Published Paper

Inserted: 30 apr 2003
Last Updated: 25 jul 2003

Journal: C.R. Mecanique
Volume: 331
Pages: 503-508
Year: 2003

Abstract:

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 $u_{0}$ of the electric potential, obtained as the period of the microscopic structure approaches zero, solves the equation \begin{equation} -{\rm Div}\Big(\sigma_0\nabla{x} u_{0} + A^{0} \nabla_{x} u_{0} + \int_{0}^{t} A^{1}(t-\tau) \nabla_{x} u_{0}(x,\tau) d\tau -{\mathcal F}(x,t)\Big)=0 \,, \end{equation} where $\sigma_0>0$ and the matrices $A^{0}$, $A^{1}$ depend on geometric and material properties, while the vector function ${\mathcal F}$ keeps trace of the initial data of the original problem. Memory effects explicitly appear here, making this elliptic equation of non standard type.

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