZ), and showed the presence of two stable counter-rotating spirals at low intensities, whereas the waves were found to undergo a symmetry instability that led to the suppression of one spiral wave at high intensities. Panfilov et al. [8] have demonstrated numerically that spiral waves in cardiac tissue can be eliminated by the application of multiple shocks of external current, and Forstova et al. [14] have shown experimentally that the controlled switching on/off of electric fields can lead to the formation of a variety of complex spatiotemporal patterns. On the other hand, the presence of obstacles affects the propagation of spiral waves in excitable media, although these waves may be distorted by the presence of obstacles [11]. Moreover, spiral wave suppression has been achieved by employing forcing along horizontal or vertical bands in two-dimensional studies of reaction diffusion equations in excitable media [11]. Spiral waves can become unstable in various ways, e.g., they can experience core and far-field instabilities, begin to meander and drift, break up, etc. [15 17]. For example, Wellner et al. [16] considered the drift of stable, meandering spiral waves in a singly diffusive FitzHugh Nagumo medium caused by a weak time-independent gradient or convection in the fast-variable equation and proposed a semiempirical solution to the drift of spiral waves that depends on the period of rotation and the value of the fast variable at the center of the spiral wave. Biktashev and Holden [18] and Zhang and Holden [15] have explained the hypermeander of spiral waves as a chaotic attractor that leads to a motion of the spiral wave tip analogous to that of a Brownian particle. On the other hand, Biktashev et al. [17,19] considered an excitable medium in two dimensions with a cubic nonlinearity given by the FitzHugh Nagumo system and a shear characterized by a velocity field in the x-direction which is either a linear or a sinusoidal function of the y-coordinate, and showed that the shear can distort and then destroy spiral waves. Such breaks were found to result in a chain reaction of spiral wave births and deaths. The velocity fields employed by Biktashev et al. [17] are one-directional and solenoidal, but not irrotational, and they do not satisfy the no-penetration condition at the boundaries of the domain; in fact, these authors used periodicity conditions for the sinusoidal

3 J.I. Ramos / Appl. Math. Comput. 146 (2003) velocity field. Elkin et al. [20] have studied numerically the movement of excitation wave breaks, while Biktashev and Holden [21] analyzed the resonant drift of autowave vortices in 2D and the effects of boundaries and inhomogeneities. Other numerical studies have shown that the propagation of spiral waves in two-dimensional excitable media depends on the applied velocity field, its rotation and its straining, as well as the boundary conditions on the velocity field [5 7]. Most of the numerical studies on spiral wave propagation performed to date have considered homogeneous Neumann boundary conditions. The main objective of this paper is to present a rather extensive numerical study, based on the two-equation Oregonator model, of the effects of homogeneous Robin boundary conditions on spiral wave propagation in two-dimensional excitable media as a function of the transfer coefficients of both the activator and the inhibitor. The transfer coefficients which are analogous to the film transfer coefficient in convection heat transfer, are assumed to be uniform along each boundary although their values may be different on different boundaries. The second objective of the numerical study presented here is to assess the influence of the boundary conditions on the stability, persistence and/or break-up of spiral waves in two-dimensional excitable media. 2. Governing equations The numerical study presented here is based on the BZ reaction which is often modelled by the Oregonator equations [1,22] and may be written as ou ot ¼ d ur 2 u þ F u ; ð1þ ov ot ¼ d vr 2 v þ F v ; ð2þ where t is time, u and v denote the concentrations of the activator and the inhibitor, respectively, d u and d v are the diffusion coefficients for u and v, respectively, and the source terms in Eqs. (1) and (2) can be written as F u ¼ 1 u u 2 fv u q ; F v ¼ u v; ð3þ u þ q where, unless stated otherwise, ¼ 0:01, f ¼ 1:4 and q ¼ 0:002, and are the same as those employed in the BZ model. In this paper, it is assumed that d u ¼ 1 and d v ¼ 0:6, and, for these values, it is known that the two-equation Oregonator model has spiral wave solutions if homogeneous Neumann boundary conditions are applied at the boundaries [10].

