Excitable systems and spiral waves

I recently came across the concept of excitable systems. They appear to play a role in the signalling processes of Dictyostelium, which was my point of entry to the subject, but at the same time they seem to be of importance for modelling a wide range of systems in nature. Examples which are frequently considered include the Field-Noyes model of the Belousov-Zhabotinskii oscillatory chemical reaction and the FitzHugh-Nagumo model which is a simplified version of the Hodgkin-Huxley model of nerve conduction. Coming back to Dictyostelium, there is the Martiel-Goldbeter model for aggregation. The general set-up is as follows. There is a system of ODE with certain qualitative features which are described by the word ‘excitability’. Then diffusion terms may be added to one or more of these equations to get a system of PDE. On an intuitive level the characteristic features of excitability as a property of the ODE system are as follows. There is a stationary solution which is stable. Thus, by definition, sufficiently small perturbations of this solution stay small. On the other hand larger perturbations are such that the corresponding solutions go a long way away (whatever that means) from the stationary state before returning there. It is often said that there is a progression from an excited state to a refractory state and then back to the excited state. In the PDE context fronts occur separating an excited from a refractory region. The spatial variation within these fronts resembles the temporal variation in the ODE system. There are many discussions of excitable systems in the literature. A source which helped me to grasp some of the basic ideas are the early parts of the article ‘Pattern formation in excitable media’ by Ehud Meron (Phys. Rep. 218, 1-66). In comparison to some other accounts I have seen this paper supplies more conceptual explanation so as to allow the novice (like myself) to understand the meaning of the figures which are often presented.

One of the most interesting features of excitable systems is the occurrence of spiral waves. This concerns the case of systems of PDE in two space dimensions. The typical scenario for the formation of a spiral wave is that a plane wave (i.e. a linear front) gets interrupted in some way. Then the broken end starts to curl around, forming a tip. The front then rotates about this tip. These waves are fascinating on account of their visual form but it is even more interesting to ask how this kind of phenomenon can be described mathematically. There is an extensive literature and I have just started to scratch the surface of it. By means of a singular limit which I have not yet understood an equation can be derived which describes the motion of the front itself. This equation is related to mean curvature flow. Actually it would be more appropriate to talk about curve-shortening flow since it is this two-dimensional special case which is relevant when starting from equations in two space dimensions. In mean curvature flow a hypersurface in Euclidean space flows in the normal direction with a speed which is equal to its mean curvature. This prescription defines a parabolic equation about whose solutions much is known in the meantime. To my knowledge the relations between excitable systems and geometric evolution equations have not been explored very much up to now.

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