Spiral wave motions

M. Georgi and N. Jangle. Spiral wave motion in reaction-diffusion systems. In... 

Spiral waves also arise in the oxidation of carbon-monoxide on platinum surfaces [10]. In 1972 they have been discovered by Winfree [79] in the photosensitive Belousov-Zhabotinsky (BZ) reaction, see for recent investigations for example [83, 84, 87]. Both reactions are studied in the SFB 555. The classical BZ reaction is a catalytic oxidation of malonic acid, using bromate in an acidic environment. Experimentally it exhibits well reproducible drift, meander and chaotic motions of the spiral wave and its tip. 

Fig. 3.1. Outward moving spiral wave (left) with different associated tip motions (right rigid rotation, drift and meander).

Varying the light intensity enables effective control of the motion of spiral waves, see [7, 37, 75, 88]. Controlling the motion of spiral waves is an important challenge, for example, for the defibrillation of cardiac tissue. The elimination of spiral waves by multiple shocks of external current has been investigated numerically in [64]. 

The range of has an interesting interpretation It represents the vector fields of all possible motions of the perturbed spiral wave tip z(t), to leading order in e. Indeed, a vector field h in the range of allows one to find a... 

We repeat that the construction of the functional H in [25] involves quantities such as the group orbit 5 (2)u of the unperturbed spiral wave. In particular, H will be a nonlocal functional, in general. Keeping in mind that the experimentalist has only a few control parameters at hand, it remains a challenging task to adjust the light intensity pattern in order to obtain the desired spiral tip motion. For specific results in this direction see [38, 84, 85, 87, 88], and our brief discussion in section 3.4. 

Keeping in mind that the dynamics of (3.31) represents the motion of the perturbed spiral wave tip, this example shows that there are indeed open sets of initial positions z(0) for which the spiral tip z t) finally gets pinned to the sink A. Coexisting are regions for which the spiral tip never gets attracted to any specific point and undergoes a drifting motion on the... 

A. Mikhailov, V. Davydov, and V. Zykov. Complex dynamics of spiral waves and motion of curves. Physica D, 70(l-2) l-39, 1994. 

The light-sensitive BZ reaction demonstrates all basic features of excitable media of quite different nature and represents a very suitable experimental system to study controlled motion of spiral waves. In the experiments reported below an open gel reactor is used [29-31]. The catalyst is immobilized in a silicahydrogel layer of 0.5 mm thickness prepared on a plate... 

The autonomous system with I t) = 0.06 has a steady state which is stable with respect to small perturbations. However, a supra-threshold perturbation, once locally applied, gives rise to a concentric wave propagating through the medium. Near to the center of the simulated domain a spiral wave rotating counterclockwise was created by a special choice of initial conditions. The spiral tip performed a compound rotation (meandering motion) including at least two different frequencies. The oscillation period measured far enough from the core center of unperturbed trajectory was... 

Variation of the chemical composition of the BZ reaction has remarkable effects on the tip motion of a spiral wave [19-22]. Here the effects induced by lowering the acidity (and thus the excitability) of the medium are discussed. Figure 9A depicts the path of the spiral tip for five different proton concentrations. The traces were obtained by copying the spiral tip from the video screen on a transparency during a slow motion playback of the recorded video movies. The trajectories may be considered as prolate hypocycloids [39] observing by definition Agladze s rule [20], e.g., a counterclockwise overall motion superimposed by a clockwise motion within the loops or vice versa. In the following the trajectories will be discussed in terms of epicycles. 

Figure lOA) of a sequence of 15 half-tone images gives an impression of the dynamics of the complex tip motion during the formation of a loop. One clearly sees the variation of the shape of the tip and the intensity modulation in the interior of the mold formed by the innermost portion of the spiral wave. 

Other characteristics of the center are described by the ratios of the wave amplitudes and frequencies at the center and other locations in the territory excited by the spiral wave. In most cases the chemical cycle at the center has the highest frequency and lowest amplitude (Figure 15). Frequency ratios n/n-l and nfn-2 have been observed, where n is the number of petals in a pattern. The amplitude ratio shows a branching behavior probably correlated with the Hopf bifurcation revealed by Barkley etal. [50]. Careful evaluation of the initial values derived from solutions for / < 1.75 and / > 1.75 allows the calculation of the solutions in the vicinity of the bifurcation point (Figure 16B). The gap around / = 1.75 results from the limitations in computing power, since the transient time of the system from the initial state to steady motion is extremely long in this region of / values [51]. 

The comparison of the experimental data with numerical simulations of high spatial resolution shows a remarkable agreement regarding the symmetry and tip dynamics of pairs of spiral waves. The reason why in most experiments pairs of counter-rotating spirals appear is found in the experimental procedure that normally generates two open ends of a wave which both curl up to a pair of spirals. An analysis of experimental data showed that the spirals evolve independently of each other without any synchronisation of their motion. The... 

Actually, the motion of the curves is not completely independent. A propagating excitation front is followed in excitable media by the recovery tail. When the distance between the waves moving in the same direction is shorter than the length of such a tail the waves interact. Such recovery effects become important for spiral waves when their rotation period is comparable to the characteristic refractory time of the medium. This happens when spiral waves are no longer sparse, i.e. the conditions of weakly excitable media are... 

As we already mentioned, when a spiral wave is sparse it can be described at a low resolution by a single curve, not distinguishing the front and the back of the excitation zone (this approach was taken also in [6]). Then the tip of the wave is represented by the end point of such a curve. To define the motion of a curve with a free end, the law of the tangential motion of its end point should be additionally specified. 

SPIRAL WAVES IN WEAKLY EXCITABLE MEDIA 125 3. Motion of Curves with Free Ends... 

