Spiral wave drift

J.M. Davidenko, A.V. Pertsov, R. Salomonsz, W. Baxter, and J. Jal-ife. Stationary and drifting spiral waves of excitation in isolated cardiac muscle. Nature, 355 349-351, 1992. 

Either approach has advantages and disadvantages when tackling rotating, meandering and drifting spiral wave phenomena. 

B. Sajidstede and A. Scheel. Superspiral Structures of Meandering and Drifting Spiral Waves. Phys. Rev. Lett., 86 171-174, 2001. 

C. Wulff. Theory of Meandering and Drifting Spiral Waves in Reaction-Diffusion Systems. PhD thesis, FU Berlin, 1996. 

Stationary and drifting spiral waves of excitation in isolated ctirdiac muscle. Nature 355 349-51. 

Davidenko, J. M. Pertsov, A. V. Salomonsz, R. Baxter, W. Jalife, J. 1992. Stationary and Drifting Spiral Waves of Excitation in Isolated Cardiac Muscle, Nature 355, 349-351. 

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).

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... 

As mentioned in the introduction, meandering and resonant drift of spiral waves in the photosensitive BZ reaction can be achieved experimentally by a periodically changing light intensity [9, 90]. The Doppler effect imposes superspiral structures in the case of spiral wave dynamics other than rigid rotation. These superspiral structures have been observed in experiments see for example [11, 41, 55, 56, 89]. See [70] for a mathematical analysis based on linearized analysis and eigenfunctions. 

V. Zykov, H. Brandtstadter, G. Bordiougov, and H. Engel. Interference patterns in spiral wave drift induced by a two-point feedback. Phys. Rev. E, 72 1-4, 2005. 

Resonant drift of a spiral wave under periodic parameter modulation. 250... 

Spiral wave drift near a straight line detector.256... 

The control methods we have in mind base on the resonant drift of spiral waves in response to a periodic change in the excitability of the whole... 

Resonant drift of a spiral wave is a displacement of the spiral wave center induced by a periodic modulation of the medium excitability. This phenomenon has been predicted for a kinematical model of weakly excitable media [14], confirmed experimentally as well as in numerous numerical simulations [16, 19, 20, 45], and explained as a generic property of excitable media [15, 17]. 

It is important to stress, that if aim = nut, where n is an integer, n > 1, the result of n subsequent displacements, i.e, after one rotation period of the spiral wave, is equal to zero, because Zk+n = Zk- A long term drift is absent in this case. 

Thus, the velocity of the resonant drift induced by the periodic modulation is determined by the ratio h/Tm- Under resonant forcing, ujm = the drift is along a straight line whose direction depends on the initial orientation of the spiral wave o and on the constant ip. More generally, if the parameter modulation is given by... 

Let us assume that the drift is slow and determine its direction. Obviously, the phase of the pulse sequence depends on the spiral location. Indeed, if the spiral wave center is placed at the point z = x + iy, the wave front specified by Eq. (9.5) crosses the detector point located at the origin of the coordinate system each time ti satisfying the following equation ... 

Application of one-channel feedback control to spiral waves in the light-sensitive BZ system allows to observe the discrete set of stable resonant attractors experimentally [21, 30, 43, 46]. Note, that Eq. (9.23) for the radius of the resonance attractor contains only one medium dependent parameter (p, which specifies the direction of the resonance drift. To avoid a rather complicated experimental procedure to determine this value, the obtained experimental data were fitted to the theoretically predicted linear dependence (9.23) using p = —0.31. The results are shown in Fig. 9.3 by dashed lines. Then, the boundaries of the basin of attraction were specified in accordance with Eq. (9.24) (solid lines in Fig. 9.3). 

Fig. 9.4(b) shows the spiral tip trajectory obtained experimentally under this feedback control. After a short transient the spiral core center drifts in parallel to the line detector. The asymptotic drift trajectory reminds the resonance attractor observed under one-channel control, because a small variation of the initial location of the spiral wave does not change the final distance between the detector and the drift line. To construct the drift velocity field for this control algorithm an Archimedean spiral approximation is used again. Assume the detector line is given as a = 0 and an Archimedean spiral described by Eq. (9.5) is located at a site x,y) with a > 0. A pure geometrical consideration shows that the spiral front touches the detector each time ti satisfying the following equation ... 

Fig. 9,4. Spiral wave drift near a virtual line detector, (a) Snapshot of a spiral wave in a thin layer of the light-sensitive BZ reaction. The overlaid dashed line represents the detector. The white curve shows the spiral tip trajectory in the absence of feedback, (b) Trajectory of the spiral wave tip induced by the feedback.

Fig. 9.5. Velocity field for spiral wave drift induced near a straight line detector (a) and a line segment (b). Solid lines show trajectories of the spiral center computed for the Oregonator model (9.1). r = 0.

