Drift velocity of rotating spiral waves in the weak deformation approximation

Transcription

1 JOURNAL OF CHEMICAL PHYSICS VOLUME 119, NUMBER 8 22 AUGUST 2003 Drift velocity of rotating spiral waves in the weak deformation approximation Hong Zhang a) Bambi Hu and Department of Physics, University of Houston, Houston, Texas Gang Hu b) Department of Physics, Beijing Normal University, Beijing , China Jinghua Xiao Received 14 May 2003; accepted 29 May 2003 The drift velocities of spiral waves driven by a periodic mechanic deformation or a constant or periodic electric field are obtained under the weak deformation approximation around the spiral wave tip. An approximate formula is derived for these drift velocities and some significant results, such as the drift of spiral waves induced by a mechanical deformation with 3 0, are predicted. Numerical simulations are performed demonstrating qualitative agreement with the analytical results American Institute of Physics. DOI: / Spiral waves are one of the most striking patterns in excitable and oscillatory media. They have been the subject of extensive research in a large variety of systems, which include the oxidation of CO on platinum, nerve impulse, reacting chemical systems like the Belousov Zhabotinsky BZ reaction, and cardiac muscle. 1 4 In particular, it is generally believed that the spirals in cardiac muscle play an essential role in heart diseases such as arrhythmia and fibrillation, the latter being the leading cause of death in the industrialized world. Therefore, the control of spirals is of crucial importance. An extremely interesting as well as practical point in this regard is that spiral waves are often subjected to some form of external forcing which might occur, for example, in the daily variation of sunlight. Thus, it is important to understand the role such external driving will play in these systems. In the study of the control of spiral waves, an active field of recent investigation concerns the resonant drift of the spiral core induced by applying an external field, such as a dc or an ac electric field. 5 For rotating waves, resonance corresponds to a net drift of the rotation centers along a straight line. To address the problem of how cardiac muscle contraction affects the dynamics of rotating spiral waves, Muñuzuri et al. 6 designed an elastic excitable medium by incorporating the BZ reaction into a polyacrylamide-silica gel to investigate the effect of mechanical deformation on spiral waves. They reported that for equal frequencies of deformation and spiral rotation, spirals will drift. Moreover, the direction of the drift does not coincide with the stretching direction and can be varied by changing the phase shift between the deformation and the spiral rotation. In their paper, the authors also suggested a simple kinematical model providing an intuitive a Electronic mail: hzhang@phys.hkbu.edu.hk; jinlinone@sina.com b Electronic mail: ganghu@bnu.edu.cn understanding of the main features observed. However, this kinematical model is not based on the original partialdifferential equation and so far no theoretical results have been obtained. Therefore, analytical derivations directly from the reaction-diffusion equation and its spiral wave solution are of crucial importance for a deeper and more comprehensive understanding of the mechanism of the drift and for a wider application of spiral wave control. In this paper, we will introduce a new approach to compute the velocity of the spiral tip based on the characteristics of spiral waves. We start with a general reaction-diffusion equation and its spiral wave solution. By applying an approximation of a weak spiral wave deformation around the tip, we derive an approximate formula for the drift velocities of the spiral tip induced by an arbitrary weak perturbation field that includes the mechanical deformation perturbation and the dc and ac electric field perturbations as special cases. This approximate formula can not only explain a variety of phenomena experimentally observed, but also predict some new phenomena. For instance, for a mechanical deformation we find a new phenomenon: the spiral can drift when the frequency of mechanical deformation is equal to the triple frequency of the spiral rotation. For the dc driven system, the drift velocity parallel to the electric field is identified simply to 0.5ME where M is the ion mobility and E is the intensity of the dc field, independent of the particular models generating spiral waves. Numerical computations are carried out as well, and they agree with the main qualitative conclusions of our analytical results. Let us begin with a general two-variable reactiondiffusion system u t f u,v D u 2 u, v t g u,v D v 2 v. 1 Here variables u and v represent the concentrations of the reagents or the temperature, the electric potential, etc.; D u /2003/119(8)/4468/5/$ American Institute of Physics

