The cardiac bidomain model and homogenization

TL;DR

This paper presents a simplified proof of homogenization for the cardiac bidomain model, deriving it from cellular models using two-scale convergence and boundary unfolding, addressing nonlinearities and complex boundary conditions.

Contribution

It introduces a novel, streamlined homogenization proof for the bidomain model, incorporating nonlinear membrane dynamics and oscillating boundary conditions.

Findings

Homogenization of the bidomain model achieved using two-scale convergence.

Inclusion of nonlinear membrane models in the homogenization process.

Handling of complex boundary conditions with boundary unfolding operator.

Abstract

We provide a rather simple proof of a homogenization result for the bidomain model of cardiac electrophysiology. Departing from a microscopic cellular model, we apply the theory of two-scale convergence to derive the bidomain model. To allow for some relevant nonlinear membrane models, we make essential use of the boundary unfolding operator. There are several complications preventing the application of standard homogenization results, including the degenerate temporal structure of the bidomain equations and a nonlinear dynamic boundary condition on an oscillating surface.