Meandering Spiral Waves Induced by Time-Periodic Coupling Strength

Commun. Theor. Phys. 60 (2013) 545 550 Vol. 60, No. 5, November 15, 2013 Meandering Spiral Waves Induced by Time-Periodic Coupling Strength WANG Mao-Sheng ( á), SUN Run-Zhi (ê ì), HUANG Wan-Xia (á ), TU Yu-Bing ( Ï), and ZHANG Ji-Qian ( ) Department of Physics, Anhui Normal University, Wuhu 241000, China (Received May 6, 2013; revised manuscript received August 16, 2013) Abstract Effects of time-periodic coupling strength (TPCS) on spiral waves dynamics are studied by numerical computations and mathematical analyses. We find that meandering or drifting spirals waves, which are not observed for the case of constant coupling strength, can be induced by TPCS. In particular, a transition between outward petal and inward petal meandering spirals is observed when the period of TPCS is varied. These two types of meandering spirals are separated by a drifting spiral, which can be induced by TPCS when the period of TPCS is very close to that of rigidly rotating spiral. Similar results can be obtained if the coupling strength is modulated by a rectangle wave. Furthermore, a kinetic model for spiral movement suggested by Di et al., [Phys. Rev. E 85 (2012) 046216] is applied for explaining the above findings. The theoretical results are in good qualitative agreement with numerical simulations. PACS numbers: 05.45.-a, 82.40.Ck, 89.75.-k Key words: meandering spiral wave, drifting spiral wave, time-periodic coupling strength 1 Introduction Spiral waves have been observed in a wide variety of excitable systems in physics, chemistry, and biology. For example, they occur in aggregating slime-mold cells, [1] CO oxidation on platinum, [2] and Belousov Zhabotinsky (BZ) reaction. [3] Another important example is in the heart, for which spiral waves play an essential role in heart diseases such as arrhythmia and fibrillation. [4 6] Both experimental and theoretical studies show that understanding and controlling spiral wave dynamics has a very important practical significance. [7 8] Spiral wave dynamics is primarily characterized by the motion of its core (i.e., the trajectory of the spiral wave tip, defined to be a phase singularity). [9 10] The simplest case is a periodic motion called rigid rotation with a spiral tip moving along a circle. Winfree s [11] careful examination revealed that the tips of spirals could trace out epicycloidlike orbits (i.e., flowerlike orbits with inward petals) and hypocycloidlike orbits (i.e., flowerlike orbits with outward petals), which are called outward meandering and inward meandering, respectively. In particular, the spiral tip appears to drift in a straight line. [9,12] Drift, characterized by the displacement of the core position with time, is a possible underlying mechanism for polymorphic ventricular tachycardia, [13 14] which is also believed to be related to ventricular fibrillation. Therefore, understanding the mechanisms leading to spiral drift is not only a problem of fundamental interest in the physics of nonlinear dynamical systems, but also has potential clinical significance. [15] Several methods have emerged to control or induce drifting spiral wave. Studies [16 18] have shown that a spiral wave in a heterogeneous medium generally drifts toward the territory with longer refractory period. However, anomalous drift of spiral waves (drift toward regions having shorter period or stronger coupling) in heterogeneous excitable media was also reported. [19] Recently, the drifting dynamics of spiral wave have been intensively studied. Di et al. developed a set of kinematic equations to describe the spiral wave drift under external electric field, [20] or with heterogeneous [21] or periodic excitability. [22] Spatial thermal gradient can cause the spiral patterns to drift in heterogeneous CO oxidation on platinum (110). [23] Yuan has studied numerically the drifting behaviors of spiral waves in excitable medium with excitability modulated by rectangle wave. [24] In particular, spiral waves will drift in response to symmetry-breaking perturbation, temporally [22,24] or spatially, [19,21,23] associated with a certain model parameters. Diffusion is also a crucial factor in pattern formation of reaction-diffusion systems. In most studies mentioned above, constant coupling strength (or diffusion coefficient) has always been assumed. However, in many cases of real systems, coupling strength can be time or space dependent. For example, in neural networks, the coupling among neurons (via electrical and/or chemical synapses) can be adaptive. In Ref. [19], gradient coupling strength was introduced and anomalous drift of spiral waves was observed. Very recently, Birzu et al. [25] studied the dy- Supported by the National Natural Science Foundation of China under Grant No. 21103002 and the Natural Science Foundation of Education Bureau of Anhui Province under Grant No. KJ2010A129 E-mail: maosheng@ustc.edu c 2013 Chinese Physical Society and IOP Publishing Ltd http://www.iop.org/ej/journal/ctp http://ctp.itp.ac.cn

