# Electrical Conduction in Biological Tissues - Homogenization Techniques and Asymptotic Decay for Linear and Nonlinear Problems

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

#### Abstract

We collect some results concerning electrical conduction problems in biological tissues. These problems are set in a finely mixed periodic medium and the unknown electric potentials solve standard elliptic equations set in different conductive regions (the intracellular and extracellular spaces), separated by an interface (the cell membrane), which exhibits both a capacitive and a conductive behavior. As the spatial period of the medium goes to zero, the problems approach a homogenization limit. The macroscopic models are obtained by using the technique of asymptotic expansions, in the case where the conductive behavior of the cell membrane is linear, and by means of two-scale convergence, in the case where, due to its biochemical structure, the cell membrane performs a strongly nonlinear conductive behavior. The asymptotic behavior of the macroscopic potential for large times is investigated, too.

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#### Paper Citation

#### in Harvard Style

Amar M., Andreucci D., Bisegna P. and Gianni R. (2013). **Electrical Conduction in Biological Tissues - Homogenization Techniques and Asymptotic Decay for Linear and Nonlinear Problems** . In *Proceedings of the 3rd International Conference on Simulation and Modeling Methodologies, Technologies and Applications - Volume 1: BIOMED, (SIMULTECH 2013)* ISBN 978-989-8565-69-3, pages 696-703. DOI: 10.5220/0004634006960703

#### in Bibtex Style

@conference{biomed13,

author={M. Amar and D. Andreucci and P. Bisegna and R. Gianni},

title={Electrical Conduction in Biological Tissues - Homogenization Techniques and Asymptotic Decay for Linear and Nonlinear Problems},

booktitle={Proceedings of the 3rd International Conference on Simulation and Modeling Methodologies, Technologies and Applications - Volume 1: BIOMED, (SIMULTECH 2013)},

year={2013},

pages={696-703},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0004634006960703},

isbn={978-989-8565-69-3},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 3rd International Conference on Simulation and Modeling Methodologies, Technologies and Applications - Volume 1: BIOMED, (SIMULTECH 2013)

TI - Electrical Conduction in Biological Tissues - Homogenization Techniques and Asymptotic Decay for Linear and Nonlinear Problems

SN - 978-989-8565-69-3

AU - Amar M.

AU - Andreucci D.

AU - Bisegna P.

AU - Gianni R.

PY - 2013

SP - 696

EP - 703

DO - 10.5220/0004634006960703