2 J. Chem. Phys., Vol. 119, No. 8, 22 August 2003 Drift velocity of rotating spiral waves 4469 and D v are the diffusion coefficients of both variables; 2 ( 2 / x 2 2 / y 2 ) is the Laplacian operator; f (u,v) and g(u,v) are the reaction functions. For certain parameter values the system 1 has a dense and rigidly rotating spiral wave solution, and in the neighborhood of the tip the solution can be generally expressed as 7,8 û u 0 a 1 r cos 0 t b 1 r 2 cos 2 0 t b 1r 2, vˆ v 0 a 2 r cos 0 t b 2 r 2 cos 2 0 t b 2r 2, where u 0, v 0, a 1, a 2, b 1, b 2, b 1, and b 2 are eight coefficients and,,, and are four phase shifts; 1 is the chirality of the spiral ( 1 for counterclockwise rotating spiral, while 1 for clockwise one ; 0 is the rotating frequency; x xˆ 0(t) r cos, y ŷ 0 (t) r sin, where (xˆ 0(t),ŷ 0 (t)) is the location of the possibly moving tip. In following discussions, we will not calculate the values of these coefficients, but we will apply the general form 2 to the derivation of the spiral drift problems. Now we consider the influence of two arbitrary external fields 1 (x,y,t) and 2 (x,y,t) on the system motion, u t f u,v D u 2 u 1 x,y,t, v t g u,v D v 2 v 2 x,y,t. Supposing 1 and 2 induce a drift of the spiral wave tip with velocity V(t) V 1 (t),v 2 (t), we can then rewrite the perturbed Eq. 3 in the moving frame of V as u t f u,v D u 2 u V" u 1, v t g u,v D v 2 v V" v 2. We assume that for small 1 and 2, the deformation of the spiral solution around the tip is weak, 9 and all the values of u, v and their derivatives at the tip in the moving frame remain unchanged compared with those values of the unperturbated spiral 2 at the tip, that is, in Eq. 4, u t û t tip, f (u,v) f (û,vˆ ) tip, D u 2 u D u 2 û tip, u û tip, v t vˆ t tip, g(u,v) g(û,vˆ ) tip, D v 2 v D v 2 vˆ tip, and v vˆ tip. Submitting these results into Eq. 4 and remembering û t f (û,vˆ ) D u 2 û tip and vˆ t g(û,vˆ ) D v 2 vˆ tip according to Eq. 1, we obtain finally V" û 1 tip 0, V" vˆ 2 tip 0, i.e., V 1 t 1 vˆ y 2 û y / û x vˆ y û y vˆ x tip, V 2 t 1 vˆ x 2 û x / û x vˆ y û y vˆ x tip, where û x tip a 1 cos( 0 t ), û y tip a 1 sin( 0 t ), vˆ x tip a 2 cos( 0 t ), vˆ y tip a 2 sin( 0 t ), and û x vˆ y û y vˆ x tip a 1 a 2 sin( ). Thus an approximate drift velocity formula in terms of external fields and spiral solutions is given. In this paper we will use the Bär model 10 a modified FitzHugh Nagumo model for numerical simulations, where D u 1, D v 0, f (u,v) 1 u(u 1) u (v b)/a, and g(u,v) v, if 0 u 1/3; g(u,v) u(u 1) 2 v, if 1/3 u 1; g(u,v) 1 v, ifu For comparing with numerical simulations of the Bär model, we set D u 1 and D v 0 for following derivations. From the approximate drift velocity formula 5, we will first discuss the case of mechanical deformation of the medium. The medium stretching can be modeled by an operation where any fixed point x of the medium is changed to x (t). Here, we consider a simple oscillation 6 x (t) x 1 A cos( t ) and Eq. 1 is then modified to u t f u,v u x x u yy, v t g u,v. 6 Using the relations u x x x x u x x x x x x u, x x 1 A cos t 1, one can reduce the forced equation 6 to u t f u,v 1 A cos t 2 u xx u yy, v t g u,v, which is identical to Ref. 6: the stretching of the medium can be modeled by changing the size of the grid in the x direction in the numerical simulation of Eq. 1. Considering A 1 and taking the first-order effect: 1 A cos( t ) 2 1 2A cos( t ), we have u t f (u,v) 1 2A cos( t ) u xx u yy. Now 1 2A cos( t )û xx and 2 0 in Eq. 5 and the drift velocity induced by the periodic mechanical forcing then reads V 1 (t) V (t) 2A cos( t )û xx vˆ y /(û x vˆ y û y vˆ x) tip and V 2 (t) V (t) 2A