THE DIFFERENTIAL GEOMETRY OF SCROLL WAVES

Traveling waves of excitation organize physical, chemical, and biological systems in space and time. In the biological context they serve to communicate information rapidly over long distances and to coordinate the activity of tissues and organs. An example of particular beauty, complexity and importance is the three-dimensional rotating scroll wave observed in the Belousov-Zhabotinskii reaction and in the ventricle of the heart. A scroll wave rotates around a filamentous phase singularity that weaves through the three-dimensional medium. At any instant of time the geometry of the scroll wave can be reduced to the spatial arrangement of a ribbon whose edges are the singular filament and the tip of the scroll wave. This ribbon, when it closes on itself, must satisfy the topological constraint L = Tw + Wr, where L is the linking number of the two edges of the ribbon, Tw is the total twist of the ribbon, and Wr is the writhing number of the singular filament. We discuss the origin of this equation and its implications for scroll wave statics and dynamics.