Spiral wave pinned

The main disadvantage of this kinematical construction is that it corresponds to spiral waves pinned to a hole but fails to describe freely rotating spiral waves in uniform excitable media which are frequently observed both experimentally as well as in numerical simulations of the underlying reaction-diffusion equations. 

Fig. 12. The rotation periods To offree spiral waves (crosses) and r ofthe spiral waves pinned by the minimal hole (black dots) obtained by numerical simulation of the reaction-diffusion model (60)-(62) at different values of the parameter s. Curve 1 gives the dependence of To on in the kinematical approximation when the recovery effects are neglected. Curve 2 shows the dependence of the rotation period T of the spiral wave pinned by the minimal hole in the same approximation. Curve 3 gives the dependence of T on e calculated using the kinematical approximation including the effects of recovery. The dashed curve shows the computed dependence of the minimal period Tmin of stable propagation of pulses in the same model. (From [27])...

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... 

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. 

We consider further the phenomenon of the breakup in rotating spiral waves. Our analysis is restricted to the spiral waves which are pinned on the obstacles. The numerical simulations performed in [41] have revealed that, as the radius R of the obstacle reaches some critical value, the wave breaks at the border of the obstacle and a free tip is produced. After a transient, the steady rotation of the free tip around a circular core of a larger radius sets in. 

The radius R of the minimal hole that is still able to maintain a pinned spiral wave and the rotation period T of this pinned wave are important properties of an excitable medium. Figure 12 shows (curve 3) the rotation period T as a function of e, computed in [27] using the kinematical description. We see that it fits well the data (black dots) of the numerical simulation of the respective reaction-diffusion model. 

Note also that, as we see from Figure 10, the critical curvature kd for the breakup of a pinned spiral wave vanishes at a higher value of the parameter e, i.e. in the medium with the lower excitability, than the critical curvature kc-It means that in this interval one can observe steadily rotating pinned spiral waves while no free spirals are possible. 

Recently, the breakup of free spiral waves has been observed in the numerical simulations of reaction-diffusion models [42-46]. This effect is principally of the same nature as breaking (or spontaneous depinning) of a pinned spiral wave. The main question is here what induces breaking of the wave front at a certain distance from the free tip. In some of the stimulations (e.g. [42-45]), the breakup is preceded by the onset of meandering which might actually produce the inhomogeneities of the residual inhibitor concentration that force the wave to break. 

The appearance of the sinks is a typical property of wave patterns on the surfaces with the positive Gaussian curvature. We consider next a spiral wave on the spherical surface. Suppose that the sphere has a small hole of radius R at one of its poles and the spiral wave is pinned by this hole. The solution for a steadily rotating pinned spiral wave on the sphere was found in the Wiener-Rosenblueth approximation (i.e. for ) = 0) in [24]. It has the form... 

Spiral waves have been studied in most detail on Pt(l 10), both under conditions of excitability and of double metastability [108]. Under identical external conditions the spirals did not exhibit a fixed period and wavelength, rather a continuous distribution of these quantities was observed (Figure 18). It is well known that spirals can be pinned to artificial nonexcitable cores [109, 110]. Such artificial cores can be formed by surface defects consequently the continuous distribution of rotation periods simply reflects the fact that defects of various size exist on the surface. Comparison of the dispersion relation computed from the model [111] with the experimentally observed periods and wave velocities (Figure 19) allows conclusions to be drawn about the size distribution of surface defects by using the Tyson-Keener formula [112], which gives the relationship between core size and period. For the data... 

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