Spatio-temporal patterns in
two-dimensional excitable media subject
to Robin boundary conditions
J.I. Ramos
E.T.S. Ingenieros Industriales,Universidad de Maalaga, Room I-325-D, Plaza El Ejido, s/n 29013 Maalaga, 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 boun-dary 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
E-mail address:[email protected] (J.I. Ramos).
0096-3003/$ - see front matter Ó 2002 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(02)00515-5
Applied Mathematics and Computation 146 (2003) 55–72
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–Zhabo-tinskii (BZ), and showed the presence of two stable counter-rotating spirals at low intensities, whereas the waves were found to undergo a symmetry insta-bility 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 Forsstovaa 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 spatio-temporal 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 ex-plained 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 thex-direction which is either a linear or a sinusoidal function of they-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
velocity field. Elkin et al. [20] have studied numerically the movement of ex-citation wave breaks, while Biktashev and Holden [21] analyzed the resonant drift of autowave vortices in 2D and the effects of boundaries and inhomo-geneities. 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 ob-jective 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 ¼dur 2uþF
u; ð1Þ
ov
ot ¼dvr 2vþF
v; ð2Þ
where t is time, u and v denote the concentrations of the activator and the inhibitor, respectively,du and dv are the diffusion coefficients foru andv, re-spectively, and the source terms in Eqs. (1) and (2) can be written as
Fu¼
1
u
u2fvuq
uþq
; Fv¼uv; ð3Þ
where, unless stated otherwise, ¼0:01, f ¼1:4 andq¼0:002, and are the same as those employed in the BZ model.
In this paper, it is assumed thatdu¼1 anddv¼0:6, 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].
Eqs. (1) and (2) were solved in the spatial domainX¼ ½Lx;Lx ½Ly;Ly withLx¼Ly ¼7:5, subject to homogeneous Robin boundary conditions on all the boundaries, i.e.,
ou
onþkuu¼0;
ov
onþkvv¼0; ð4Þ
wheren denotes the coordinate normal to the boundary pointing away from the domain, andkuandkv 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.ku¼kv¼0 correspond to ho-mogeneous Neumann boundary conditions, whereasku¼kv¼ 1correspond to homogeneous Dirichlet boundary conditions.
The initial conditions inXfor Eqs. (1) and (2) are
u¼0 for 0<h<0:5; u¼qðf þ1Þ=ðf 1Þ elsewhere; ð5Þ
v¼qf þ1
f 1þ
h
8pf; ð6Þ
wherehis the angle with respect to the origin of coordinates measured coun-terclockwise from the positive x-axis. This initial condition results in the for-mation 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-lin-earized, second-order accurate (in both space and time) finite difference method [23]. This method factorizes the elliptic equations that result upon discretiza-tion 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 102102 point equally spaced mesh and a time step of 104. Computations were also
per-formed with equally-spaced meshes of 202202 and 502502 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 102102 point equally-spaced meshes and a time step equal to 104 are presented.
3. Presentation of results
In this section, some sample results illustrating the effects of the transfer coefficientsku andkv which control the homogeneous Robin’s boundary con-ditions employed in the study are presented. For the sake of brevity, we have introducedK ðlu;lv;ru;rv;bu;bv;tu;tvÞwherel,r,bandtrefer to the values of
kat the left, right, bottom and top boundaries, respectively, and the subscripts
u andvdenote 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 ofK are smaller than or equal to one, and, therefore, the influence ofkuandkvon 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 ofKfor 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;
25dÞwhered¼15=101, are nearly independent of the magnitude of the com-ponents ofKprovided that this magnitude is smaller than or equal to one. It should also be noted that the maximum values ofupresented in Table 1 for the periodic solutions, i.e., spiral waves, occur at nearly the same time, i.e.,
t¼98:95.