4 58 J.I. Ramos / Appl. Math. Comput. 146 (2003) Eqs. (1) and (2) were solved in the spatial domain X ¼½ L x ; L x Š½ L y ; L y Š with L x ¼ L y ¼ 7:5, subject to homogeneous Robin boundary conditions on all the boundaries, i.e., ou on þ k uu ¼ 0; ov on þ k vv ¼ 0; ð4þ where n denotes the coordinate normal to the boundary pointing away from the domain, and k u and k v are constants which could depend on the boundary and are referred to as transfer coefficients because they do play the same role as the film coefficient in convective heat transfer. k u ¼ k v ¼ 0 correspond to homogeneous Neumann boundary conditions, whereas k u ¼ k v ¼1correspond to homogeneous Dirichlet boundary conditions. The initial conditions in X for Eqs. (1) and (2) are u ¼ 0 for 0 < h < 0:5; u ¼ qðf þ 1Þ=ðf 1Þ elsewhere; ð5þ v ¼ q f þ 1 f 1 þ h 8pf ; ð6þ where h is the angle with respect to the origin of coordinates measured counterclockwise from the positive x-axis. This initial condition results in the formation of a spiral wave which rotates counter-clockwise if homogeneous Neumann boundary conditions are applied on all the boundaries [10]. Eqs. (1) and (2) were solved numerically by means of an implicit, time-linearized, second-order accurate (in both space and time) finite difference method [23]. This method factorizes the elliptic equations that result upon discretization of time at each time level, into two one-dimensional boundary value problems and employs an iterative technique to account for the approximate factorization errors. Computations were performed on a point equally spaced mesh and a time step of Computations were also performed with equally-spaced meshes of and points and different 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 point equally-spaced meshes and a time step equal to 10 4 are presented. 3. Presentation of results In this section, some sample results illustrating the effects of the transfer coefficients k u and k v which control the homogeneous Robin s boundary conditions employed in the study are presented. For the sake of brevity, we have introduced K ðl u ; l v ; r u ; r v ; b u ; b v ; t u ; t v Þ where l, r, b and t refer to the values of

5 J.I. Ramos / Appl. Math. Comput. 146 (2003) k at the left, right, bottom and top boundaries, respectively, and the subscripts u and v denote the activator and inhibitor, respectively, and summarized our findings in Table 1. The first five rows and the first entry of the sixth row of Table 1 indicate that a periodic spiral wave propagates in two-dimensional excitable media, when all the components of K are smaller than or equal to one, and, therefore, the influence of k u and k v on the spiral wave propagation is small if these values are uniform along each boundary of the computational domain and equal to or smaller than one. For those values of K for which the dynamics is periodic, the results summarized in Table 1 clearly indicate that both the maximum concentration of the activator and the period of the spiral wave at two different monitoring locations, i.e., ðx; yþ ¼ ð20d; 20dÞ and ð25d; 25dÞ where d ¼ 15=101, are nearly independent of the magnitude of the components of K provided that this magnitude is smaller than or equal to one. It should also be noted that the maximum values of u presented in Table 1 for the periodic solutions, i.e., spiral waves, occur at nearly the same time, i.e., t ¼ 98:95. Table 1 also indicates that periodic spiral waves can also be observed in twodimensional excitable media subject to homogeneous Robin boundary conditions in all the boundaries if either l u ¼ r u ¼ 10, 100, 1000 or b u ¼ t u ¼ 10, 100, 1000, provided that the other parameters of K are equal to one. This implies that, provided that the values of either k u or k v have the same magnitude on opposite boundaries, periodic dynamics are observed. However, almost periodic motions characterized by small differences between successive peaks of either u or v result when either l v ¼ r v ¼ 10, 100, 1000 or b v ¼ t v ¼ 10, 100, 1000, provided that the other parameters of K are equal to one. These almost periodic motions are characterized by phase diagrams of ðu; vþ at ðx; y; tþ ¼ ð20d; 20d; tþ and ð25d; 25d; tþ which show very little spreading about a main closed curve. Table 1 also shows that nonperiodic motions are observed if all the components of ðl u ; l v ; r u ; r v Þ, ðb u ; b v ; t u ; t v Þ or K are larger than one. These nonperiodic behaviour is characterized by phase diagrams which exhibit peaks of different magnitude of u and v, a large number of closed loops, and Fourier spectra that present many peaks. As it will be shown below, this behaviour is associated with the break-up of spiral waves which may form islands, merge with other spiral waves or approach the boundaries of the computational domain where the concentrations of both the activator and the inhibitor may be large. The entry 10 3 K a in Table 1 almost corresponds to homogeneous Dirichlet boundary conditions and indicates that for this value, and also for 10K a and 10 2 K a, nonperiodic motions result; moreover, the differences between the magnitudes of successive peaks of u, the number of loops in the phase diagrams of ðu; vþ at ðx; y; tþ ¼ð20d; 20d; tþ and ð25d; 25d; tþ and the frequency contents of the Fourier spectra of both u and v at ðx; y; tþ ¼ð20d; 20d; tþ and ð25d; 25d; tþ