cos( t )û xx vˆ x /(û x vˆ y û y vˆ x) tip. Inserting Eq. 2 into V (t) and V (t), we obtain the formula of the drift velocity of the tip: V t C cos t 2b 1 sin 0 t b 1 sin 0 t 2 b 1 sin 3 0 t 2, V t C cos t 2b 1 cos 0 t b 1 cos 0 t 2 b 1 cos 3 0 t 2, where C 2A/ a 1 sin( ). Thus an approximate formula is obtained directly from the original partial differential equation for the drift of mechanical deformation where the effects of the driving phase and the chirality of the spiral are given explicitly. To get a net drift velocity, should take the resonant frequency 0 or 3 0. For 0, the constant or average drift velocity reads 7 8 V Cb 1 sin 0.5Cb 1 sin 2, 9 V Cb 1 cos 0.5Cb 1 cos 2, where the oscillatory parts in Eq. 8 are averaged out. It is clear that there exists a drift velocity V orthogonal to the stretching direction, and the direction of the drift can be adjusted by changing the phase shift of the stretching pertur-

3 4470 J. Chem. Phys., Vol. 119, No. 8, 22 August 2003 Zhang et al. FIG. 1. Dependence of the drift by mechanical deformations on the phase shift at resonant frequency 0 a, b and at 3 0 c, d. a, c counterclockwise ( 1) spiral; b, d clockwise ( 1) spiral. Parameters in the forced FitzHugh Nagumo equation 7 are a 0.84, b 0.07, 0.02, A Grid points, and t t.u. are used in our simulations. The system size is L L, L 35. bation. In Figs. 1 a and 1 b, we present the tip motion by numerical simulation of Eq. 7 for counterclockwise ( 1) and clockwise ( 1) spirals at 0 for different phase shifts. The formula 9 agrees with the numerical observations and explains the experimental observations mentioned in the introduction. A significant feature predicted in Eq. 8 is that the constant drift velocity can also appear at triple frequency of the mechanical deformation, i.e., for 3 0. The net drift velocity for 3 0 is V 0.5Cb 1 sin 2, V 0.5Cb 1 cos In Figs. 1 c and 1 d, we show the numerical results of Eq. 7 for the drift of counterclockwise ( 1) and clockwise ( 1) spirals at 3 0 for various phase shifts. They confirm the results of Eq. 10. In our simulations, a constant drift has not been observed for 2 0,4 0,5 0,6 0, and so on. From Eqs. 9 and 10, some relations can be computed without knowing the particular constants of Eq. 2. These relations are interesting because they are independent of the special models and should be of general significance. Defining the constant drift velocity amplitude V and the drift angle as V 2 V 2 V, tan V /V 11 we find in Eq. 10 for the triple-frequency resonance that V 0.5Cb 1, which is independent of, and is linear in, i.e., ( /2 2 ). But, these simple relations definitely do not hold for the 0 resonance according to Eq. 9, and such predictions are confirmed in Fig. 2. The general result of Eq. 5 can also be applied to the case of the spiral wave drift of a homogeneous electric field, which has been a topic of extensive investigation. 5 In this case Eq. 1 becomes u t f u,v 2 u MEu x, v t g u,v, 12

4 J. Chem. Phys., Vol. 119, No. 8, 22 August 2003 Drift velocity of rotating spiral waves 4471 FIG The quantities V a, c and b, d defined in Eq. 11 vs. 3 0 for a and b, and 0 for c and d. These numerical results confirm the results of Eqs. 9 and 10 also see the discussions after Eq. 11. where M is the ion mobility and E the electric field strength. For ME 1, we can insert 1 MEû x, 2 0, and Eq. 2 to Eq. 5 and then obtain an approximate formula for the drift velocity: V t 0.5ME 0.5ME csc sin 2 0 t, 13 V t 0.5ME cot 0.5ME csc cos 2 0 t. A clear conclusion in Eq. 13 is that the spiral wave undergoes a directional drift when the electric field frequency is zero dc field or twice the spiral frequency. When a dc electric field is applied, only the first components in Eq. 13 contribute to the constant drift of the spiral tip, and V FIG. 3. The numerical results of Eq. 12 for counterclockwise ( 1) spiral wave. The dc electric field is along the x direction, ME 0.002, a V, b is fixed at 0.07; b V, a is fixed at The dashed lines in a and b show the values of the theoretical prediction of Eq. 13 for dc electric field. The ac electric field ME , with 2 0, c V vs ; d vs. These numerical results agree with the conclusions of Eq. 13 for the drift induced by electric fields. Parameters in Eq. 12 are a 0.84, b 0.07, 0.02.

