ZEUS: Numerical methods to detect quasi-particles describing excitable media

Many complex systems with societal relevance are excitable, including brain tissue, cardiac tissue, heat waves, and epidemic spread. In cardiac tissue, arrhythmias often arise when electrical conduction is blocked by incomplete recovery after a previous stimulation. We recently presented a topological theory for excitable media that also captures conduction blocks. Therein, points where a wave front or a wave back spatially connects to a conduction block were shown to be topologically preserved, and denoted heads and tails, respectively. Here, we introduce algorithms to automatically localize heads and tails in excitation patterns on triangle meshes. We describe two variants, depending on the co-dimension of the forbidden zone in state space. As a result, the conduction block region is rendered either as a line of conduction block or an extended conduction block region. A key operation is to apply a bitwise OR operation onto states of a local vertex according to a partitioning into forbidden zone (Z), excited (E), unexcited (U), and their respective signed variants (S), which we abbreviate as ZEUS methods. The methods are applied to visualize heads, tails, and conduction blocks in simulated data and optical voltage mapping recordings of ventricular tachycardia in rabbit hearts. We compare the outcome to classical phase singularity analysis. Theoretical relations between the different options and advantages of each method are discussed. Robust algorithms are presented and made publicly available to identify certain topologically preserved points in excitation patterns. These methods can be used for automated analysis and classification.