6 Table 1 Separation (T) between and largest amplitude (u M ) of the pulses in the activator s concentration u as a function of K ðl u ; l v ; r u ; r v ; b u ; b v ; t u ; t v Þ at ðx; yþ¼ð20d; 20dÞ for homogeneous Robin s boundary conditions K K a ¼ð1; 1; 1; 1; 1; 1; 1; 1Þ ð0:5; 1; 1; 1; 1; 1; 1; 1Þ ð1; 0:5; 1; 1; 1; 1; 1; 1Þ ðt ; u M Þ ð1:60; 0:77075Þ a ð1:59; 0:77065Þ a ð1:59; 0:77008Þ a K ð1; 1; 0:5; 1; 1; 1; 1; 1Þ ð1; 1; 1; 0:5; 1; 1; 1; 1Þ ð1; 1; 1; 1; 0:5; 1; 1; 1Þ ðt ; u M Þ ð1:59; 0:77075Þ a ð1:59; 0:77075Þ a ð1:59; 0:77068Þ a K ð1; 1; 1; 1; 1; 0:5; 1; 1Þ ð1; 1; 1; 1; 1; 1; 0:5; 1Þ ð1; 1; 1; 1; 1; 1; 1; 0:5Þ ðt ; u M Þ ð1:59; 0:77040Þ a ð1:59; 0:77075Þ a ð1:59; 0:77075Þ a K ð0:5; 1; 0:5; 1; 1; 1; 1; 1Þ ð1; 0:5; 1; 0:5; 1; 1; 1; 1Þ ð1; 1; 1; 1; 0:5; 1; 0:5; 1Þ ðt ; u M Þ ð1:59; 0:77065Þ a ð1:59; 0:77008Þ a ð1:59; 0:77068Þ a K ð1; 1; 1; 1; 1; 0:5; 1; 0:5Þ ð0:5; 0:5; 0:5; 0:5; 1; 1; 1; 1Þ ð1; 1; 1; 1; 0:5; 0:5; 0:5; 0:5Þ ðt ; u M Þ ð1:59; 0:77040Þ a ð1:59; 0:77000Þ a ð1:59; 0:77032Þ a K 0.5K a ð10 5 ; 10 5 ; 1; 1; 1; 1; 1; 1Þ ð1; 1; 10 5 ; 10 5 ; 1; 1; 1; 1Þ ðt ; u M Þ ð1:59; 0:76955Þ a NP b AP c K ð10 2 ; 1; 10 2 ; 1; 1; 1; 1; 1Þ ð1; 1; 1; 1; 10 2 ; 1; 10 2 ; 1Þ ð1; 10 2 ; 1; 10 2 ; 1; 1; 1; 1Þ ðt ; u M Þ ð1:59; 0:77204Þ a ð1:59; 0:77184Þ a AP d K ð1; 1; 1; 1; 1; 10 2 ; 1; 10 2 Þ ð1; 1; 1; 1; 10 2 ; 10 2 ; 10 2 ; 10 2 Þ ð10 2 ; 10 2 ; 10 2 ; 10 2 ; 1; 1; 1; 1Þ ðt ; u M Þ AP e NP f NP g K 10 2 K a ð10 3 ; 1; 10 3 ; 1; 1; 1; 1; 1Þ ð1; 1; 1; 1; 10 3 ; 1; 10 3 ; 1Þ ðt ; u M Þ NP h ð1:59; 0:77208Þ a ð1:59; 0:77187Þ a 60 J.I. Ramos / Appl. Math. Comput. 146 (2003) K ð1; 10 3 ; 1; 10 3 ; 1; 1; 1; 1Þ ð1; 1; 1; 1; 1; 10 3 ; 1; 10 3 Þ ð1; 1; 1; 1; 10 3 ; 10 3 ; 10 3 ; 10 3 Þ ðt ; u M Þ AP i AP j NP k K ð10 3 ; 10 3 ; 10 3 ; 10 3 ; 1; 1; 1; 1Þ 10 3 K a ð10; 1; 10; 1; 1; 1; 1; 1Þ ðt ; u M Þ NP l NP m ð1:59; 0:77167Þ a

