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 u₀, which keeps the prescribed boundary data, and solves the equation ${\rm div}\, [B^0\nabla_x u_0+\int_0^t A^1(t-\tau)\nabla_x u_0(\tau)d\tau-\mathcal{F}]=0$. This is an elliptic equation containing a term depending on the history of the gradient of u₀; the matrices B⁰, A¹ 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 ℱ keeping trace of the initial data. [ABSTRACT FROM AUTHOR]