A hierarchy of models for the electrical conduction in biological tissues via twoscale convergence: the nonlinear case

We study a hierarchy of electrical conduction problems in biological tissues. These problems are set in a finely mixed periodic medium, and the unknown electric potentials solve standard elliptic equations set in different conductive regions (the intracellular and extracellular spaces), separated by an interface (the cell membranes), which exhibits a capacitive and a nonlinear conductive behavior, due to its biochemical structure. Different scalings in the interface condition correspond to different problems in the hierarchy. As the spatial period of the medium goes to zero, the problems approach a homogenization limit depending on the initial scaling. The macroscopic models are obtained by using the technique of two-scale convergence.