Use of topological charge to determine filament location and dynamics in a numerical model of scroll wave activity

The unique time course of an excitable element in cardiac tissue can be represented as the phase of its trajectory in state space. A phase singularity is defined as a spatial point where the surrounding phase values changes by a total of 2 pi, thereby forming the organizing center for a reentrant excitatory wave, a phenomenon which occurs in cardiac fibrillation. In this paper, we describe a methodology to detect the singular filament in numeric simulations of three-dimensional (3-D) scroll waves by using the concept of topological charge. Here, we use simple two-variable models of cardiac activity to construct the state space, generate the phase field, and calculate the topological charge as a summation of 3-D convolution operations. We illustrate the usage of the algorithm on the basic dynamics of vortex ring filament behavior as well as the more complex spatiotemporal behavior observed in fibrillation. We also compare the motion of filament wavetips as determined by the phase field produced by two-variable state space and single-variable, time-delay embedded state space. Finally, we examine the state spaces produced by a more complex three-variable model. We conclude that the use of state-space analysis, along with the unique properties of topological charge, allows for a novel means of filament localization.