Discussion.

In this paper we have derived motion equations both for the crest line of the excitation wave and for its tip, within a single perturbation procedure and using only assumptions of smallness of typical wave curvature and proximity to the manifold of stationary propagating half-wave solutions in parametric space, supposedly corresponding to Winfree's boundary . These motion equations are obtained here for the first time. The motion equations depend upon coefficients determined by the properties of the linearised operator at these basic solutions.

It is interesting to compare our results with the traditional `kinematic' theory of [10, 11], built partly from asymptotical and partly from phenomenological consideration. The traditional tip motion equations (2), (3) do not coincide, nor are they a partial case of our new equations (36), as (3) has different functional form. However, if we restrict consideration to stationary solutions, they can be considered as a special case. How special is it? We answer this questions in terms of relative codimension, i.e. number of additional equalities for parameters required to obtain this case. If we look for conditions when the spiral wave solutions are identical to those obtained from the traditional approach, then the only additional condition (61) is required, and in this sense the traditional approach has relative codimension one. However, as it was noted in Section 3.3, this provides only formal correspondence of the solutions, and even the small parameters in these solutions have different physical sense.

Another possible interpretation of this question is, when the equations rather than solutions become identical to the traditional ones. To see it, during derivation of (36) we had retained more terms than were really used; and some terms did not play any role because of their additional smallness caused by slow variation of curvature along the crestline, not accounted for by ansatz (19). Let us consider now the minimal cut-off system, obeying the following requirements: (i) it has all the terms necessary to achieve the generic solution of Section 3.2, (ii) it has all the terms necessary to achieve the `non-growing' solution of Section 3.3, formally equivalent to the `traditional' spiral wave solutions, and (iii) it has all the terms necessary to have the `traditional' tip motion equations (2), (4) as a special case. This minimal system is (we drop symbols

 

Now, consider a special case defined by the following five conditions

 

and make a change of parameters

 

Then the minimal system gets the form

 

coinciding with that of (4), (18), (2). Thus, in terms of motion equations, the traditional case has the much higher codimension five. This means that, while there is a certain probability that in a particular system condition (61) may be fulfilled with reasonable accuracy, and stationary spiral wave solution would have the properties predicted by the traditional equations, there is much less hope that the five conditions (68) would be fulfilled simultaneously, even approximately, and so the usefullness of the traditional equations for the study of parametric dependencies is much more doubtful. The applicability of those equations for nonstationary regimes is still more restricted, as in that case the `traditional' equations do not match the new ones.

Having found spiral wave solutions in the vicinity of the manifold of stationary propagating half-wave solutions, we now can rethink its relationship with . The following properties of the line are similar to that of :

• Spiral wave solution exist only to one side of this manifold, which is expressed by the condition (58).
• Approaching this manifold is accompanied by growth of spiral wave period and core radius as .

Some new properties of this parametric region are predicted by the new theory, for instance

• The tip angle should be varying along this boundary, being always positive. In other words, growing half-waves can give birth to spiral waves via perturbation of parameters.
• There may be a codimension 2 submanifold on this boundary (e.g. a point in a two-dimensional parametric space) where , corresponding to the half-wave neither growing nor shrinking. Near this submanifold, the asymptotics of spiral wave solutions are different, e.g. period and radius grow as rather than (this is the `traditional' solution).
• Analytical continuation of through , if any, is no longer a `rotor boundary'. In other words, perturbation of shrinking half-waves does not produce spiral waves. So, , generically, is a manifold with a border.

These new properties can be tested by numerical experiment; however, due to their asymptotical nature, it may be of considerable computational cost. A qualitative prediction of existence of should be easier to test. It is not observed in Winfree's [15] parametric map of the FitzHugh-Nagumo system, nor in Barkley's [16] map of his `vertical isoclines' system. However, it is worth noticing that while in [16] is well separated from the meander boundary , i.e. the boundary between rigidly rotating spirals and biperiodic spirals, in [15] the rotor boundary after some point, goes very close to , and to distinguish them reliably, very careful computations are required. So, we can put forward a hypothetical alternative interpretation of that diagram, that at , manifolds and join, and what goes next and shown as and going close to each other, is actually single boundary between meandering (biperiodic) spiral waves and the absence of any spiral waves solutions. Joining and would mean that in the vicinity of , there is transition from simple (rigid) to compound (meandering) rotation not due to interaction of the wave tip with the refractory tail (which is commonly considered as physical mechanism of meander). The possibility of such a transition has been recently hypothesized by Starobin & Starmer [17]. Note, that in this case, the transition would be described within the new kinematic theory, and seen as Hopf bifurcation of a stationary solution in the evolution equations for the `natural equation' K(s,t) of the crest line. We believe that all these question deserve further study.

From the viewpoint of practical importance of the kinematic theory, the assumption of the proximity to the boundary may seem rather exotic. However, in terms of properties of cardiac tissue, it corresponds to reduced excitability and/or shortened action potential and refractoriness, which are features of certain pathological conditions, and this makes this limit interesting from the practical viewpoint. For instance, numerical experiments of Efimov et al. [18] with a model of ventricular tissue show that transition through (which is the way leading to the boundary in Winfree's [15] diagram) can be achieved by reducing the number of functioning Na channels, and such a reduction is known to correlate with certain cardiac pathologies, such as ischaemia and influence of some pharmaceutical agents.

In this respect, it is interesting to compare the ``kinematic'' theory considered here with the ideologically close approach of [12, 13, 14]. Despite the fact that these two approaches are close relatives and even have equation (1) in common, the ``Fife limit'' theory is based on the consideration that excitation wave has a sharp front and a sharp back, and description is made in terms of motion of these two lines. Though the notion of sharp front is quite relevant to cardiac excitation wave, the notion of the waveback is rather questionable, the tip is not a junction of the front with the back and the ``Fife'' theory is hardly applicable to heart tissue at all. On the contrary, the kinematic approach needs neither a sharp back nor even a sharp front, and only assumes that the crest line remains smooth and pulse profile across this line perturbed slightly, and these conditions may be relevant to certain conditions in heart.