Spiral wave formation

H. Levine, I. Aranson, L. Tsimring, and T. V. Truong, Positive genetic feedback governs cAMP spiral wave formation in Dictyostelium. Proc. Natl. Acad. Sci. USA 93, 6382-6386 (1996). 

V. Perez-Munuzuri, M. Gomez-Gesteira, and V. Perez-Villar. A geometrical-kinematical approach to spiral wave formation super-spiral waves. Physica D Nonlinear Phenomena, 64 420-430, 1993. 

Figure 3.23. Snapshots showing the initial stages of the spiral wave formation. (E.I. Latkin, V. 1. Elokhin, V.V. Gorodetskii, Chemical Engineering Journal 91 (2003) 123-131).

FIGURE 8.9. A sequence of PEEM images from a Pt(l 10) surface during spiral wave formation in CO oxidation taken at intervals of 30 s. T=448K, poj =... 

FIGURE 8.1 o. Sequence of patterns leading to spiral wave formation starting with a disordered configuration of the adsorbates resulting from numerical solution of the PDEs modeling CO oxidation on Pt(l 0 0) [25]. 

J. Lauzeral, J. Halloy, and A. Goldbeter, Desynchronization of cells on the developmental path triggers the formation of spiral waves of cAMP during Dictyostelium aggregation. Proc. Natl. Acad. Sci. USA 94, 9153-9158 (1997). 

To see through the eyes of this theory is to see one s place in the spiral scheme and to know and anticipate when the transition to new epochs will occur. One sees this in the physical world. The planet is five or six billion years old. The formation of the inorganic universe occupies the first turn of the spiral wave. Then life appears. If one examines this planet, which is the only planet we can examine in depth, one finds that processes are steadily accelerating in both speed and complexity. 

The ring waves do not appear in the filtered BZ reagent, but are generated on repeated contamination of the solution with dust. Apparently, the process of their formation is initiated on nonhomogeneities present in the solution (heterogeneity of the system). In contrast, the spiral waves... 

In this respect, feedback-mediated parametric modulation seems to be a more promising control strategy, since in this case the modulation period always coincides exactly with the actual rotation period of the spiral wave [20, 21]. Another important motivation to study feedback-mediated dynamics of spiral waves is related to the fact that a feedback is either naturally present or can be easily implemented in many excitable media [22-25]. For example, recent experimental investigations performed with the BZ medium [26, 27] and during the catalytic CO oxidation on platinum single crystal surfaces [28] reveal that global feedback can provide an efficient tool for the control of pattern formation. 

We have also discussed the formation of spatio-temporal patterns in non-variational systems. A typical example of such systems at nano-meter scales is reaction-diffusion systems that are ubiquitous in biology, chemical catalysis, electrochemistry, etc. These systems are characterized by the energy supply from the outside and can exhibit complex nonlinear behavior like oscillations and waves. A macroscopic example of such a system is Rayleigh-Benard convection accompanied by mean flow that leads to strong distortion of periodic patterns and the formation of labyrinth patterns and spiral waves. Similar nano-meter scale patterns are observed during phase separation of diblock copolymer Aims in the presence of hydrodynamic effects. The pattern s nonlinear dynamics in both macro- and nano-systems can be described by a Swift-Hohenberg equation coupled to the non-local mean-flow equation. 

A typical feature of a non-potential systems is the non-stationary oscillatory behavior that usually manifests itself in the propagation of waves. We have shown that the nonlinear evolution of waves near the instability threshold is described by the complex Ginzburg-Landau (CGL) equation. This equation is capable of describing various kinds of instabilities of wave patterns, like the Benjamin-Feir instability. In two dimensions, the CGL equation describes the formation of spiral waves that are observed in many biological and chemical systems characterized by the interplay of diffusion and chemical reactions at nano-scales. 

Newell-Whitehead-Segel equation, 23 Non-potential effects, 41 Orientational instability, 283 Pattern formation, 1,11 Phase field, 168 Polymerization wave, 235, 239 Polymerization waves, 236, 238 Propagating front, 260-261 Quantum dots, 123-124 Rayleigh-Benard convection, 61 SHS, 247-248 Smectics, 57 Spiral wave, 47 Stochastic oscillations, 92 Stripes, 2, 10 Surface diffusion, 126... 

Arthur Winfree, a biologist with an interest in spatial and temporal patterns, had attended the Prague conference and had decided to pursue the study of pattern formation in the BZ reaction. In 1972, the cover of Science magazine featured Winfree s photo of spiral wave patterns in the BZ reaction (Winfree, 1972). Figure 1.7 shows such patterns. Field and Noyes immediately saw how to understand the development of such patterns with the aid of the FKN mechanism, and they published an explanation the same year (Field and Noyes, 1972). 

Figure 1 (a) A spiral wave formed in a thin gel layer of the Belousov-Zhabotinsky reaction (from Belmonte and Flesselles, Ref. 5. (b) Formation of a labyrinthine pattern in the bistable region of the iodine-ferrocyanide-sulfite chemical reaction in a gel reactor (from Lee and Swinney, Ref. 6). 

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. 

A related effect is the formation of a bound state of two spiral waves [21]. Suppose that we have a pair of spiral waves which rotate in the opposite directions. This wave pattern is symmetric in respect to the mirror reflection about a straight line which is orthogonal to the line connecting the centres of the two spirals and is located at an equal distance from each of them. This symmetry implies that the wave fronts are orthogonal to the central line when they come to it. But this is exactly the same condition which must be satisfied near a physical boundary of the medium. Hence, if we put a flat physical boundary at the central symmetry line the system would not be able to detect the difference. It means that the behaviour of each spiral in the pair is exactly the same as if it were placed at a certain distance from a physical straight boundary in the medium. 

Usually, spiral waves are studied without taking into account the finite size effects of the system, i.e., by neglecting the boundary effects. Nevertheless, in nature, spiral structures appear in finite size systems. For instance, reeent experiments by Lechleiter et al. [2] showed the propagation of spiral waves of intracellular calcium concentration in oocytes. In these experiments the diameter of the living cells is about five wavelengths of the spiral waves. Also recently, experimental studies have been performed on the spiral formation in small pieces of cardiac tissue [3]. When the size of a system containing a spiral becomes sufficiently small, boundary effects become important and new behaviors may be observed, which would not be present in an infinite system. Moreover, when the finite size effects appear, the geometrical shape of the system boundaries may also influence the evolution of a spiral wave. In this section, we consider these types of effects. 

In experiments, Zhabotinsky and Zaikin [2] observed that a break of the wave front results in formation of a spiral wave when a circular chemical wave encounters a region with a nonuniform negative gradient of excitability (Figure 3). 

Fig. 5.26. A photo-emission electron microscopy image of the formation of spiral waves in catalytic oxidation of CO on a platinum (110) surface.

Apart from the theory of combustion, the concept of self-sustaining waves (now sometimes called autowaves), developed in works by A. N. Kolmogorov, I. G. Petrovskii, N. S. Piskunov,2 R. Fisher3 and the present paper, has proved to be extraordinarily fruitful for many problems in chemistry and biophysics. Some of these applications may be found in the review by V. A. Vasiliev et al Let us note waves of a fundamentally new type spiral, self-sustaining vortex formations,... 

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