Linear and nonlinear models for electrical conduction in biological tissues

We collect some results concerning electrical conduction problems in biological tissues. Indeed, it is well known that electric potentials can be used in diagnostic devices to investigate the properties of biological tissues. Besides the well-known diagnostic techniques such as magnetic resonance, X-rays and so on, it plays an important role a more recent, cheap and noninvasive technique known as {\it electric impedance tomography} (EIT). Such a technique is essentially based on the possibility of determining the physiological properties of a living body by means of the knowledge of its electrical behavior. The models we present here are described by means of a finely mixed periodic medium, with a very small characteristic length, made by two different conductive regions (the intracellular and extracellular spaces), separated by an interface (the cell membrane), which exhibits both a capacitive and a conductive behavior. Due to the biochemical structure of the cell membrane, its conductive properties can perform a linear or a nonlinear behavior. We will discuss these two different situations, producing in each case a ``macroscopic'' or ``homogenized'' model, obtained letting the spatial period of the medium go to zero. The asymptotic behavior of the macroscopic potential for large times is investigated, too.