Figure 2: Simple scroll wave, in the FHN medium 
s.s. large. Shown is a snapshot of the excitation wave front, defined as
the surface u=0, coloured depending on the value of the other dynamic
variable, v. Fore-front (smaller v) is red and semi-transparent,
back-front (higher v) is completely transparent and invisible. Surface
near the edge of the excitation wave, for intermediate values of v,
is greenish and solid. This edge region is a visualisation of the scroll
filament. The solution is virtually independent on the vertical coordinate.
Topological classification of possible three-dimensional AW patterns has been considered by Winfree and Strogatz [1983,1984], and numerical experiments have revealed a rich variety of different scroll wave behaviours (see e.g. the review by Panfilov [1991]). An analytical approach for description of scroll wave dynamics was proposed by Keener [1988]. It was an asymptotics assuming that the characteristic spatial scale of the filament is much greater than the characteristic scale of a spiral wave, and in any cross-section orthogonal to the filament, the scroll wave is close to the 2-D spiral wave. The result of the singular perturbation theory, valid for generic reaction-diffusion system not only the FHN system, are equations for slow evolution of the filament and of the distribution of spiral wave rotation phase along the filament. Subsequent analysis of these equations [Biktashev et al. 1994] has shown that in the main order of magnitude, the equation of the filament motion is independent on the phase distribution, and has the form
where 
describes the period-averaged position of the filament at the time moment
t with parameter 
chosen so that points with equal
move orthogonally to the filament, and arbitrary in other respects (note
that the arc length s may not obey this property). Then the arc
length differentiation operator 
is defined as
A simple fact from differential geometry is that rate of change of total
length of a moving curve is equal (disregarding fringe effects) to the
integral over the curve of the scalar product of the curve motion velocity 
and the vector of curvature 
. Thus, an elementary but important property of Eq. (3)
is that the evolution of the total length is monotonic decreasing if 
is positive, and monotonic increasing if
is negative. Biktashev et al. [1994]
have suggested the term ``filament tension'' for this important medium
characteristic. Its heuristical value is that it predicts qualitatively
different behaviour for 3-D AW media with positive and negative filament
tensions. If the tension is positive, then a straight filament (simple
scroll) should be stable, and the vortex ring should shrink and collapse.
On the contrary, if the tension is negative, the vortex ring should expand
rather than collapse, and straight filaments are unstable. Despite obvious
limitations of the asymptotical theory, numerical experiments described
in [Biktashev et al. 1994] have shown
that these predictions do catch the main qualitative features of the 3-D
vortex dynamics.
Equation (3) is analogous to Da Rios
equations of motion of vortex lines in fluids [Ricca 1991;
1992], obtained in so called localised-induction
approximation (LIA); in fact, these equations are a partial case of (3)
for 
. The specifics of hydrodynamical vortices is that LIA procedure involves
logarithmic divergence (this problem does not exist for the AW vortices),
and that their filament tension
is always exactly zero, so that this term is sometimes used for the other
parameter, 
, which is not a medium constant but characteristic of the vortex magnitude.
The assumptions of the asymptotic theory require that the filaments are smooth and far from each other and from medium boundaries. Naturally, evolution of filaments with negative tension will lead to violation of all these assumptions, as lengthening of the curve will increase both its curvature and `concentration' within the medium. Thus, this asymptotical theory only predicts that the behaviour of such AW media will be unusual and complicated, but fails to predict the details. These details can be revealed by numeric experiments.