Suppose that the curvature at the free end of a curve is equal to the critical curvature kc. Then the conditions for the curve s motion would be effectively the same as if a circular obstacle were present in the medium and the curve s tip were rotating around it. The curve represents then a spiral described by (28) where ko must be replaced by kc. Its free tip rotates around an effective circular obstacle which may be called a core of the spiral wave. The core radius i o can be found by substituting kc instead of ko in (27) and solving the resulting equation, which yields... 

The set of equations which includes the general kinematical equation (10) supplemented by boundary condition (11) and Equations (12, (13) and (15) of motion of the end point describes not only a steadily rotating spiral. It can also be used to determine the evolution of a curve starting from an arbitrary initial condition. Moreover, if the parameters of these equations are some functions of time and/or of spatial coordinates, the same set of equations describes the behaviour of spiral waves in time-dependent or nonuniform weakly excitable media. 

If the properties of a medium change in time and/or in space, the parameters of the kinematical model (i.e. Vo. D, 7 and Gq or kc = G0/7) become certain functions of the spatial coordinates and/or of time. In the quasi-steady approximation the motion of the tip is influenced only by the dependence of the properties of the medium along its trajectory. It means that the kinematical parameters in the Equations (38)-(41) would represent certain functions of Xo and Yq. When such functions are known, the system of these equations can be solved to determine the trajectory of motion of the tip, and, hence, the behaviour of the entire spiral wave. 

If one tries to develop a perturbation theory proceeding directly from the reaction-diffusion equations this meets with serious difficulties. They arise because translation and rotation perturbation modes for the spiral wave are not spatially localized. We bypass such difficulties by using the quasi-steady approximation formulated in the previous section. In this approximation the trajectory of the tip motion can be calculated by solving a system of ordinary differential equations which depend only on the local properties of the medium in the vicinity of the tip. The perturbations which originate outside a small neighbourhood of the end point propagate quickly to the periphery and do not influence the motion of the tip. The evolution of the entire curve can then be calculated in the WR approximation using the known trajectory of the tip motion as a dynamic boundary condition. 

After putting kc(t) = Go t)/j into Equation (38) of the quasi-steady approximation and linearizing this equation, one can find variations of the curvature ko at the free end of the curve which are induced by the modulation of Gq. Substitution of these variations into the linearized equation (39) yields the time dependence of the angle cco- The motion of the tip of the spiral wave can then be calculated from (40) and (41). We find that in the linear... 

The above results for the motion of the tip of a spiral wave induced by the periodic modulation of the properties of an excitable medium were derived, in a slightly different form, in 1986 in [22] and later extensively discussed in [13, 26, 28, 29]. They show that the tip perforjns a cycloidal motion that represents a superposition of a rotation around a circle of radius Ro and of a circular motion with radius i i (hence the momentary rotation centre of the spiral wave moves along a circle of this radius). The type of the cycloid depends on the relationship between the two frequencies. If ujo > u>t, the centre of the spiral wave moves in the direction which is opposite to the direction of its rotation (i.e. clockwise if the spiral wave rotates in the counterclockwise direction, see Figure 5). When wq > wi, these two rotation directions coincide. 

Under the condition of the complete resonance the rotation centre of a spiral wave moves along a straight line, so that its tip performs the motion described by... 

A related effect for spiral waves inside small disks of excitable medium has been recently investigated [18]. The numerical simulations of a reaction-diffusion system showed that the dynamical regime with a spiral wave steadily rotating around the centre of the disk is unstable in small disks. A slight shift of the initial position of the spiral wave leads to the motion of the wave s core towards the boundary of the disk and then the spiral waves begins to migrate along this boundary at a certain distance from it. 

Generally, such interactions result in the breakdown of rigid rotation and the emergence of the meandering regimes for spiral waves. These regimes are characterized by the complex motion of the wave tip. The trajectory of such motion is sensitive to the parameters of the medium. Near the onset of meandering it represents a cycloid obtained by a superposition of two independent circular motions. 

It follows from (74) that the motion of curves is sensitive to the local properties of the surface only if its Gaussian curvature is nonvanishing. Since the Gaussian curvature is zero for such surfaces as cylinders and cones (because one of the two principal curvatures vanishes) the motion of curves and the properties of spiral waves on these surfaces are identical to those on the plane. 

Fig. 13. The trajectory of the tip motion of a spiral wave on the surface of a prolate spheroid obtained by the numerical integration of the reaction-diffusion equations (60)- 62) with additional diffusion of the inhibitor Du = Du = ). (From [28])...

If the sphere is breathing, so that its radius periodically changes with time, the motion of the tip of the spiral wave is periodically perturbed. As was pointed out in [51], this leads to the drift of the spiral wave over the sphere which is similar to the resonance of a spiral wave on the plane. 

When the Gaussian curvature of the surface is not constant (for example, for a prolate spheroid) the wave s tip experiences varying curvature F as it moves over the surface. This results in the systematic drift of spiral waves on the nonuniformly curved surfaces [51 ]. To check this prediction, the numerical simulation of the spiral wave on the surface of a prolate spheroid has been performed in [28] using full reaction-diffusion equations of the model (60)-(62). Figure 13 shows the computed trajectory of motion of the tip of a spiral wave on the coordinate plane (0, ) where 6 and are the spherical angles. We see that the spiral wave drifts approximately along the equator of the spheroid. 

This section is devoted to numerical experiments on the motion of spiral waves as described previously, in finite systems of square and circular geometries. Both types of geometries are considered in order to investigate which properties are generic for systems of limited size and which effects are due to the particular shape of the boundaries. Moreover, two different dynamics will be considered. The oscillatory units are described either by the CGL equation of by relaxational oscillators. 

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