This discussion of higher-dimensional waves is merely the tip of the iceberg (or tip of the spiral), and there are many interesting unresolved questions.

While the spirals that are observed in physiological systems share certain qualitative similarities, their details are certainly different. The FitzHugh-Nagumo model discussed here shows only the qualitative behavior for generic excitable systems and so has little quantitative relevance. An analysis for the two-pool model of calcium wave propagation (Chapter 12) is similar in style, but much more difficult in detail because

Dispersion curve

Frequency

Frequency

Figure 10.3 Critical curve (10.33) and the approximate dispersion curve (9.89) using piecewise linear dynamics f(v, w) = H(v — a) — v — w, g(v, w) = v — yw with a = 0.1, y = 0, e = 0.05.

the eikonal-curvature relationship is nonlinear and the dispersion curve is more difficult to obtain. Other physiological systems are likely governed by other dynamics. For example, a model for spreading cortical depression in the cortex has been proposed by Tuckwell and Miura (1978; Miura, 1981), and numerical simulations have shown rotating spirals. However, a detailed mathematical study of these equations has not been given.

This analytical calculation for the FitzHugh-Nagumo system is based on singular perturbation theory and therefore is not mathematically rigorous. In fact, as yet there is not a rigorous proof of the existence of spiral waves in an excitable medium. While the approximate solution presented here is known to be asymptotically valid, the structure of the core of the spiral is not correct. This problem has been addressed by Pelce and Sun (1991) and Keener (1992) for FitzHugh-Nagumo models with a single diffusing variable, and by Keener (1994) and Kessler and Kupferman (1996) for FitzHugh-Nagumo models with two diffusing variables, relevant for chemical reaction systems.

A second issue of concern is the stability of spirals. This also is a large topic, which is not addressed here. The interested reader should consult the work of Winfree (1991), Jahnke and Winfree (1991), Barkley (1994), Karma (1993, 1994), Panfilov and Hogeweg (1995), Kessler and Kupferman (1996).

Because the analytical study of excitable media is so difficult, simpler models have been sought, with the result that finite-state automata are quite popular. A finite-state automaton divides the state space into a few discrete values (for example v = 0 or 1), divides the spatial domain into discrete cells, and discretizes time into discrete steps. Then, rules are devised for how the states of the cells change in time. Finite-state automata are extremely easy to program and visualize. They give some useful insight into the behavior of excitable media, but they are also beguiling and can give "wrong"

answers that are not easily detected. The literature on finite-state automata is vast (see, for instance, Moe et al., 1964; Smith and Cohen, 1984; and Gerhardt et al., 1990).

The obvious generalization of a spiral wave in a two-dimensional region to three dimensions is called a scroll wave (Winfree, 1973, 1991; Keener and Tyson, 1992). Scroll waves have been observed numerically (Jahnke et al., 1988; Lugosi and Winfree, 1988), in three-dimensional BZ reagent (Gomatam and Grindrod, 1987), and in cardiac tissue (Chen et al., 1988), although in experimental settings they are extremely difficult to visualize. In numerical simulations it is possible to initiate scroll waves with interesting topology, including closed scroll rings, knotted scrolls, or linked pairs of scroll rings.

The mathematical theory of scroll waves is also in its infancy. Attributes of the topology of closed scrolls were worked out by Winfree and Strogatz (1983a,b,c; 1984), and a general asymptotic theory for their evolution has been suggested (Keener, 1988b) and tested against numerical experiments on circular scroll rings and helical scrolls. There is not sufficient space here to discuss the theory of scroll waves. However, scroll waves are mentioned again briefly in later chapters on cardiac waves and rhythmicity.

10.3 EXERCiSES

1. What is the eikonal-curvature equation for (10.1) when the medium is anisotropic and D is a symmetric matrix, slowly varying in space?

Hint: Generalize (10.11) by calculating V ■ (DVu) using components of D as dij, and use this to generalize (10.12).

Answer: (10.14) becomes lt = VVl ■ DVl(c0Vk + V ■ (vV|VoVl)) •

Hint: Use Cramer's rule to find the inverse of a matrix with three column vectors, say t1,t2, and t3. Use the fact that the determinant of such a matrix is t1 ■ (t2 x t3). Then apply this to the transpose of the Jacobian matrix .

3. Verify that in two spatial dimensions, V ■ (jUj) is the curvature of the level surface of the function l.

Hint: If x = X(t),y = Y(t) is the parametric representation of a smooth level-surface curve, then l(X(t), Y(t)) = 0. Use this and derivatives of this expression with respect to t to show that V ( VS ^ - +ytXtt—YttXt that V' (jVljJ =±(Xx+Yy)3/2 .

4. The following are the rules for a simple finite automaton on a rectangular grid of points:

(a) The state space consists of three states, 0, 1, and 2, 0 meaning at rest, 1 meaning excited, and 2 meaning refractory.

(b) A point in state 1 goes to state 2 on the next time step. A point in state 2 goes to 0 on the next step.

(c) A point in state 0 remains in state 0 unless at least one of its nearest neighbors is in state 1, in which case it goes to state 1 on the next step.

Write a computer program that implements these rules. What initial data must be supplied to initiate a spiral? Can you initiate a double spiral by supplying two stimuli at different times and different points?

5. (a) Numerically simulate spiral waves for the Pushchino model of Chapter 4, Exercise 13. (b) Numerically simulate spiral waves for the Pushchino model with

Use the parameters V1 = 0.0026, V2 = 0.837, W1 = 1.8, C1 = 20, C2 = 3, C3 = 15, a = 0.06, r1 = 75, t2 = 1.0, t3 = 2.75, and k = 3. What is the difference between these spirals and those for the previous model?

Answer: There are no stable spirals for this model, but spirals continually form and break apart, giving a "chaotic" appearance.

C1V when V<V1, f(V) = _C2V + a when V1 <V<V2, C3(V _ 1) when V>V2,

t1 when V < V1, w > w1, t2 when V >V2, t3 when V < V1, w <w1.

CHAPTER 11

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