Rotating spiral waves occur naturally in a wide variety of biological, physiological and chemical contexts. A class which has been extensively studied is those which arise in the Belousov–Zhabotinskii reaction. Relatively, it is a very much simpler system than those which arise in physiology where we do not know the detailed mechanism involved unlike the Belousov–Zhabotinskii mechanism. Experimental work on spiral solutions in this reaction has been done by many people such as Winfree (1974), one of the major figures in their early study and subsequent application of the concepts to cardiac prob-lems, Krinskii et al. (1986) and by M¨uller et al. (1985, 1986, 1987). The latter’s novel experimental technique, using light absorption, highlights actual concentration levels quantitatively. Figure 1.16 as well as Figures 1.19 and 1.20, show some experimentally observed spiral waves in the Belousov–Zhabotinskii reaction; refer also to Figure 1.1(b).
Although the spirals in these figures are symmetric, this is by no means the only pat-tern form; see, for example, Winfree (1974) in particular, and M¨uller et al. (1986), who exhibit dramatic examples of complex spiral patterns. A lot of work has gone into the mathematical study of spiral waves and in particular the diffusion version of the FKN model system. Keener and Tyson (1986) analysed spiral waves in excitable reaction dif-fusion systems with general excitable kinetics. They applied their technique to the FKN model with diffusion and the results are in good agreement with experiment. Although in a different context, see Figure 1.18 for other examples of nonsymmetric, as well as symmetric, spirals. There are numerous examples of spiral waves in the book by Keener and Sneyd (1998). General discussions of spiral waves have been given, for example, by Keener (1986), who presents a geometric theory, Zykov (1988) in his book on wave processes in excitable media and in the book by Grindrod (1996).
Much novel and seminal work on chemical spiral waves has been carried out by Showalter and his colleagues. In the paper by Amemiya et al. (1996), for example, they use a Field–K¨or¨os–Noyes (FKN) model system for the Belousov–Zhabotinskii reaction (similar to that studied in Chapter 8, Volume I) which exhibits excitability kinetics, to investigate three-dimensional spiral waves and carry out related experiments to back up their analysis; see other references there.
There are many other important occurrences of spiral waves. Brain tissue can ex-hibit electrochemical waves of ‘spreading depression’ which spread through the cortex
Figure 1.16. Spiral waves in a thin (1 mm) layer of an excitable Belousov–Zhabotinskii reaction. The section shown is 9 mm square. (Courtesy of T. Plesser from M¨uller et al. 1986)
1.8 Spiral Waves 55
Figure 1.17. (a) Evolution of spiralling reverberating waves of cortical spreading depression about a lesion (a thermal coagulation barrier) in the right hemisphere of a rat cerebral cortex. The waves were initiated chemically. The shaded regions have different potential from the rest of the tissue. (After Shibata and Bure˘s 1974) (b) Rotating spiral waves experimentally induced in rabbit heart (left atria) muscle: the numbers repre-sent milliseconds. Each region was traversed in 10 msec with the lettering corresponding to the points in the heart muscle on the right. The right also shows the isochronic lines, that is, lines where the potential is the same during passage of the wave. (Reproduced from Allessie et al. 1977, courtesy of M.A. Allessie and the American Heart Association, Inc.)
of the brain. These waves are characterised by a depolarisation of the neuronal mem-brane and decreased neural activity. Shibata and Bure˘s (1972, 1974) studied this phe-nomenon experimentally and demonstrated the existence of spiral waves which rotate about a lesion in the brain tissue from the cortex of a rat. Figure 1.17(a) schematically shows the wave behaviour they observed. Keener and Sneyd (1998) discuss wave mo-tion in general and in particular the types of wave propagamo-tion found in the Hodgkin–
Huxley equations and their caricature system, the FitzHugh–Nagumo equations. They also describe cardiac rhythmicity and wave propagation and calcium waves; some of the wave phenomena discussed are quite different to those covered in this book, such as wave curvature effects. A general procedure, based on an eikonal approach, for in-cluding curvature effects, particularly on curved surfaces is given by Grindrod et al.
(1991).
