Homogenization of syncytial tissues.

This article derives a continuum representation of a multicellular, syncytial tissue directly from its microstructure and basic physical principles. The final equations for the homogenized syncytium contain the bidomain model as a special case. The derivation begins with an idealized, periodic representation of the tissue microstructure. Potentials inside the cells and in the extracellular fluid are governed by the Laplace equation. Electrical properties of the membrane separating those two regions are represented by the boundary conditions on the potentials. An homogenization process based on a two-scale asymptotic expansion converts this microscopic, pointwise description into an averaged, continuum representation by two reaction-diffusion equations. The same process also yields formulas for the effective conductivities of the tissue in terms of its microstructure and specific conductivities of cytoplasm and extracellular fluid. The validity of the homogenized syncytium model is assured deep in the tissue for autonomous processes, such as propagation, and in the presence of external fields that are nearly uniform and limited in strength. The derived model is not formally valid on the surface of tissue, in the proximity of sources, and under strong or rapidly changing electrical fields.