Frustrated drift of an anchored scroll-wave filament and the geodesic principle

We investigate anchored scroll-wave filaments in an excitable medium whose diffusivity matrix, including its determinant, is spatially nonuniform. The study is motivated by cardiological applications where scroll-wave behavior in the presence of diffusivity gradients is believed to play an important role in the development of severe arrhythmias. A diffusivity gradient is expected to make the filament drift, unless drift is prevented ("frustrated") by anchoring to localized defects in the propagation medium. The resulting stationary filament is a geodesic curve, as demonstrated here in the case of a nonzero but constant gradient. That is, the diffusivity matrix has a determinant that varies in space, in contrast to what was assumed in earlier work. Here, we show that the filament shape results from a metric tensor of the form (det D)D{-1} , where D is the diffusivity tensor. The filament's shape is solely determined by the diffusivity tensor and is independent of the equation's reaction terms. We derive the analytic solution for the filament and determine conditions for the existence of that solution. The theory is in excellent agreement with numerical simulations.