Evolution and memory effects in the homogenization limit for electrical conduction in biological tissues

We study a problem set in a finely mixed periodic medium, modelling electrical conduction in biological tissues. The unknown electric potential solves standard elliptic equations set in different conductive regions (the intracellular and extracellular spaces), separated by a dielectric surface (the cell membranes), which exhibits both a capacitive and a nonlinear conductive behaviour. Accordingly, dynamical conditions prevail on the membranes, so that the dependence of the solution on the time variable t is not only of parametric character. As the spatial period of the medium goes to zero, the electric potential approaches in a suitable sense a homogenization limit u(o), which keeps the prescribed boundary data, and solves the equation div [B(o)del(x)u(o) + integral(o)(t) A(1)(t - tau)del(x)u(o)(tau) dtau - F] = 0. This is an elliptic equation containing a term depending on the history of the gradient of uo; the matrices B(o), A(1) in it depend on the microstructure of the medium. More exactly, we have that, in the limit, the current is still divergence-free, but it depends on the history of the potential gradient, so that memory effects explicitly appear. The limiting equation also contains a term T keeping trace of the initial data.