There are new phenomena and new applications of reaction diffusion models and spiral waves that are continuing to be discovered. A good place to start is with papers by
Winfree and the references given in them. For example, Winfree et al. (1996) relate trav-elling waves to aspects of movement in heart muscle and nerves which are ‘excitable.’
They obtain complex periodic travelling waves which resemble scrolls radiating from vortex rings which are organising centres. Winfree (1994b) uses a generic excitable re-action diffusion system and shows that the general configuration of these vortex lines is a turbulent tangle. The generic system he used is
∂u
When death results from a disruption of the coordinated contractions of heart mus-cle fibres, the cause is often due to fibrillation. In a fibrillating heart, small regions undergo contractions essentially independent of each other. The heart looks, as noted before, like a handful of squirming worms — it is a quivering mass of tissue. If this disruption lasts for more than a few minutes death usually results. Krinskii (1978) and Krinskii et al. (1986), for example, discussed spiral waves in mathematical models of cardiac arrhythmias. Winfree (1983a,b) considered the possible application to sudden cardiac death. He suggested there that the precursor to fibrillation is the appearance of rotating waves of electrical impulses. Figure 1.17(b) illustrates such waves induced in rabbit heart tissue by Allessie et al. (1977). These authors (Allessie et al. 1973, 1976, Smeets et al. 1986) also carried out an extensive experimental programme on rotating wave propagation in heart muscle.
Winfree (1994a, 1995) put forward the interesting hypothesis that sudden cardiac death could involve three-dimensional rotors (spiral-type waves) of electrical activity which suddenly become unstable when the heart thickness exceeds some critical value;
see there references to other articles in this general area. He has made extensive studies of the complex wave phenomena in muscle tissue over the past 20 years. For example, electrical aspects, activation fronts, anisotropy, and so on are important in cardiac phys-iology and have been discussed in detail by Winfree (1997) within a reaction diffusion context. He also discusses the roles of electrical potential diffusion, electrical turbu-lence, activation front curvature and anisotropy in heart muscle and their possible role in cardiac failure. His work suggests possible scenarios for remedial therapy for seri-ous cardiac failures such as ventricular fibrillation. With his series of papers modelling muscle tissue activity in the heart Winfree has greatly enhanced our understanding of sudden cardiac death and changed in a major way previously held (medical) beliefs.
The spirals that arise in signalling patterns of the slime mould Dictyostelium dis-coideum are equally dramatic as seen in Figure 1.18. A model for these, based on an experimentally motivated kinetics scheme, was proposed by Tyson et al. (1989a,b).
It is really important (and generally) that, although the similarity between Fig-ures 1.1(b) and 1.18 is striking, one must not be tempted to assume that the model for the Belousov–Zhabotinskii reaction is then an appropriate model for the slime mould patterns—the mechanisms are very different. Although producing the right kind of
pat-1.8 Spiral Waves 57
Figure 1.18. Spiral signalling patterns in the slime mould Dictyostelium discoideum which show the increas-ing chemoattractant (cyclic AMP) signallincreas-ing. The photographs are taken about 10 min apart, and each shows about 5× 107amoebae. The Petri dish is 50 mm in diameter. The amoebae move periodically and the light and dark bands which show up under dark-field illumination arise from the differences in optical properties between moving and stationary amoebae. The cells are bright when moving and dark when stationary. The patterns eventually lead to the formation of bacterial territories. (Courtesy of P. C. Newell from Newell 1983)
terns is an important and essential aspect of successful modelling, understanding the basic mechanism is the ultimate objective.
The possible existence of large-scale spirals in interacting population situations does not seem to have been considered with a view to practical applications, but, given the reaction diffusion character of the models, they certainly exist in theory.
From a mathematical point of view, what do we mean by a spiral wave? In the case of the Belousov–Zhabotinskii reaction, for example, it is a rotating, time periodic, spatial structure of reactant concentrations; see Figures 1.17 and 1.20. At a fixed time a snapshot shows a typical spiral pattern. A movie of the process shows the whole spiral pattern moving like a rotating clock spring. Figure 1.19 shows such a snapshot and a superposition of the patterns taken at fixed time intervals. The sharp wavefronts are contours of constant concentration, that is, isoconcentration lines.
