The cardiac bidomain model and homogenization

Erik Grandelius; Kenneth H. Karlsen; Erik Grandelius; Kenneth H. Karlsen

doi:10.3934/nhm.2019009

Networks and Heterogeneous Media

2019, Volume 14, Issue 1: 173-204. doi: 10.3934/nhm.2019009

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The cardiac bidomain model and homogenization

Erik Grandelius ^,,
Kenneth H. Karlsen

Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, N–0316 Oslo, Norway

Received: 01 August 2018 Revised: 01 November 2018
Primary: 35K57, 35B27; Secondary: 35K65, 92C30

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.
- Bidomain equations,
- cardiac electric field,
- reaction-diffusion system,
- degenerate,
- weak solution,
- homogenization,
- two-scale convergence,
- unfolding operator
Citation: Erik Grandelius, Kenneth H. Karlsen. The cardiac bidomain model and homogenization[J]. Networks and Heterogeneous Media, 2019, 14(1): 173-204. doi: 10.3934/nhm.2019009

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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.

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