7 K ð1; 1; 1; 1; 10; 1; 10; 1Þ ð1; 10; 1; 10; 1; 1; 1; 1Þ ð1; 1; 1; 1; 1; 10; 1; 10Þ ðt ; u M Þ ð1:59; 0:77148Þ a AP n AP o K ð1; 1; 1; 1; 10; 10; 10; 10Þ ð10; 10; 10; 10; 1; 1; 1; 1Þ 10K a ðt ; u M Þ NP p NP q NP r (AP ¼ almost periodic; NP ¼ nonperiodic). a Periodic. b u M ¼ 0:82316 and at t ¼ 197:37 and , respectively. c u M ¼ 0:73053 and at t ¼ 97:71 and 99.11, respectively. d u M ¼ 0:73707 and at t ¼ 100:39 and , respectively. e u M ¼ 0:73709 and at t ¼ 100:39 and , respectively. f u M ¼ 0:77603 and at t ¼ 197:64 and , respectively. g u M ¼ 0:73489 and at t ¼ 198:05 and , respectively. h u M ¼ 0:79718 and at t ¼ 198:11 and , respectively. i u M ¼ 0:73797 and at t ¼ 100:49 and , respectively. j u M ¼ 0:73770 and at t ¼ 100:49 and , respectively. k u M ¼ 0:70391 and at t ¼ 198:43 and , respectively. l u M ¼ 0:75312 and at t ¼ 198:54 and , respectively. m u M ¼ 0:77095 and at t ¼ 97:34 and 98.94, respectively. n u M ¼ 0:74530 and at t ¼ 100:36 and , respectively. o u M ¼ 0:74552 and at t ¼ 98:61 and , respectively. p u M ¼ 0:78000 and at t ¼ 197:59 and , respectively. q u M ¼ 0:74887 and at t ¼ 197:23 and , respectively. r u M ¼ 0:81847 and at t ¼ 198:02 and , respectively. J.I. Ramos / Appl. Math. Comput. 146 (2003)

8 62 J.I. Ramos / Appl. Math. Comput. 146 (2003) increases as K is increased, i.e., as the boundary conditions become Dirichlet s type. Some sample results illustrating the nonperiodic dynamics discussed above are presented in Figs. 1 3; the periodic dynamics correspond to a spiral wave similar to the one found in numerical studies with homogeneous Neumann boundary conditions at all the boundaries and is not presented here [10]. Fig. 1 corresponds to K ¼ð10 5 ; 10 5 ; 1; 1; 1; 1; 1; 1Þ, i.e., the boundary conditions on the left boundary of the computational domain for both the activator and the inhibitor are nearly of the Dirichlet type, and the concentration of the activator is nearly zero at and large near to the left boundary. This figure exhibits several arms where the activator s concentration is high; some of these arms break up into islands which then reconnect with other arms and interact with the boundaries. After this interaction is completed, some of the arms move in a counter-clockwise manner, whereas other arms rotate clockwise, merge and break up into new islands where the activator s concentration is high. It is worthwhile to point out that, even though both the activator s concentration and its gradient near to the left boundary are usually large, there are instances Fig. 1. Concentration of the activator u at (from left to right, from top to bottom) t ¼ 200:02, , , , , , , and ðrobin s boundary conditions; K ¼ð10 5 ; 10 5 ; 1; 1; 1; 1; 1; 1ÞÞ.

9 J.I. Ramos / Appl. Math. Comput. 146 (2003) Fig. 2. Concentration of the activator u at (from left to right, from top to bottom) t ¼ 200:02, , , , , , , and ðrobin s boundary conditions; K ¼ð1; 1; 1; 1; 10 3 ; 10 3 ; 10 3 ; 10 3 ÞÞ. when the activator s concentration is very small in certain parts of this boundary as shown in the third, sixth, seventh and eighth frames of Fig. 1. Fig. 2 corresponds to K ¼ð1; 1; 1; 1; 10 3 ; 10 3 ; 10 3 ; 10 3 Þ, i.e., almost Dirichlet boundary conditions for u and v at the top and bottom boundaries as indicated by the high values of u near to these boundaries. Fig. 2 shows that an almost vertical arm characterized by a high activator s concentration is attached to and breaks away from the top boundary forming an island. This island merges with a right-travelling front that appeared on the left boundary, but the resulting structure breaks up into an island and several arms which are attached to the bottom and right boundaries, while several arms emerge form the left boundary. The arms attached to the right boundary approach the top and bottom boundaries and, eventually, the one on the bottom boundary separates from this boundary and interacts with the top boundary, from which it detaches forming an island. The complex spatio-temporal patterns exhibited in Fig. 2 indicate that curved travelling fronts, arms rotating clock- and anticlockwise, islands, front break-up and connection to and disconnection from the boundaries

10 64 J.I. Ramos / Appl. Math. Comput. 146 (2003) Fig. 3. Concentration of the activator u at (from left to right, from top to bottom) t ¼ 200:02, , , , , , , and ðrobin s boundary conditions; K ¼ð10 3 ; 10 3 ; 10 3 ; 10 3 ; 1; 1; 1; 1ÞÞ. characterize the nonperiodic dynamics of two-dimensional excitable media subject to Robin boundary conditions when the transfer coefficients k u and k v are constant and equal on the top and bottom boundaries and their magnitude is larger than unity. By way of contrast, the periodic solutions reported in Table 1 correspond to spiral waves which rotate in a counter-clockwise manner, and only u is large on the boundaries where the spiral arm emerges [10]. Although not shown here, the results presented in Fig. 2 differ markedly from those corresponding to K ¼ð1; 1; 1; 1; 10 2 ; 10 2 ; 10 2 ; 10 2 Þ and K ¼ð1; 1; 1; 1; 10; 10; 10; 10Þ. In the first case, it is observed that a curved from which connects the left and top boundaries emerges from the upper left corner of the domain, advances towards the bottom right corner, merges with an arm which emerges from the right boundary and travels leftwards and upwards and then detaches from the left and top boundaries. After detachment, the resulting pattern breaks up into two arms and an island; the island moves towards the bottom boundary and the arm which was created at the right boundary grows in size and moves downwards, but remains attached to that boundary. For K ¼ð1; 1; 1; 1; 10; 10; 10; 10Þ arms of spirals rotating clock- and anticlockwise

