Homogenization limit for electrical conduction in biological tissues in the radio-frequency range

We study an evolutive model for electrical conduction in biological tissues, where the conductive intra-cellular and extracellular spaces are separated by insulating cell membranes. The mathematical scheme is an elliptic problem, with dynamical boundary conditions on the cell membranes. The problem is set in a finely mixed periodic medium. We show that the homogenization limit u₀ of the electric potential, obtained as the period of the microscopic structure approaches zero, solves the equation - div (σ0∇xu0+A0∇xu0+∫0tA1(t-τ)∇xu0(x,τ)dτ-ℱ(x,t))=0 where σ₀>0 and the matrices A⁰, A¹ depend on geometric and material properties, while the vector function ℱ keeps trace of the initial data of the original problem. Memory effects explicitly appear here, making this elliptic equation of non standard type.