Jacobian-determinant vector method of identifying filaments in a three-dimensional medium

Scroll waves in three-dimensional excitable media rotate around singular filaments, which are topological defects analogous to phase singularities in two dimensions. Methods for precise localization of these filaments are essential for studies of filament dynamics in modeling and experimental studies. Practical applications of these methods include understanding the mechanisms of cardiac arrhythmias, most of which have recently been shown to have a three-dimensional nature. In this study, we propose a novel Jacobian-determinant vector method for filament identification based on the topological current theory. We define a Jacobian-determinant vector field, which we construct so that its magnitude peaks at filament locations, while its direction shows the local filament orientation, guiding further filament detection. We validate the method using the Aliev-Panfilov model across complex filament configurations, including Hopf links and trefoil knots, and demonstrate better performance compared to convolution-based and zero-normal-velocity methods, particularly under noisy and slightly fibrotic environments. This method, thus, provides a useful tool for studying scroll-wave dynamics in both theoretical and applied contexts.