11 J.I. Ramos / Appl. Math. Comput. 146 (2003) are observed; these waves merge, connect with the boundaries and break up into islands and complex spatio-temporal patterns. It has also been observed that, for K ¼ð1; 1; 1; 1; 10 3 ; 10 3 ; 10 3 ; 10 3 Þ, the activator s concentration near the top and bottom boundaries was higher than for K ¼ð1; 1; 1; 1; 10 2 ; 10 2 ; 10 2 ; 10 2 Þ and K ¼ð1; 1; 1; 1; 10; 10; 10; 10Þ, although for these three sets of parameters there are regions at the top and bottom boundaries where the activator s concentration is rather small. These results are consistent with the fact that, as k u and k v tend to infinity, the Robin boundary conditions tend to homogeneous Dirichlet ones and, therefore, in this limit, the concentrations of both the activator and inhibitor tend to zero at the boundaries. The presence of layers near to the boundaries where the activator s concentration is high indicates that the derivative of u normal to the boundaries is high and, although the reaction rates are small at these boundaries, the reaction may be significant slightly away from them. An entirely different behaviour to that presented in Fig. 2 is observed in Fig. 3 which corresponds to K ¼ð10 3 ; 10 3 ; 10 3 ; 10 3 ; 1; 1; 1; 1Þ, i.e., almost Dirichlet boundary conditions for u and v on the left and right boundaries as indicated by the high values of u near to these boundaries. A comparison between the results presented in Figs. 2 and 3 clearly indicates that the dynamics of twodimensional excitable media depends strongly on the boundary conditions and the magnitude of the transfer coefficients for both the activator and the inhibitor where the Robin s boundary conditions are applied. Fig. 3 shows that a curved front attached to the right boundary travels upwards, and attaches to the left boundary. This front is followed by another curved one which is attached to the bottom boundary. The first front detaches from the left and right boundaries and merges with an arm that emerges from the top boundary; the resulting pattern moves in a clockwise manner and results in a high activator s concentration region near to the upper right corner, whereas the tip of the second front attaches to the right boundary, detaches from the bottom one, and connects the left and right boundaries. The results presented in Fig. 3 differ substantially from those corresponding to K ¼ð10 2 ; 10 2 ; 10 2 ; 10 2 ; 1; 1; 1; 1Þ and K ¼ð10; 10; 10; 10; 1; 1; 1; 1Þ. In the first case, the numerical results indicate that the activator s concentration is high along layers near to the left and right boundaries where k u and k v are large. These layer develop corrugations and two arms (one rotating clockwise and the other anticlockwise) emerge from the top and bottom boundaries and move downwards and upwards, respectively. Eventually, the arms become connected with the left and right boundaries, detach from the top and bottom boundaries and create two fronts moving in opposite directions which, in turn, merge and result in a thicker arm connected to the left boundary. In the second case, clockwise and anticlockwise rotating arms of spiral waves are observed. These arms interact and merge with each other and with the layers formed along the boundary, and then break up into a number of islands which are arms of spiral

