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

TL;DR

This paper derives a macroscopic tridomain model of cardiac electrical activity from a detailed microscopic model using unfolding homogenization, addressing complex nonlinear and degenerate features.

Contribution

It introduces a homogenization approach for a detailed microscopic tridomain model with dynamic gap junctions, leading to a new macroscopic model for cardiac conduction.

Findings

Successful derivation of macroscopic model from microscopic system

Handling of nonlinear boundary conditions using boundary unfolding

Addressing degeneracy in temporal structure of equations

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.