Table 1 also indicates that periodic spiral waves can also be observed in two-dimensional excitable media subject to homogeneous Robin boundary condi-tions in all the boundaries if eitherlu¼ru¼10, 100, 1000 orbu¼tu¼10, 100, 1000, provided that the other parameters ofKare equal to one. This implies that, provided that the values of eitherku or kv have the same magnitude on opposite boundaries, periodic dynamics are observed. However, almost peri-odic motions characterized by small differences between successive peaks of either u or v result when either lv¼rv ¼10, 100, 1000 or bv¼tv¼10, 100, 1000, provided that the other parameters ofKare 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 com-ponents of ðlu;lv;ru;rvÞ, ðbu;bv;tu;tvÞ or Kare larger than one. These nonpe-riodic behaviour is characterized by phase diagrams which exhibit peaks of different magnitude ofu andv, 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 103K
a in Table 1 almost corresponds to homogeneous Dirichlet boundary conditions and indicates that for this value, and also for 10Ka and 102Ka, nonperiodic motions result; moreover, the differences between the
magnitudes of successive peaks ofu, the number of loops in the phase diagrams
ofðu;vÞatðx;y;tÞ ¼ ð20d;20d;tÞandð25d;25d;tÞand the frequency contents of
the Fourier spectra of bothu andvat ðx;y;tÞ ¼ ð20d;20d;tÞand ð25d;25d;tÞ J.I. Ramos / Appl. Math. Comput. 146 (2003) 55–72 59
Table 1
Separation (T) between and largest amplitude (uM) of the pulses in the activator’s concentrationuas a function ofK ðlu;lv;ru;rv;bu;bv;tu;tvÞat ðx;yÞ ¼ ð20d;20dÞfor homogeneous Robin’s boundary conditions
K Ka¼ ð1;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;uMÞ ð1:60;0:77075Þa ð1:59;0:77065Þa ð1:59;0:77008Þa
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;uMÞ ð1:59;0:77075Þa ð1:59;0:77075Þa ð1:59;0:77068Þa
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;uMÞ ð1:59;0:77040Þa ð1:59;0:77075Þa ð1:59;0:77075Þa
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;uMÞ ð1:59;0:77065Þa ð1:59;0:77008Þa ð1:59;0:77068Þa
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;uMÞ ð1:59;0:77040Þa ð1:59;0:77000Þa ð1:59;0:77032Þa
K 0.5Ka ð105;105;1;1;1;1;1;1Þ ð1;1;105;105;1;1;1;1Þ
ðT;uMÞ ð1:59;0:76955Þa NPb APc
K ð102;1;102;1;1;1;1;1Þ ð1;1;1;1;102;1;102;1Þ ð1;102;1;102;1;1;1;1Þ
ðT;uMÞ ð1:59;0:77204Þa ð1:59;0:77184Þa APd
K ð1;1;1;1;1;102;1;102Þ ð1;1;1;1;102;102;102;102Þ ð102;102;102;102;1;1;1;1Þ
ðT;uMÞ APe NPf NPg
K 102Ka ð103;1;103;1;1;1;1;1Þ ð1;1;1;1;103;1;103;1Þ
ðT;uMÞ NPh ð1:59;0:77208Þa ð1:59;0:77187Þa
K ð1;103;1;103;1;1;1;1Þ ð1;1;1;1;1;103;1;103Þ ð1;1;1;1;103;103;103;103Þ
ðT;uMÞ APi APj NPk
K ð103;103;103;103;1;1;1;1Þ 103Ka ð10;1;10;1;1;1;1;1Þ
ðT;uMÞ NPl NPm ð1:59;0:77167Þa
60
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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;uMÞ ð1:59;0:77148Þa APn APo
K ð1;1;1;1;10;10;10;10Þ ð10;10;10;10;1;1;1;1Þ 10Ka
ðT;uMÞ NPp NPq NPr
(AP¼almost periodic; NP¼nonperiodic).
aPeriodic.
bu
M¼0:82316 and 0.76978 att¼197:37 and 198.90, respectively. cu
M¼0:73053 and 0.73173 att¼97:71 and 99.11, respectively. d
uM¼0:73707 and 0.73712 att¼100:39 and 101.89, respectively. e
uM¼0:73709 and 0.73702 att¼100:39 and 101.89, respectively. f
uM¼0:77603 and 0.77528 att¼197:64 and 199.16, respectively. g
uM¼0:73489 and 0.73879 att¼198:05 and 199.53, respectively. h
uM¼0:79718 and 0.78433 att¼198:11 and 199.72, respectively. i
uM¼0:73797 and 0.73803 att¼100:49 and 101.99, respectively. j
uM¼0:73770 and 0.73803 att¼100:49 and 101.99, respectively. k
uM¼0:70391 and 0.68702 att¼198:43 and 199.84, respectively. l
uM¼0:75312 and 0.74000 att¼198:54 and 199.93, respectively. m
uM¼0:77095 and 0.77891 att¼97:34 and 98.94, respectively. n
uM¼0:74530 and 0.74493 att¼100:36 and 101.88, respectively. o
uM¼0:74552 and 0.74547 att¼98:61 and 100.57, respectively. p
uM¼0:78000 and 0.76531 att¼197:59 and 199.15, respectively. qu
M¼0:74887 and 0.76965 att¼197:23 and 198.75, respectively. ru
M¼0:81847 and 0.77575 att¼198:02 and 199.67, respectively.