12 66 J.I. Ramos / Appl. Math. Comput. 146 (2003) waves and result in complex spatio-temporal patterns characterized by islands, layers of high activator s concentration along the boundaries of the domain, corner layers, etc. Similar patterns to the ones described for K ¼ð10 2 ; 10 2 ; 10 2 ; 10 2 ; 1; 1; 1; 1Þ have also been observed for K ¼ð1; 10 2 ; 1; 10 2 ; 1; 1; 1; 1Þ and K ¼ð1; 10 3 ; 1; 10 3 ; 1; 1; 1; 1Þ. On the other hand, for K ¼ð1; 1; 1; 1; 1; 10 2 ; 1; 10 2 Þ and ð1; 1; 1; 1; 1; 10 3 ; 1; 10 3 Þ, it was found that two arms of spiral waves (one rotating clockwise and the other anticlockwise) emerge from the left boundary, whereas only an arm which rotates clockwise emerges from the right one. The first two arms detach from the left boundary and form a curved front which propagates towards the right boundary; this front merges with the arm attached to the right boundary and a V-shaped region which propagates upwards is formed. Similar trends to those presented in Fig. 3 have also been observed for K ¼ð1; 1; 10 5 ; 10 5 ; 1; 1; 1; 1Þ, i.e., when the boundary conditions on the right boundary are almost Dirichlet s type, and, as discussed previously, they are quite different from those presented in Fig. 1 which corresponds to almost Dirichlet boundary conditions at the left boundary. In view of these results, it may be stated that the dynamics of two-dimensional excitable media are rather sensitive to both the values of k u and k v and the boundaries where Robin s boundary conditions are imposed. For K ¼ð10 3 ; 10 3 ; 10 3 ; 10 3 ; 10 3 ; 10 3 ; 10 3 ; 10 3 Þ, i.e., when k u ¼ k v ¼ 10 3 along all the boundaries of the computational domain and, therefore, almost homogeneous Dirichlet boundary conditions for both u and v apply on all the boundaries, it has been found that the activator s concentration is high in a layer along the boundaries, except in some regions where it has very small values, and that an arm of an anticlockwise rotating spiral wave emerges from the left boundary from which it detaches and then reattaches to the bottom boundary. This arm may attach to the right boundary and then break up into an island and a free rotating arm which is not connected to the boundaries of the domain. Similar trends to the just described have also been observed for K ¼ð10 2 ; 10 2 ; 10 2 ; 10 2 ; 10 2 ; 10 2 ; 10 2 ; 10 2 Þ, whereas those for K ¼ð10; 10; 10; 10; 10; 10; 10; 10Þ are characterized by thick corner layers, multiple tongues and arms where the activator s concentration is high, and multiple attachments to and detachments from the boundaries. The results presented in previous paragraphs indicate that complex spatiotemporal patterns exist in two-dimensional excitable media for large values of k u and k v. For the homogeneous Robin boundary conditions considered in this paper, the limits k u!1and k v!1correspond to homogeneous Dirichlet boundary conditions and raise the following questions: Can similar patterns be observed when nonhomogeneous Dirichlet boundary conditions are applied on all the boundaries? How do the results for K ¼ð10 3 ; 10 3 ; 10 3 ; 10 3 ; 10 3 ; 10 3 ; 10 3 ; 10 3 Þ differ from those corresponding to k u!1and k v!1? The answers to these questions are summarized in Table 2 where the subscripts l, r, t and b

13 J.I. Ramos / Appl. Math. Comput. 146 (2003) Table 2 Separation (T) between and largest amplitude (u M ) of the pulses in the activator s concentration u as a function of U ðu l ; v l ; u r ; v r ; u b ; v b ; u t ; v t Þ at ðx; yþ ¼ð20d; 20dÞ for Dirichlet s boundary conditions Parameter U ¼ 0 u l ¼ 0:2 u r ¼ 0:2 u l ¼ u r ¼ 0:2 ðt ; u M Þ NP a NP b NP c NP d Parameter u b ¼ 0:2 u t ¼ 0:2 u b ¼ u t ¼ 0:2 u l ¼ u r ¼ u b ¼ u t ¼ 0:2 ðt ; u M Þ (1.59, ) e NP f NP g (1.60, ) e Parameter v l ¼ 0:2 v r ¼ 0:2 v l ¼ v r ¼ 0:2 v b ¼ 0:2 ðt ; u M Þ NP h NP i NP j AP k Parameter v t ¼ 0:2 v b ¼ v t ¼ 0:2 v l ¼ v r ¼ v b ¼ v t ¼ 0:2 ðt ; u M Þ NP l AP m (1.60, ) e (AP ¼ almost periodic; NP ¼ nonperiodic). a u M ¼ 0:78431 and at t ¼ 97:34 and 98.94, respectively. b u M ¼ 0:76960 and at t ¼ 97:36 and 98.95, respectively. c u M ¼ 0:77982 and at t ¼ 97:35 and 98.92, respectively. d u M ¼ 0:77311 and at t ¼ 97:36 and 98.94, respectively. e Periodic. f u M ¼ 0:78932 and at t ¼ 97:29 and 98.91, respectively. g u M ¼ 0:76611 and at t ¼ 97:33 and 98.92, respectively. h u M ¼ 0:79041 and at t ¼ 97:34 and 98.96, respectively. i u M ¼ 0:77694 and at t ¼ 97:35 and 98.95, respectively. j u M ¼ 0:78488 and at t ¼ 97:35 and 98.95, respectively. k u M ¼ 0:77212 and at t ¼ 97:33 and 98.92, respectively. l u M ¼ 0:77785 and at t ¼ 97:33 and 98.93, respectively. m u M ¼ 0:77031 and at t ¼ 97:35 and 98.94, respectively. refer to the left, right, top and bottom boundaries, respectively, homogeneous Dirichlet boundary conditions correspond to U ðu l ; v l ; u r ; v r ; u b ; v b ; u t ; v t Þ¼0, and the different rows in this table indicate the component of U which is different from zero, e.g., u b ¼ 0:2 corresponds to U ð0; 0; 0; 0; 0:2; 0; 0; 0Þ. Table 2 indicates that only periodic behaviour is observed for u b ¼ 0:2, u l ¼ u r ¼ u b ¼ u t ¼ 0:2 and v l ¼ v r ¼ v b ¼ v t ¼ 0:2; in other cases, the dynamics are almost periodic or nonperiodic. Fig. 4 corresponds to homogeneous Dirichlet boundary conditions on all the boundaries and shows layers along the boundaries where the activator concentration is high and large gradients at the boundaries where u ¼ 0. This figure also shows that an anticlockwise rotating spiral wave is initially attached to the top boundary, breaks up upon interaction with the left boundary, becomes a free spiral wave which then attaches to both the left and the bottom boundaries, breaks up upon interaction with the right boundary, and, finally, attaches to the top boundary. Fig. 4 also shows that there are patches along the boundaries where the activator s concentration is small, and the location of these patches

