Scroll waves

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... 

T. Bretschneider, F. Siegert, and C. J. Weijer, Three-dimensional scroll waves of cAMP could direct cell movement and gene expression in Dictyostelium slugs. Proc. Natl. Acad. Sci. USA 92, 4387 391 (1995). 

Fig. 9.7 Snapshots of a re-entrant scroll wave in a 6x6x3 cm isotropic and homogeneous epicardial cuboid (left), and a wedge model of the human left ventricular free wall with similar dimensions, and fiber and sheet orientations giving orthotropy of propagation (right). Both examples use the ten Tusscher— Noble—Noble—Panfilov model [42] for excitation kinetics. In the wedge model these kinetics are spatially heterogeneous, with endocardial, mid-myocardial and epicardial tissue occupying approximately equal fractions of the transmural distance. In the cuboid, the diffusion coefficient was 0.154 mm2 ms-1 in all directions. The diffusion coefficient in the wedge was set to 0.154 mm2 ms-1 in the fiber axis direction, with a ratio of...

J. P. Keener, The dynamics of three-dimensional scroll waves in excitable media , Physica, D31, 269 (1988). 

We first comment on several aspects of eikonal meanders see section 3.3.3. Specifically we mention the relation to tip control experiments [38, 84, 85, 87, 88], discuss the occurrence and interpretation of shock waves, the role of the signs of G and kq, possible generalizations to dynamics of scroll wave filaments in three dimensions, and the role of diffusion and viscous regularization in the sense of section 3.3.2. We then return to the reaction-diffusion view point of section 3.2 and discuss the chances of a reduced... 

In addition to the direct solution of PDEs corresponding to reaction-diffusion equations, in recent years attention has begun to be focused on the use of coupled lattice methods. In this approach, diffusion is not treated explicitly, but, rather, a lattice of elements in which the kinetic processes occur are coupled together in a variety of ways. The simulation of excitable media by cellular automata techniques has grown in popularity because they offer much greater computational efficiency for the two- and three-dimensional configurations required to study complex wave activity such as spirals and scroll waves. 

Winfree (1972) has reported convenient conditions for observing target patterns and scroll waves,. 

Scroll waves and target patterns (as well as solitary trigger waves) share the common property of dependence on the interaction of reaction and diffusion, as witnessed by the fact that they are all blocked by impermeable barriers. 

That these waves all have the same period, Winfree takes as compelling evidence that they are all manifestations of the same thing a three-dimensional scroll wave whose axis threads through the thin layer of medium from one interface to another. What we see is a projection of the scroll wave, which lies at various angles in the medium. On this basis the various transformations described in the last paragraph are explained as in Fig. 5. 

It is not immediately obvious how scroll waves arise from reaction-diffusion equations (Chapter I)... 

Fig. 5. Scroll waves in a thin layer of Z reagent (from Winfree, 1974b).

Simple scroll waves are the most frequent early structures with wavelength of nearly 3.5mm and each wave segment propagating vertically to the scroll length at a velocity of 0.04mm s l, ambient temperature 22°C. Minor discontinuities and gas bubble punctures of the waves are smoothed as the structures evolve and show with time a general tendency to increased symmetry. 

A Theory of Rotating Scroll Waves in Excitable Media... 

Periodic traveling waves of excitation display certain characteristic patterns. In one spatial dimension, one would observe a train of impulses, such as the periodic action potentials that emanate from the oscillatory respiratory centers in the medulla of vertebrates. Two-dimensional excitable media exhibit two topologically distinct patterns expanding concentric circles and rotating spirals. In three dimensions, the corresponding structures are expanding concentric spherical waves and rotating scroll waves. 

Fig. 1. Scroll wave filaments (dashed curves) move slowly through space as the scroll rotates, (a) An elongated spiral becomes symmetric, and (b) an elongated ring becomes circular and then disappears (after Winfree [10]). (c) A scroll ring shrinks and disappears, and (d) a figure-eight ring splits into two circular rings which then shrink and disappear (after Welsh [17]).

In this review we shall concentrate our attention on scroll waves (Figure la). As the scroll rotates, almost all points in space undergo periodic oscillations in the oxidation state of iron ions, which allows us to define a phase function 4> x,y,z,t) almost everywhere. However, the phase func-... 

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