546 Communications in Theoretical Physics Vol. 60 namics of a population of globally coupled FitzHugh Nagumo oscillators with time-periodic coupling strength (TPCS). Wang et al. [26 28] have also studied the effects of TPCS on the dynamics of scale-free or small-world neural networks, and rich coherence resonance behaviors have been observed. The effects of mechanical deformations on spiral wave in elastic excitable medium have been reported by Munuzuri et al., [29] and the drift velocities of spiral waves driven by a periodic mechanic deformation or a constant or periodic electric field have been obtained by Zhang et al. [30] The mechanical deformation of the medium can be modeled by changing the size of the grid in the x direction in the numerical simulation, [29] which corresponds to periodic change of coupling strength in this direction. [30] One notes that TPCS introduces a kind of broken of the system s time translational symmetry, and it is thus interesting for us to investigate the effects of TPCS on the drift of spiral waves in excitable media. In this study, based on Barkley model, [12,31] we have studied the effects of TPCS on the dynamics of spiral wave. Spiral tip trajectories under the condition of TPCS were recorded, and a transition between epicycloidlike and hypocycloidlike orbits was observed. When the period of TPCS is very close to the period of rigidly rotating spiral, spiral tip travels in a straight line (i.e., resonance drift of spiral wave). Effects of rectangle wave modulate coupling strength have also been studied, and similar results were observed. Furthermore, a kinetic model for spiral movement suggested by Di et al. [20 22] was employed to explain our findings, and the numerical results are in good accordance with the theory. 2 Meandering Spiral Waves Induced by TPCS and Analyses The spiral dynamics was studied by means of the Barkley model: u t = 1 ( ε u(1 u) u v + b ) + d 2 u, a v t = u v. (1) Here, the variables u and v describe the fast and slow variables of excitable systems, respectively. ε 1 is a small positive parameter, characterizing the excitability of the medium and reflecting the disparate time scales of variables u and v. 2 = 2 / 2 x + 2 / 2 y is the Laplacian operator. Numerical simulation was taken on a twodimensional square sheet with 128 128 space sites and no-flux boundary condition was used. Equation (1) was integrated by an explicit Euler method with time step t = 0.002 44 and spatial step x = y = 0.3125, with the initial state to be a single spiral wave. The parameters are a = 0.55, b = 0.05, and ε = 0.02 if not otherwise stated. d is the diffusion coefficient of u, which also stands for the coupling strength among the space sites. For fixed d = 1, the spiral wave rotates stably in clockwise direction with the tip on a rigid circle (as shown in Fig. 1). Here we define the spiral tip to be the intersection of the two contours u = 0.55 and v = 0.154. Fig. 1 (Color online) Spiral wave of variable u and tip trajectory (the red rigid circle) for d = 1. Fig. 2 (Color online) Dependence of radius R traj of the spiral wave s tip orbit on the coupling strength d. Figure 2 shows the effects of coupling strength d on the radius (R traj ) of the tip orbit. Here, radius R traj is defined as R traj = (x max x min )/2. x max and x min are the maximum and minimum x-coordinates of the tip trajectory. The insets in Fig. 2 show the tips trajectories for d = 0.5 and d = 1.5, respectively. It is well known that changing the diffusion coefficient is equivalent to changing the length scale. Hence, the theoretical curve of R traj versus d can be simply given by R traj (d) = R traj d=1 d. (2) Moreover, the spiral period will be independent of d. We would like to point out here that R traj d=1 = 4.867 18 and the rotation period is about 5.388 s, i.e., T 0 2208 time steps in our simulations. Pervious simulated results showed that the coupling strength d has important influence on the spiral wave dynamics. However, the coupling strength d considered so far is constant. Now, let us divert our attention to the effects of TPCS on the tip trajectory. To this end, we assume: d = 1 + 0.05 sin(2πt/(t t)), (3)

No. 5 Communications in Theoretical Physics 547 where T (unit: time steps) is the period of the coupling strength. We now study the effects of TPCS on spiral dynamics by investigating the spiral trajectories for different T, and the simulation results are shown in Fig. 3. In the region of T < T 0 (see Figs. 3(a) to 3(d)), the spiral tips move in hypocycloidlike orbits (with outward petals). On the other hand, when T > T 0 (see the Figs. 3(f) to 3(i)), the spiral tips move in epicycloidlike orbits (with inward petals). For a certain critical value T = 2210, which is approximately equal to the value of T 0 = 2208, the meandering radius [32] (also be called radius of secondary circle [9] ) diverges yielding a drifting spiral wave (or traveling wave [9,32] ). When T is close to T 0 (see Figs. 3(d) to 3(f)), the meandering radius of the spiral is so large that the tip will touch the boundary and then move along the boundary. Fig. 3 Tip trajectories of spiral wave for different T of TPCS. (a) (d) correspond to the case of T < T 0, the tip meandering with outward petals. (f) (i) correspond to the case of T > T 0, the tip meandering with inward petals. In (e) the T of TPCS is approximately equal to T 0, traveling spiral is observed. As mentioned above, periodic changing of the coupling strength leads to a kind of broken of time translational symmetry. One may thus expect that some other type of time modulation can also result in similar effects. For example, we have also studied the case where the coupling strength is modulated by rectangle wave, i.e.: 1.05, t [nt t, n(t + T/2) t), d = (n = 0, 1, 2,...), (4) 1, t [n(t + T/2) t, (n + 1)T t), where T (unit: time steps) stands for the period of the rectangle wave. The simulation results are displayed in Fig. 4, which are qualitatively the same as that depicted in Fig. 3. Again, the system undergoes a transition from outward petal (see Figs. 4(a) 4(d)) to inward petal (see Figs. 4(f) 4(i)) meandering spiral waves as T is increased, and drifting spirals are observed for T = 2207 (see Fig. 4(e)).