14 68 J.I. Ramos / Appl. Math. Comput. 146 (2003) Fig. 4. Concentration of the activator u at (from left to right, from top to bottom) t ¼ 100:04, , , , , , , and ðdirichlet s boundary conditions; U ðu l ; v l ; u r ; v r ; u b ; v b ; u t ; v t Þ¼0Þ. depends on time. Despite the periodic-like behaviour observed in Fig. 4, it has been found that the magnitude of the largest activator s concentration is not constant, the ðu; vþ-phase diagram at the two monitor locations mentioned above contains many loops and the Fourier spectra of u at the monitor locations are broad. Similar results to those presented in Fig. 4 have also been observed for u l ¼ 0:2, u r ¼ 0:2, u l ¼ u r ¼ 0:2, u b ¼ 0:2, u t ¼ 0:2, u b ¼ u t ¼ 0:2 oru l ¼ u r ¼ u b ¼ u t ¼ 0:2 and homogeneous Dirichlet boundary conditions at the remaining boundaries, thus indicating that the boundary on which a nonhomogeneous Dirichlet boundary condition for the activator s concentration is applied does not influence significantly the dynamics of spiral waves in twodimensional excitable media provided that homogeneous Dirichlet boundary conditions for the inhibitor are imposed on all the boundaries. When homogeneous Dirichlet boundary conditions are imposed on the activator s concentration on all the boundaries, similar patterns to the ones presented in Fig. 4 have been observed except that the activator s concentration was nil at the left, right, left and right, bottom, top, and bottom and top boundaries for v l ¼ 0:2, v r ¼ 0:2, v l ¼ v r ¼ 0:2, v b ¼ 0:2, v t ¼ 0:2 and

15 J.I. Ramos / Appl. Math. Comput. 146 (2003) v b ¼ v t ¼ 0:2, respectively, whereas the activator s concentration was high on layers along the other boundaries; therefore, the boundary conditions for the inhibitor s concentration play a much more important role in determining the activator s concentration near the boundaries of the domain than those of the activator. Moreover, for example, for v b ¼ 0:2, it was found that the spiral wave flattened when it approached, but it never became in contact with the bottom boundary; by way of contrast, the spiral wave observed for u b ¼ 0:2 interacted with and became in contact with the bottom boundary. The dynamics of spiral waves subject to v l ¼ v r ¼ v b ¼ v t ¼ 0:2, i.e., when the inhibitor s concentrations at the boundaries are equal to 0.2 is exhibited in Fig. 5 which clearly shows a spiral wave that does not become in contact with the boundaries, is stable and persistent, and may break up into an island and a spiral wave. A comparison between the results presented in Figs. 4 and 5 indicates that the spiral wave s dynamics for U ¼ 0 are more complex than that for v l ¼ v r ¼ v b ¼ v t ¼ 0:2 due to both the presence of layers along most the domain s boundaries and the interactions between the spiral wave and these layers. Fig. 5. Concentration of the activator u at (from left to right, from top to bottom) t ¼ 100:04, , , , , , , and ðdirichlet s boundary conditions; U ðu l ; v l ; u r ; v r ; u b ; v b ; u t ; v t Þ¼0:2Þ.