Consider now a spiral wave rotating around its centre. If you stand at a fixed po-sition in the medium it seems locally as though a periodic wavetrain is passing you by since every time the spiral turns a wavefront moves past you.
As we saw in Chapter 9, Volume I, the state or concentration of a reactant can be described by a function of its phase,φ. It is clearly appropriate to use polar coordinates
(a) (c)
(b)
Figure 1.19. (a) Snapshot (4.5 mm square) of a spiral wave in a thin (1 mm) layer of an excitable Belousov–
Zhabotinskii reagent. The grey-scale image is a measure of the level of transmitted light intensity (7 intensity levels were measured), which in turn corresponds to isoconcentration lines of one of the reactants. (b) The grey-scale highlights the geometric details of the isoconcentration lines of one of the reactants in the reaction.
(c) Superposition of snapshots (4.5 mm square) taken at three-second intervals, including the one in (a).
The series covers approximately one complete revolution of the spiral. Here six light intensity levels were measured. Note the small core region. (From M¨uller et al. 1985 courtesy of T. Plesser and the American Association for the Advancement of Science: Copyright 1985 AAAS)
1.8 Spiral Waves 59
r andθ when discussing spiral waves. A simple rotating spiral is described by a periodic function of the phaseφ with
φ = t ± mθ + ψ(r), (1.131)
where is the frequency, m is the number of arms on the spiral and ψ(r) is a func-tion which describes the type of spiral. The± in the mθ term determines the sense of rotation. Figure 1.20 shows examples of 1-armed and 3-armed spirals including an ex-perimental example of the latter. Suppose, for example, we setφ = 0 and look at the steady state situation; we get a simple geometric description of a spiral from (1.131): a 1-armed spiral, for example, is given byθ = ψ(r). Specific ψ(r) are
θ = ar, θ = a ln r (1.132)
with a > 0; these are respectively Archimedian and logarithmic spirals. For a spiral about a central core the corresponding forms are
θ = a(r − r0), θ = a ln (r − r0). (1.133)
(d)
Figure 1.20. (a) Typical 1-armed Archimedian spiral. The actual spiral line is a line of constant phaseφ, that is, a line of constant concentration. (b) Typical 3-armed spiral. (c) Three-dimensional spiral. These have a scroll-like quality and have been demonstrated experimentally by Welsh et al. (1983) with the Belousov–
Zhabotinskii reaction. (d) Experimentally demonstrated 3-armed spiral in the Belousov–Zhabotinskii reac-tion. (From Agladze and Krinskii 1982 courtesy of V. Krinskii)
Figure 1.20(a) is a typical Archimedian spiral with Figure 1.20(b) an example with m= 3.
A mathematical description of a spiral configuration in a reactant, u say, could then be expressed by
u(r, θ, t) = F(φ), (1.134)
where F(φ) is a 2π-periodic function of the phase φ given by (1.131). If t is fixed we get a snapshot of a spiral, the form of which puts certain constraints onψ(r) in (1.131); ar and a ln r in (1.132) are but two simple cases. A mixed type, for example, hasψ(r) = ar + b ln r with a and b constants. In (1.134) with φ as in (1.131), if we fix r and t and circle around the centre we have m-fold symmetry where m is the number of arms; an example with m = 3 is shown in Figure 1.20(b) and in Figure 1.20(d), one obtained experimentally by Agladze and Krinskii (1982). If we fix r andθ, that is, we stay at a fixed point, we see a succession of wavefronts as we described above.
If a wavefront passes at t = t0 with say,φ = φ0, the next wave passes by at time t= t0+ 2π/ which is when φ = φ0+ 2π.
If we look at a snapshot of a spiral and move out from the centre along a ray we see intuitively that there is a wavelength associated with the spiral; it varies however as we move out from the centre. If one wavefront is at r1and the next, moving out, is at r2, we can define the wavelengthλ by
λ = r2− r1, θ(r2) = θ(r1) + 2π.
From (1.131), with t fixed, we have, along the curveφ = constant, φθ + φr The wavelengthλ(r) is now given by
λ(r) = θ(r)+2π
The pitch of the spiral is defined by
dr dθ
φ=constant= m ψ(r)