Drift induced near a line detector represents au efficient method to move a spiral wave along a straight line and, hence, to shift it from a given initial location A to a desirable site B along the shortest pathway. Now let us... 

Fig. 9.6. Feedback-mediated drift of a spiral wave observed in a thin layer of the BZ reaction, (a) Spiral wave captured by a heterogeneity defect D. (b) Successful shift of the spiral wave from B to A near a curved detector avoiding the defect D. Overlaid dashed lines represent the applied line detector. Scale bcur 1 mm.

Extensive studies revealed the possibility of spiral wave drift near a convex-shaped one-dimensional detector of arbitrary curvature up to the limiting case of the point detector. The drift near a concave arc is more complicated. Indeed, Fig 7(a) illustrates a continuous drift around a cosineshaped detector. Since the deformation of this detector with respect to a straight line is relatively small, the spiral waves drifts along both concave as well as convex arcs of the detector. However, near a strongly concave arc the drift terminates at some site, as if the spiral wave becomes anchored at some invisible defect, compare Fig. 9.7(b). In fact, the medium is entirely uniform and does not contain any defects. 

To confirm these conclusions we have performed numerical simulations of the feedback mediated drift within the Oregonator model (9.1). The thick solid lines in Fig. 9.9 show the obtained trajectories of a spiral wave center for two different initial locations. The spiral center moves in very good agreement with the predicted drift velocity field and stops near the place where the velocity field vanishes. 

Let us assume that the center of an unperturbed spiral wave is located at site 2 = X + iy. Due to the rotation of the spiral wave, the computed integral B t z) and, hence, the modulation signal I t z) are periodic functions of time with period Too- Applying the signal I t z) to the medium, the spiral wave is forced to drift in accordance with the general rule (9.14). Direction and magnitude of drift velocity are determined by the first Fourier component of the periodic modulation, which can be expressed as... 

For 0.5 < dp/X < 1.5, the drift velocity field changes dramatically, see Fig. 9.10(b). There are three spatially unbounded fixed lines that destroy the circular-shaped attractors existing as long as dp/ < 0.5. In numerical simulations with the Oregonator model (9.1), the spiral wave center follows an approximately circular trajectory until it stops practically at a fixed line in complete agreement with the predictions from the drift velocity field. 

It is very important to stress that changes in the geometrical shape of the integration domain can induce bifurcations in the drift velocity field [31, 47, 50, 52]. Let us consider, for example, the drift velocity field computed for an elliptical domain with major axis o = 3A and minor axis b = a/1.1. As shown in Fig. 9.13(a), instead of the stable limit cycle of the resonance attractor in the circular domain of radius Rg, = 1.5A we have two pairs of fixed points where the drift velocity vanishes. In each pair, one fixed point is a saddle and the other one is a stable node. Depending on the initial conditions, the spiral wave approaches one of the two stable nodes. Trajectories of the spiral center obtained by numerical integration of the Oregonator model (9.1) are in perfect agreement with the predicted drift... 

The theoretically predicted destruction of the resonance attractor in response to deviations from the circular shape of the integration domain has been confirmed experimentally within the light-sensitive BZ medium. A spiral wave was exposed to uniform illumination proportional to the total gray level obtained in an elliptical integration domain. Fig. 9.13(b) shows the resonant drift mediated during global feedback control. The spiral wave drifts towards a stable node of the drift velocity field. Close to this fixed point the drift velocity becomes very slow. Thus, the experimentally observed termination of the spiral drift at certain positions in a uniform medium is explained in the framework of the developed theory of feedback-mediated resonant drift. 

We have shown that all existing methods of spiral wave control can be considered in the framework of a unified theoretical approach. This approach is based on the well established phenomenon of resonant drift induced by a periodic parametric modulation at the rotation frequency of a spiral wave [14-20]. The direction of resonant drift depends on the initial orientation of the spiral and on the phase of the first Fourier component of the periodic modulation. To specify the spiral orientation we propose to... 

It was extremely important to demonstrate that because the phase of the feedback signal depends on the spiral orientation, the direction of feedback-induced resonant drift is determined only by the spiral location and does not depend on its initial orientation. Thus, under feedback control the dynamics of spiral waves can be described by a drift velocity field [47, 53]. 

The deeper understanding of the dynamics of feedback-induced drift allows us to propose robust and efficient methods to move a spiral center from a given initial location along an arbitrary path to a desired final position. In this way, we can avoid eventually existing defects in the medium that could pin the spiral wave. This result might be important for such possible application as low-voltage defibrillation of cardiac tissue. 

A very interesting observation is the termination of spiral drift in spatially uniform media. Pinning of spiral waves is nowadays the subject of very general mathematical considerations [55]. A spiral wave can stay fixed at a certain site as if it is anchored at an invisible defect. In fact, this behavior is a consequence of the superposition principle for feedback-induced drift discussed in section 9.4.1. There is a deep analogy between this phenomenon and the destructive interference of linear waves. 

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