## M. Amar - D. Andreucci - P. Bisegna - R. Gianni

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

created on 26 Nov 2002

modified by amar on 13 Dec 2004

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BibTeX]

*Published Paper*

**Inserted:** 26 nov 2002

**Last Updated:** 13 dec 2004

**Journal:** Mathematical Models and Methods In Applied Sciences

**Volume:** 14

**Number:** 9

**Pages:** 1261-1295

**Year:** 2004

**Abstract:**

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 eletric potential approaches a homogenization limit
$u_{0}$, solving
\begin{equation**}
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 the
properties of the tissue, and the vector function $\mathcal{F}$ keeps trace
of the initial data of the original problem.
In the limit, the current, given as the term in square brackets in the PDE
above, is still divergence-free, but it depends on the history of the
potential gradient, so that memory effects explicitly appear.

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