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increases asKis 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 K¼ ð105;105;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 activatoruat (from left to right, from top to bottom)t¼200:02, 200.04, 200.06, 200.08, 200.10, 200.12, 200.14, 200.16 and 200.18.ðRobin’s boundary conditions; K¼ ð105;105;1;1;1;1;1;1ÞÞ.
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 toK¼ ð1;1;1;1;103;103;103;103Þ, i.e., almost Dirichlet
boundary conditions foruandvat the top and bottom boundaries as indicated by the high values ofunear 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
Fig. 2. Concentration of the activatoruat (from left to right, from top to bottom)t¼200:02, 200.04, 200.06, 200.08, 200.10, 200.12, 200.14, 200.16 and 200.18.ðRobin’s boundary conditions; K¼ ð1;1;1;1;103;103;103;103ÞÞ.
characterize the nonperiodic dynamics of two-dimensional excitable media subject to Robin boundary conditions when the transfer coefficientsku andkv 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 man-ner, and onlyu 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 toK¼ ð1;1;1;1;102;102;102;102Þ andK¼ ð1;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
K¼ ð1;1;1;1;10;10;10;10Þarms of spirals rotating clock- and anticlockwise
Fig. 3. Concentration of the activatoruat (from left to right, from top to bottom)t¼200:02, 200.04, 200.06, 200.08, 200.10, 200.12, 200.14, 200.16 and 200.18.ðRobin’s boundary conditions; K¼ ð103;103;103;103;1;1;1;1ÞÞ.
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, forK¼ ð1;1;1;1;103;103;103;103Þ, the activator’s concentration near the
top and bottom boundaries was higher than for K¼ ð1;1;1;1;102; 102;102;
102Þand K¼ ð1;1;1;1;10;10;10;10Þ, although for these three sets of
param-eters 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
kuandkvtend to infinity, the Robin boundary conditions tend to homogeneous Dirichlet ones and, therefore, in this limit, the concentrations of both the ac-tivator 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 toK¼ ð103;103;103;103;1;1;1;1Þ, i.e., almost Dirichlet
boundary conditions foruandvon 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 two-dimensional excitable media depends strongly on the boundary conditions and the magnitude of the transfer coefficients for both the activator and the in-hibitor 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 re-sults 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 toK¼ ð102;102;102;102;1;1;1;1ÞandK¼ ð10;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 whereku and kv 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
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 K¼ ð102;102;102;102;1;1;1;1Þ
have also been observed forK¼ ð1;102;1;102;1;1;1;1ÞandK¼ ð1;103;1;103;
1;1;1;1Þ. On the other hand, for K¼ ð1;1;1;1;1;102;1;102Þandð1;1;1;1;1;
103;1;103Þ, 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 to-wards 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
K¼ ð1;1;105;105;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 ku and kv and the boundaries where Robin’s boundary conditions are imposed.
ForK¼ ð103;103;103;103;103;103;103;103Þ, i.e., whenk
u¼kv¼103along all the boundaries of the computational domain and, therefore, almost ho-mogeneous 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
K¼ ð102;102;102;102;102;102;102;102Þ, whereas those for K¼ ð10;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 spatio-temporal patterns exist in two-dimensional excitable media for large values of
kuandkv. For the homogeneous Robin boundary conditions considered in this paper, the limitsku! 1 and kv! 1 correspond 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 K¼ ð103;103;103;103;103; 103;
103;103Þdiffer from those corresponding tok
u! 1andkv! 1? The answers to these questions are summarized in Table 2 where the subscriptsl,r,tandb
refer to the left, right, top and bottom boundaries, respectively, homogeneous Dirichlet boundary conditions correspond toU ðul;vl;ur;vr;ub;vb;ut;vtÞ ¼0,
and the different rows in this table indicate the component ofUwhich is dif-ferent from zero, e.g.,ub¼0:2 corresponds toU ð0;0;0;0;0:2;0;0;0Þ. Table
2 indicates that only periodic behaviour is observed for ub¼0:2, ul¼ur¼ub¼ut¼0:2 and vl¼vr¼vb¼vt¼0:2; in other cases, the
dy-namics 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 con-centration is high and large gradients at the boundaries whereu¼0. 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
Table 2
Separation (T) between and largest amplitude (uM) of the pulses in the activator’s concentration u as a function of U ðul;vl;ur;vr;ub;vb;ut;vtÞ at ðx;yÞ ¼ ð20d;20dÞ for Dirichlet’s boundary conditions
Parameter U¼0 ul¼0:2 ur¼0:2 ul¼ur¼0:2
ðT;uMÞ NPa NPb NPc NPd
Parameter ub¼0:2 ut¼0:2 ub¼ut¼0:2 ul¼ur¼ub¼ut¼0:2
ðT;uMÞ (1.59,0.76926)e NPf NPg (1.60,0.76859)e Parameter vl¼0:2 vr¼0:2 vl¼vr¼0:2 vb¼0:2
ðT;uMÞ NPh NPi NPj APk Parameter vt¼0:2 vb¼vt¼0:2 vl¼vr¼vb
¼vt¼0:2
ðT;uMÞ NPl APm (1.60,0.77590)e (AP¼almost periodic; NP¼nonperiodic).
