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This paper discusses theoretical aspects of the modeling of the sources of the EEG (i.e., the bioelectromagnetic inverse problem or source localization problem). Using the Helmholtz decomposition (HD) of the current density vector (CDV) of the primary current into an irrotational (

The irrotational source model was explicitly proposed as a biophysically constrained model for the EEG in [

While the irrotational source model is a rather trivial consequence of basic textbook results, it can, nevertheless, lead to an apparent contradiction when we try to place it within the context of EEG models. The basic problem arises because we intuitively expect the generators (i.e., the neural currents) to be confined to the brain volume. Yet, irrotational currents are different from zero whenever the conductivity is different from zero. Consequently, the irrotational sources are not confined to the gray cortical matter but rather extend into tissues where no primary currents are expected as, for example, white matter, bone, or the cerebrospinal fluid. We will here show that this observation is not in contradiction with the physical and physiological validity of the irrotational source model (ISM) of ELECTRA. In fact, we will formally show that the spatial extent of the generators is defined by its divergence rather than by the spatial extent of its irrotational component. We will also show that the divergence of the irrotational component of focal (dipolar) sources remains confined to the dipole center while the divergence of the irrotational component of extended sources confines itself to the brain region. This observation helps to understand the measure we should minimize within inverse problems to restrict the spatial extent of biological sources.

To formally introduce the problem let us start with the following example fully justified by results in [

If we interpret region

In particular, we will demonstrate that, as suggested by (

The paper is structured as follows: In Section

Unless otherwise stated, we will denote vectors and matrices by bold uppercase letters except the spatial variables that will be denoted by bold lowercase letters (e.g.,

(Also called the fundamental theorem of vector calculus): let

Typically stated for isotropic media it has been also demonstrated for anisotropic media [

The existence of HD is closely related to the solution of Poisson’s equation (

In the electrodynamics literature it is usual to associate

In lay terms a dipole at location

We remind the reader that in this paper the term “the dipole” designates “the current density vector of the dipole” as expressed by (

Since the dipole has a single singular point (at

In the following we assume that we have a region

Equation (

Note that the divergence of

In general the solenoidal part can be obtained by subtraction of the irrotational part (

We would like to remark that while the results of this section are, a posteriori (i.e., once we see them), not very surprising but expected for static fields, their mathematical proof using the Helmholtz decomposition of the dipole and/or the delta calculus is missing from the literature. The formal treatment presented here helps to clarify aspects of the source itself within the framework of the HD. All the formulas in this section were verified with the symbolic Matlab toolbox except for the integrals (

In short, results in this section show that from the decomposition of dipole as

theoretically, the only part of a dipole able to generate electric potentials (

while the dipole denotes a source confined to location

the spatial extent of a dipolar primary current is defined by the divergence of its irrotational part;

even if we computed the irrotational and the solenoidal parts separately using the defining equations (

In summary, the current dipole is electrically visible because it “contains” a distributed irrotational source

The analysis of the irrotational and solenoidal components of a distributed irrotational current might be also of some neurophysiological interest. Indeed, any current expressed as the gradient of another scalar field (e.g., ionic concentrations or purely ohmic currents) is necessarily irrotational. Therefore, in this section we compute the Helmholtz decomposition of the pure (because it contains no solenoidal part) irrotational source of (

as for the dipole, a distributed irrotational source contributes to the EEG measurements via its irrotational part

as for the dipole, the irrotational component

however, according to (

since the current density vector is not (a priori) zero anywhere, the irrotational components

In summary, the irrotational part of a pure irrotational current source density vector is the same current density vector and extends all over the space. However, as for the dipole, the divergence of

The use of the quasi static approximation in the solution of the EEG inverse problem has been motivated by the range of frequencies of the EEG and the electrical properties of the media [

To start we consider an inhomogeneous (i.e.,

The conductivity and the permittivity tensors (or scalars) can be combined to obtain the complex permittivity as follows:

Since

Substituting

Equation (

In summary, from (

Section

First, most of the apparently trivial interpretations of the results given here are based on treating the delta function (i.e., the dipole) as a function in the classical sense. This is formally incorrect since the delta function is not defined as a function of the space but only for integrals that include or not the location of the delta. In other words, an equality containing the delta function is to be interpreted only after integrating both terms according to the definition and properties of the delta function (e.g., Appendix

Second, the adopted formalism seems to be the only way to clarify the apparent contradiction due to the difference between the spatial extent (i.e., the region where it is different from zero) of the primary current, the spatial extent of its irrotational part, and the spatial extent of the generators of the electric (i.e., EEG) data. In effect, independently of the spatial extension of an arbitrary primary current, its irrotational part extends as much as the potential it generates. However the solenoidal part can exist (e.g., the dipole) or not (e.g., the irrotational source) and extend all over the space (e.g., the dipole) while the generators (i.e., the divergence of the primary current) might be confined to a completely different region (e.g., the irrotational source). The apparent contradiction arises from using the wrong definition of the generators. Effectively, since the generator of the EEG is not the current density vector

From the decomposition of the dipole we observe that all over the space there is a solenoidal part that cancels out the irrotational part. However the decomposition of the pure irrotational source shows that this is not really a need but a particularity of the dipole. In general, for all electrically visible sources (i.e., that produces EEG), whenever the primary current density vector is zero, there will be a solenoidal part to cancel out the irrotational part. For the sake of the physical interpretation, in the analysis presented here we have excluded the dipole location

Section

In this paper we have discussed the sources of the EEG making emphasis on the irrotational source model (ISM) originally proposed in [

We have also shown that the ISM is compatible with more general electrodynamics models of the EEG including inhomogeneous anisotropic dispersive media and thus that it is not limited to the quasistatic approximation.

In summary, from a rigorous point of view, the irrotational source model of ELECTRA corresponds to the estimation of the irrotational part of the unknown primary current

Notations examples and definitions for the mathematical elements used in the paper and the appendices are the following:

Points in 3D Euclidian space

Vector fields in

Scalar fields in

Nabla (or Del) symbol is

A vector field

Given vectors

Let

The authors declare that there is no conflict of interests regarding the publication of this paper. There is no conflict of interests including any financial, personal, or other relationships with persons or organizations for any author related to the work described in this paper.

This work has been supported by the 3R Research Foundation Grant 119-10, funding studies aiming at reducing or replacing animal research. Thanks are due to Dr. Chloe de Balthasar and Dr. Sven Wagner for reading of and comments and suggestions on an earlier version of this paper.