Microscopic tridomain model of electrical activity in the heart with dynamical gap junctions. Part 2 – Derivation of the macroscopic tridomain model by unfolding homogenization method
Affiliations: [a] Institut de Recherche Mathématique de Rennes, UMR 6625 CNRS, Université de Rennes 1, Campus de Beaulieu, F-35042 Rennes cedex, France | [b] Institut de Mathématiques de Bordeaux and INRIA-Carmen Bordeaux Sud-Ouest, Université de Bordeaux, 33076 Bordeaux Cedex, France | [c] Laboratoire de Mathématiques Jean Leray, UMR 6629 CNRS, École Centrale de Nantes, 1 rue de Noé, 44321 Nantes, France | [d] Léonard de Vinci Pôle Universitaire, Research Center, 92 916 Paris La Défense, France | [e] Department of Mathematics, Faculty of Sciences 1, Laboratory of Mathematics-DSST, Lebanese University Hadat, Lebanon
Abstract: We study the homogenization of a novel microscopic tridomain system, allowing for a more detailed analysis of the properties of cardiac conduction than the classical bidomain and monodomain models. In (Acta Appl.Math. 179 (2022) 1–35), we detail this model in which gap junctions are considered as the connections between adjacent cells in cardiac muscle and could serve as alternative or supporting pathways for cell-to-cell electrical signal propagation. Departing from this microscopic cellular model, we apply the periodic unfolding method to derive the macroscopic tridomain model. Several difficulties prevent the application of unfolding homogenization results, including the degenerate temporal structure of the tridomain equations and a nonlinear dynamic boundary condition on the cellular membrane. To prove the convergence of the nonlinear terms, especially those defined on the microscopic interface, we use the boundary unfolding operator and a Kolmogorov–Riesz compactness’s result.
Keywords: Tridomain model, reaction-diffusion system, homogenization theory, time-periodic unfolding method, gap junctions, cardiac electric field