16 70 J.I. Ramos / Appl. Math. Comput. 146 (2003) The results presented in this paper indicate that, in the absence of forcing, convective fields, inhomogeneities, etc., spiral waves are stable and persistent for both homogeneous Neumann boundary conditions for the activator and the inhibitor, and for homogeneous Dirichlet boundary conditions for the activator and nonhomogeneous Dirichlet boundary conditions for the inhibitor, but exhibit a rich spatio-temporal behaviour when homogeneous Dirichlet boundary conditions or homogeneous Robin boundary conditions with large transfer coefficients are imposed on all the boundaries. Therefore, spiral waves are not robust to changes in boundary conditions. The results presented here for homogeneous Robin boundary conditions are consistent with those corresponding to homogeneous Dirichlet boundary conditions provided that the transfer coefficients are sufficiently large. Despite the complex patterns presented in this paper, it should be pointed out that the values of u and v at the monitor locations mentioned above are characterized by peaks whose magnitude may be a function of time. In some cases, it has been observed that the amplitude of these peaks underwent large variations during short times, thus indicating some type of intermittent behaviour if the computations were not performed for sufficiently long times, i.e., t > Conclusions The dynamics of the two-equation Oregonator model in two-dimensional excitable media subject to homogeneous Robin boundary conditions on all the boundaries has been studied numerically by means of a linearized implicit finite difference technique, and it has been shown that spiral waves propagate through the media if the transfer coefficients at all the boundaries are equal to or smaller than one. These waves are almost identical to those observed under homogeneous Neumann boundary conditions on all the boundaries. Therefore, spiral waves in two-dimensional excitable media are robust under perturbations of homogeneous Neumann boundary conditions, and exhibit a periodic behaviour, width and maximum concentration at about the same time. Complex spatio-temporal patterns which may be either almost periodic or nonperiodic are observed when the transfer coefficient for either the activator or the inhibitor is larger than one. These patterns are characterized by arms of spiral waves which may break up into islands where the activator s concentration is high, attachments to and detachments from the boundaries, layers of high activator s concentration along the boundaries, corner layers, etc. The complexity of these patterns is a strong function of both the magnitude of the transfer coefficient and the boundary where the Robin boundary conditions are applied, and increases as the transfer coefficient is increased. It has also been observed that the layers along the boundaries where the activator s concen-

17 J.I. Ramos / Appl. Math. Comput. 146 (2003) tration is high, exhibit large gradients in the direction normal to these boundaries, and that these layers may exhibit regions of low activation s concentration whose thickness and location are nonperiodic functions of time. When homogeneous Dirichlet boundary conditions are applied on all the boundaries, it has been observed that an arm of spiral waves very similar to those observed when homogeneous Neumann boundary conditions are imposed on all the boundaries, interacts with the layers along the boundaries where the activator s concentration is high and may form islands and tongues. However, when nonhomogeneous Dirichlet boundary conditions are imposed on both the activator and the inhibitor on all the boundaries, a periodic spiral wave is observed, and this wave does not interact with the boundaries, although it breaks up into an island and another spiral wave. Acknowledgements The research reported in this paper was supported by Project PB from the D.G.E.S. and Project BFM de la Direccion General de Investigacion of Spain and Fondos FEDER. References [1] R.J. Field, M. Burger (Eds.), Oscillations and Travelling Waves in Chemical Systems, John Wiley & Sons, New York, [2] A.V. Holden, M. Markus, H.G. Othmer (Eds.), Nonlinear Wave Processes in Excitable Media, Plenum Press, New York, [3] R. Kapral, R. Showalter (Eds.), Chemical Waves and Patterns, Kluwer Academic, Dordrecht, [4] A.T. Winfree, The Geometry of Biological Time, Springer-Verlag, New York, [5] J.I. Ramos, Tile patterns in excitable media subject to non-solenoidal flow fields, Chaos Solitons & Fractals 12 (2001) [6] J.I. Ramos, Convection-induced anisotropy in excitable media subject to solenoidal advective flow fields, Chaos, Solitons & Fractals 12 (2001) [7] J.I. Ramos, Spatio-temporal patterns in excitable media with non-solenoidal flow straining, Mathematics and Computers in Simulation 55 (2001) [8] A.V. Panfilov, S.C. M uller, V.S. Zykov, J.P. Keener, Elimination of spiral waves in cardiac tissue by multiple electrical shocks, Physical Review E 61 (2000) [9] I. Schebesch, H. Engel, Interacting spiral waves in the Oregonator model of light-sensitive Belousov Zhabotinskii reaction, Physical Review E 60 (1999) [10] J.I. Ramos, Propagation of spiral waves in anisotropic media: from waves to stripes, Chaos, Solitons & Fractals 12 (2001) [11] J.I. Ramos, Wave propagation and suppression in excitable media with holes and external forcing, Chaos, Solitons & Fractals 13 (2002) [12] J.I. Ramos, Dynamics of spiral waves in excitable media with local time-periodic modulation, Chaos, Solitons & Fractals 13 (2002)

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