548 Communications in Theoretical Physics Vol. 60 Fig. 4 Tip trajectories of spiral wave for different T of rectangle wave modulate coupling strength. (a) (d) correspond to the case of T < T 0, the tip meandering with outward petals. (f) (i) correspond to the case of T > T 0, the tip meandering with inward petals. In (e), T is approximately equal to T 0, traveling spiral is observed. Fig. 5 (Color online) A schematic diagram for spiral movement. To understand our findings in an analytical manner, here we adopted the kinematic model proposed by Di et al. [20 22] for spiral movements. Generally, point q, the so-called phase change point at which the wave front and wave back coincide is considered as the spiral wave tip. Another characteristic point is Q where the radial direction is tangent to the wave front, this point rotating around the center describes the boundary of circular core of the spiral wave (see Ref. [33] for more details). In the kinematic model proposed by Di et al., [20 22] point Q is called the spiral wave tip. Figure 5 shows a schematic diagram of a drifting spiral wave. In the constant coupling case, where d does not vary with time, the spiral wave rotates rigidly. The spiral tip traces a circle in the medium with a normal velocity c n = V traj (d) = 2πR traj (d)/t 0 and a radial velocity c g = 0. However, for the case of TPCS, a changing d will lead to a changing instantaneous radius, which will cause an additional radial velocity. Generally, the tip motion can be described by the following equations: ẋ = c g cosθ hc n sin θ, ẏ = c g sinθ + hc n cosθ, θ = hω, (5) where h is the chirality of the spiral wave (h = +1 for counterclockwise and h = 1 for clockwise rotation), and ω is the rotation frequency. For a constant coupling strength d, all the parameters c g, c n and ω are also constants. With TPCS, however, these parameters are also time dependent. Since the periodic perturbation is of

No. 5 Communications in Theoretical Physics 549 small amplitude as shown in Eq. (3), here we adopt a linear expansion around the reference point d = 1 to address the time dependence as follows: c n (d) = V traj (1) + λδv traj = V traj (1) + λv traj (1)δd, c g (d) = ρδr traj αδv traj = (ρr ctraj αv traj )δd, ω = 2π T 0 β c g c n, (6) where a prime on V traj and R traj denotes their derivatives with respect to parameter d. λ, ρ, α, and β are adjustable parameters, and according to Eq. (3) ( 2πt ) δd = 0.05 sin. (7) T t Fig. 6 (Color online) Tip trajectories of spiral wave from simulation and the kinematic model for different period of TPCS. (a), (d), and (g) show tip trajectories for period T = 2180, 2210, and 2240 from simulations, the corresponding results from the kinematic equations are plotted in (c), (f), and (i). The amplified views of (a), (d), and (g) are also shown in (b), (e), and (f). The spiral wave rotates clockwise (see the black arrow), and the red arrows indicate the directions of spiral wave drift. The parameters sets for the kinematic model are λ = 1.56, ρ = 2.73, α = 1.56, and β = 0.4. Combining Eqs. (5), (6), and (7), one can obtain the tip trajectories under different conditions. For example, Figs. 6(a), 6(d), and 6(g) show tip trajectories for period T = 2180, 2210, and 2240 from simulations, respectively. In order to clearly show the characteristics of tip motion, the partial enlargement of Figs. 6(a), 6(d) and 6(g) are displayed in Figs. 6(b), 6(e), and 6(f). Figures 6(c), 6(f), and 6(i) are the corresponding results for period T = 2180, 2210, and 2240 from the kinematic equations. They all show a very good agreement qualitatively. 3 Conclusion In conclusion, we have studied the effects of TPCS on spiral wave dynamics of reaction diffusion system by using the Barkley model. TPCS can make the spiral wave tip meandering, including outward petal and inward petal meandering, as well as linear drift. Similar behaviors of spiral tip can also be observed when the coupling strength is modulated by a rectangle wave. Moreover, the underlying mechanism of resonance drifts of spiral wave has been discussed based on the kinematic model introduced in Ref. [22]. The results of theoretical analysis coincide very well with the numerical simulations. The diastole or shrinkage of the heart will result in deformations of cardiac tissue. [34] These deformations in turn affect spiral wave dynamics in the heart. [35] On the other hand, the effect of media deformation can be simulated by changing the diffusion coefficients with changeable signals. [34] Our findings strongly suggest that TPCS has remarkable effects on the spiral wave dynamics in Barkley model. Finally, we hope this work can give some useful clues in understanding the meandering spiral waves of cardiac tissue.

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