a
uM¼0:78431 and 0.78372 att¼97:34 and 98.94, respectively. bu
M¼0:76960 and 0.77343 att¼97:36 and 98.95, respectively. cu
M¼0:77982 and 0.80106 att¼97:35 and 98.92, respectively. du
M¼0:77311 and 0.77901 att¼97:36 and 98.94, respectively. ePeriodic.
fu
M¼0:78932 and 0.78602 att¼97:29 and 98.91, respectively. g
uM¼0:76611 and 0.78068 att¼97:33 and 98.92, respectively. h
uM¼0:79041 and 0.77193 att¼97:34 and 98.96, respectively. i
uM¼0:77694 and 0.77028 att¼97:35 and 98.95, respectively. j
uM¼0:78488 and 0.78184 att¼97:35 and 98.95, respectively. k
uM¼0:77212 and 0.77315 att¼97:33 and 98.92, respectively. l
uM¼0:77785 and 0.78695 att¼97:33 and 98.93, respectively. m
uM¼0:77031 and 0.77401 att¼97:35 and 98.94, respectively.
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 ofuat the monitor locations are broad. Similar results to those presented in Fig. 4 have also been observed for ul¼0:2, ur¼0:2, ul¼ur¼0:2, ub ¼0:2, ut ¼0:2, ub¼ut¼0:2 or ul¼ ur¼ub¼ut ¼0:2 and homogeneous Dirichlet boundary conditions at the
re-maining boundaries, thus indicating that the boundary on which a nonho-mogeneous Dirichlet boundary condition for the activator’s concentration is applied does not influence significantly the dynamics of spiral waves in two-dimensional 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 concentra-tion was nil at the left, right, left and right, bottom, top, and bottom and top boundaries for vl¼0:2, vr¼0:2, vl¼vr¼0:2, vb¼0:2, vt¼0:2 and Fig. 4. Concentration of the activatoruat (from left to right, from top to bottom)t¼100:04, 100.06, 100.08, 100.10, 100.12, 100.14, 100.16, 100.18 and 100.20.ðDirichlet’s boundary conditions;
U ðul;vl;ur;vr;ub;vb;ut;vtÞ ¼0Þ.
vb ¼vt¼0:2, 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, forvb¼0:2, 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 forub¼0:2
interacted with and became in contact with the bottom boundary.
The dynamics of spiral waves subject tovl¼vr¼vb ¼vt¼0:2, 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 in-dicates that the spiral wave’s dynamics forU¼0are more complex than that for vl¼vr¼vb¼vt¼0:2 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 activatoruat (from left to right, from top to bottom)t¼100:04, 100.06, 100.08, 100.10, 100.12, 100.14, 100.16, 100.18 and 100.20.ðDirichlet’s boundary conditions;
U ðul;vl;ur;vr;ub;vb;ut;vtÞ ¼0:2Þ.
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 inhibi-tor, 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 cor-responding 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 ofu and vat 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 be-haviour if the computations were not performed for sufficiently long times, i.e.,
t>100.
4. 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. There-fore, spiral waves in two-dimensional excitable media are robust under per-turbations 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 concen-tration 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
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 im-posed 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 PB97-1086 from the D.G.E.S. and Project BFM2001-1902 de la Direccioon General de Investigacioon of Spain and Fondos FEDER.
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