Return current in encephalography. Variational principles

Abstract

The encephalographic problem of finding the electric potential V and the return current associated with any assumed primary current, Jp, is put in the form of a variational principle. With Jp and the conductivity specified, the correct V is one which makes an integral quantity P[V] a maximum. The terms in P[V] are related to the rates at which work is done by the electric field on the primary and return currents. It is shown that there is a unique solution for the electric field, and it satisfies the conservation of energy; this condition can serve as a check on any numerical solution. With the conductivity a different constant in different regions, the variational principle is recast in terms of the charge density on the surfaces of discontinuity. An iteration-variation method for finding the solution is outlined, and possible computational advantages over other approaches are discussed.

Stable Numerical Identification of Sources in Non-Homogeneous Media

“…For the EIP, this model can be deduced from Maxwell's equations and from some experimental results, where f represents the bioelectric source located in the brain Ω 1 , and the measurement V is the EEG on the scalp ∂Ω = S 2 . The boundary conditions (3) and ( 4) are the transmission conditions, and the boundary condition (5) indicates that air conductivity (i.e., of Ω c ) is zero [25][26][27][28]. From the Green's formulas, f satisfies the following compatibility condition:…”

Inverse electroencephalography for volumetric sources

“…According to these assumptions, the electric potential u generated in Ω satisfies the following boundary value problem [7,14]:…”

“…The analysis of inverse problem for the case of volumetric sources has been done in works [1,3,7,9,12,14], and the case of cortical sources in Refs. [4][5][6].…”

Inverse electroencephalography for cortical sources

“…The forward problem consists of determining the electrostatic potential assuming that the bioelectrical source on the cerebral cortex is known. Different statements of inverse problems for electroencephalography appear in [1,5,7,9,21], and the analysis of the case of volumetric sources has been done in [1,4,5,9,11,18,21]. The two main difficulties in the study of the IEP are its possible ill-posedness and the diversity of opinions about what could be considered as a representative characteristic of the bioelectric cortical activity [11,21], namely: the potential mapping, the charge density, the dipole density, or the distribution density of primary currents generated by the cortical cells activity.…”