5 4472 J. Chem. Phys., Vol. 119, No. 8, 22 August 2003 Zhang et al. 0.5ME, V 0.5ME cot( ). It is seen that the transverse component of the drift velocity V is not zero and its sign is determined by the ME and the chirality of the spiral. This result agrees with the experimental observations so far. 5 An interesting result in Eq. 13 is that the component of the constant drift parallel to the field, V 0.5ME, has a simple and general form. It does not depend on the particular model generating the spiral wave. This conclusion is confirmed by numerical simulations of Eq. 12 in Figs. 3 a and 3 b. When we add an ac electric field E E 0 cos( t ), E 0 0, with 2 0, the spiral will drift with the constant velocity V 0.25ME 0 csc( )sin( ), V 0.25ME 0 csc( )cos( ). From these formulas we expect that V is independent of, and is linear in. These analytical results are confirmed by the numerical computations in Figs. 3 c and 3 d. However, the numerical results of the ac drift in Ref. 11 show that V does depend on and is not linear in. The reason is clear: in Ref. 11 ME 0 is equal to 2.0, while in our case ME 0 in Figs. 3 c and 3 d is All our results are valid for weak perturbation. In conclusion, we have derived, directly from the original reaction-diffusion equation and its spiral wave solution, an approximate formula of the perturbation-induced spiral wave drift velocity that includes the drifts of a mechanical deformation and an electric field as special cases. Our analysis is performed under the approximation that the deformation of the spiral in the tip region is neglected when spiral resonantly drifts. This approximation is found to be good for weak perturbations and rigidly rotating dense reference spirals. With this approximate formula we are able to explain the main features appearing in the directional spiral-tip-drift problems, such as the constant drift of periodic mechanical deformation at 0, the dc electric field net drift and the ac electric field drift at 2 0, and the phase and chirality relations in these drift processes. In particular, we predict a triple frequency resonant drift for the mechanical deformation driving in Eq. 8 and the simple V 0.5ME relation for the dc electric field induced drift in Eq. 13. These analytical results are confirmed qualitatively by direct simulations of a reaction-diffusion model. The main results of this paper, e.g., the directional triple-frequency-drift of a mechanical deformation, etc., are also verified by numerical simulations of the Oregonator model. 12 Finally, we must point out that our work can also be directly applied to the study of the drift of spiral waves during periodic modulation of the excitability of an excitable medium, 13 which has been well investigated by Mikhailov, Davydov, and Zykov, 14 Mantel and Barkley, 15 and Hakim and Karma. 16 We will discuss this kind of drift phenomenon and compare our work with other works in detail in another paper. This work was supported in part by grants from the Hong Kong Research Grants Council RGC, the Hong Kong Baptist University Faculty Research Grant FRG, the National Nature Science Foundation of China, and the Nonlinear Science Project of China. We would like to thank Q. Ouyang, Z. L. Qu, F. G. Xie, and S. L. Zhang for useful discussions. 1 A. T. 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