  In this paper, we are dealing with a rigorous homogenization result at two
different levels for the bidomain model of cardiac electro-physiology. The
first level associated with the mesoscopic structure such that the cardiac
tissue consists of extracellular and intracellular domains separated by an
interface (the sarcolemma). The second one related to the microscopic structure
in such a way that the intracellular medium can only be viewed as a periodical
layout of unit cells (mitochondria). At the interface between intra-and
extracellular media, the fluxes are given by nonlinear functions of ionic and
applied currents. A rigorous homogenization process based on unfolding
operators is applied to derive the macroscopic (homogenized) model from our
meso-microscopic bidomain model. We apply a three-scale unfolding method in the
intracellular problem to obtain its homogenized equation at two levels. The
first level upscaling of the intracellular structure yields the mesoscopic
equation. The second step of the homogenization leads to obtain the
intracellular homogenized equation. To prove the convergence of the nonlinear
terms, especially those defined on the microscopic interface, we use the
boundary unfolding method and a Kolmogorov-Riesz compactness's result. Next, we
use the standard unfolding method to homogenize the extracellular problem.
Finally, we obtain, at the limit, a reaction-diffusion system on a single
domain (the superposition of the intracellular and extracellular media) which
contains the homogenized equations depending on three scales. Such a model is
widely used for describing the macroscopic behavior of the cardiac tissue,
which is recognized to be an important messengers between the cytoplasm
(intracellular) and the other extracellular inside the biological cells.
Contents 2020 Mathematics Subject Classification. 65N55 and 35A01 and 35B27 and
35K57 and 65M..
