Mean-Field Formulation of Maxwell Equations to Model Electrically Inhomogeneous and Isotropic Media

Claude Bédard,Alain Destexhe

doi:10.4236/jemaa.2014.610029

Claude Bédard, Alain Destexhe

Open Access

https://doi.org/10.4236/jemaa.2014.610029

Copy DOI

Abstract

Maxwell equations were originally designed to describe classic electromagnetic phenomena in any type of medium. In particular, to describe electromagnetic phenomena under the quasistatic electric approximation in media that are electrically inhomogeneous and isotropic, such as for example when there are strong spatial variations of conductivity, the formalism must be adapted according to the problem considered. We review here two approaches to this problem, first a “microscopic” model, where the spatial variations of conductivity and permittivity are explicitly taken into account. In a second “macroscopic” model, these spatial variations are taken on average by using a mean-field formulation of Maxwell equations. Both of these models can describe the electromagnetic behavior of inhomogeneous media. We illustrate this formalism to describe the electric behavior of biological media, such as brain tissue, which is typically very inhomogeneous. We show that the theory predicts that for the typical frequency range of biological phenomena (lower than about 1000 Hz), the inhomogeneous nature of the medium has a determinant influence.

Highlights

Maxwell equations of electromagnetism were initially designed to describe classic electromagnetic phenomena in arbitrary media
Other models have been proposed for 1 f scaling at high frequencies [15], but the present model is the only one that accounts for the frequency scaling of extracellular potentials over the whole frequency range. This model shows that ionic diffusion provides a physical mechanism that is capable of explaining a large range of experimental observations, and that a macroscopic model is the right formalism to investigate such properties
We have overviewed two formalisms to model the electrical behavior of electrically inhomogeneous media

Summary

Introduction

Maxwell equations of electromagnetism were initially designed to describe classic electromagnetic phenomena in arbitrary media (such as vacuum, air, water, etc). (2014) Mean-Field Formulation of Maxwell Equations to Model Electrically Inhomogeneous and Isotropic Media. The notions of electric conductivity and permittivity (different from that of vacuum) of a given medium do not have a physical sense at a sub-atomic level, and these notions only apply to a mean-field level. This is analogous to the notion of pressure and temperature in classic thermodynamics. Similar considerations apply to composite materials, such as amorphous solids [1] In such cases, Maxwell equations can still be used, but the fact that the conductivity and permittivity highly depend on space, complicates its application. Note that most of the models used in biological systems neglect the electrical inhomogeneity of the tissue

Microscopic Model of Electrically Inhomogeneous Isotropic Media

Macroscopic Model of Electrically Inhomogeneous Media

Discussion