Key aspects for effective mathematical modelling of fractional-diffusion in cardiac electrophysiology: a quantitative study

Microscopic structural features of cardiac tissue play a fundamental role in determining complex spatio-temporal excitation dynamics at the macroscopic level. Recent efforts have been devoted to the development of mathematical models accounting for non-local spatio-temporal coupling able to capture these complex dynamics without the need of resolving tissue heterogeneities down to the micro-scale. In this work, we analyse in detail several important aspects affecting the overall predictive power of these modelling tools and provide some guidelines for an effective use of space-fractional models of cardiac electrophysiology in practical applications. Through an extensive computational study in simplified computational domains, we highlight the robustness of models belonging to different categories, i.e., physiological and phenomenological descriptions, against the introduction of non-locality, and lay down the foundations for future research and model validation against experimental data. A modern genetic algorithm framework is used to investigate proper parameterisations of the considered models, and the crucial role played by the boundary assumptions in the considered settings is discussed. Several numerical results are provided to support our claims.