Spiral wave

Figure A3.14.9. Reaction-diflfiision structures for an excitable BZ system showing (a) target and (b) spiral waves. (Courtesy of A F Taylor.)...

Targets and spirals have been observed in the CIMA/CDIMA system [13] and also in dilute flames (i.e. flames close to their lean flammability limits) in situations of enlianced heat loss [33]. In such systems, substantial fiiel is left unbumt. Spiral waves have also been implicated in the onset of cardiac arrhytlnnia [32] the nomial contractive events occurring across the atria in the mannnalian heart are, in some sense, equivalent to a wave pulse initiated from the sino-atrial node, which acts as a pacemaker. If this pulse becomes fragmented, perhaps by passing over a region of heart muscle tissue of lower excitability, then spiral structures (in 3D, these are scroll waves) or re-entrant waves may develop. These have the incorrect... 

Figure A3.14.il. Spiral waves imaged by photoelectron electron microscopy for the oxidation of CO by O2 on a Pt(l 10) single crystal under UHV conditions. (Reprinted with pennission from [35], The American Institute of Physics.)...

Winfree A T 1972 Spiral waves of chemical activity Science 175 634-6... 

Pearlman FI 1997 Target and spiral wave patterns in premixed gas combustion J. Chem. Soc. Faraday Trans. 93 2487-90... 

Nettesheim S, von Oertzen A, Rotermund FI FI and ErtI G 1993 Reaction diffusion patterns in the catalytic CO-oxidation on Pt(110) front propagation and spiral waves J. Chem. Rhys. 98 9977-85... 

Figure C3.6.8 (a) A growing ring of excitation in an excitable FitzHugh-Nagumo medium, (b) A spiral wave in tlie same system.

Ca waves in systems [ike Xenopus laevis oocytes and pancreatic (3 cells fall into this category Electrochemical waves in cardiac and nerve tissue have this origin and the appearance and/or breakup of spiral wave patterns in excitable media are believed to be responsible for various types of arrhythmias in the heart [39, 40]. Figure C3.6.9 shows an excitable spiral wave in dog epicardial muscle [41]. 

The cores of the spiral waves need not be stationary and can move in periodic, quasi-periodic or even chaotic flower trajectories [42, 43]. In addition, spatio-temporal chaos can arise if such spiral waves break up and the spiral wave fragments spawn pairs of new spirals [42, 44]. 

The local dynamics of tire systems considered tluis far has been eitlier steady or oscillatory. However, we may consider reaction-diffusion media where tire local reaction rates give rise to chaotic temporal behaviour of tire sort discussed earlier. Diffusional coupling of such local chaotic elements can lead to new types of spatio-temporal periodic and chaotic states. It is possible to find phase-synchronized states in such systems where tire amplitude varies chaotically from site to site in tire medium whilst a suitably defined phase is synclironized tliroughout tire medium 51. Such phase synclironization may play a role in layered neural networks and perceptive processes in mammals. Somewhat suriDrisingly, even when tire local dynamics is chaotic, tire system may support spiral waves... 

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). 

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. 

Krinsky VI Mathematical models of cardiac arrhythmias (spiral waves) in Szekeres L (ed) Pharmacology of Antiarrhythmic Agents. Oxford, Pergamon Press, 1981, pp 105-124. 

Another form of behaviour exhibited by a number of chemical reactions, including the Belousov-Zhabotinskii system, is that of excitability. This concerns a mixture which is prepared under conditions outside the oscillatory range. The system sits at the stationary state, which is stable. Infinitesimal perturbations decay back to the stationary state, perhaps in- a damped oscillatory manner. The effect of finite, but possibly still quite small, perturbations can, however, be markedly different. The system ultimately returns to the same state, but only after a large excursion, resembling a single oscillatory pulse. Excitable B-Z systems are well known for this propensity for supporting spiral waves (see chapter 1). 

Fig. 4. Spiral wave fronts arising in the Belousov-Zhabotinski reagent (from ref. 31).

Figure 11.17 The four snapshots show the evolution and breakup of a spiral wave pattern in 2-dimensional simulated cardiac tissue (300 x 300 cells). The chaotic regime shown in the final snapshot corresponds to fibrillation. Reprinted from [587] with permission from Lippincott, Williams and Wilkins.

Undoubtedly, the most promising modehng of the cardiac dynamics is associated with the study of the spatial evolution of the cardiac electrical activity. The cardiac tissue is considered to be an excitable medium whose the electrical activity is described both in time and space by reaction-diffusion partial differential equations [519]. This kind of system is able to produce spiral waves, which are the precursors of chaotic behavior. This consideration explains the transition from normal heart rate to tachycardia, which corresponds to the appearance of spiral waves, and the fohowing transition to fibrillation, which corresponds to the chaotic regime after the breaking up of the spiral waves, Figure 11.17. The transition from the spiral waves to chaos is often characterized as electrical turbulence due to its resemblance to the equivalent hydrodynamic phenomenon. 

Qu, Z., Weiss, J., and Garfinkel, A., Cardiac electrical restitution properties and stability of reentrant spiral waves A simulation study, American Journal of Physiology, Vol. 276, No. 1(2), 1999, pp. H269-283. 

Figure 4. (Left) Six snapshots of the field 0 within the same cell, at six successive times with a delay x/6 (from left to right, top to bottom), as a result of the numerical integration of Eq. (26). Here Da 0.4 and Pe 315. Black stands for 0=1, white for 0 = 0. (Right) The same but for Da =4 and Pe — 315, x is now replaced by x L/U. Note that a spiral wave invades the interior of the cell, with a speed comparable to U.

These principles also apply to spatial distributions of populations as reported by Hassell et al. (1991). In a study using host-parasite interactions as the model, a variety of spatial patterns were developed using the Nicholson-Bailey model. Host-parasite interactions demonstrated patterns ranging from static "crystal lattice" patterns, spiral waves, chaotic variation, or extinction with the appropriate variation of only three parameters within the same set of equations. The deterministically determined patterns could be extremely complex and not distinguishable from stochastic environmental changes. 

Winfree, A. T. (1972) Spiral waves of chemical activity. Science 175, 634. 

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