The unique determination of neuronal currents in the brain via magnetoencephalography

Fokas, A. S.; Kurylev, Yaroslav; Marinakis, Vangelis

doi:10.1088/0266-5611/20/4/005

Cited by 99 publications

(112 citation statements)

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“…However, the representation of U(t) used in Fokas et al (1996Fokas et al ( , 2004 is different from the one used here. This is due to the fact that the representation for the current J p used here (see equation (3.1)) is different from the one used in Fokas et al (1996Fokas et al ( , 2004. The latter representation is quite convenient for the case that one considers only MEG, but it is inappropriate for the case that MEG and EEG are used simultaneously.…”

Section: Discussionmentioning

confidence: 99%

“…The latter representation is quite convenient for the case that one considers only MEG, but it is inappropriate for the case that MEG and EEG are used simultaneously. Indeed, in the representation of Fokas et al (1996Fokas et al ( , 2004, the radial component of J p is arbitrary and then the contributions of this part to MEG and EEG do not uncouple.…”

Section: Discussionmentioning

confidence: 99%

“…In particular, the non-uniqueness of the inverse problem is considered as the Achilles' heel of MEG. In this context, a complete answer to the non-uniqueness question for a homogeneous spherical model was presented in Fokas et al (1996Fokas et al ( , 2004 where it was shown that: (i) the only part of a continuously distributed current that can be reconstructed via MEG consists of certain moments of one of the two functions specifying the tangential component of the current (the other function specifying the tangential component, as well as the radial component of the current, is 'invisible' in the spherical model of MEG) and (ii) it is possible to reconstruct uniquely the current that minimizes the L 2 -norm. Some of these results were extended, from a spherical to a starshaped geometry in Dassios et al (2005).…”

Section: Introductionmentioning

confidence: 99%

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Electro–magneto-encephalography for a three-shell model: distributed current in arbitrary, spherical and ellipsoidal geometries

Fokas

2008

J. R. Soc. Interface.

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The problem of determining a continuously distributed neuronal current inside the brain under the assumption of a three-shell model is analysed. It is shown that for an arbitrary geometry, electroencephalography (EEG) provides information about one of the three functions specifying the three components of the current, whereas magnetoencephalography (MEG) provides information about a combination of this function and of one of the remaining two functions. Hence, the simultaneous use of EEG and MEG yields information about two of the three functions needed for the reconstruction of the current. In particular, for spherical and ellipsoidal geometries, it is possible to determine the angular parts of these two functions as well as to obtain an explicit constraint satisfied by their radial parts. The complete determination of the radial parts, as well as the determination of the third function, requires some additional a priori assumption about the current. One such assumption involving harmonicity is briefly discussed.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Electro–magneto-encephalography for a three-shell model: distributed current in arbitrary, spherical and ellipsoidal geometries

Fokas

2008

J. R. Soc. Interface.

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“…The fact that neither of these inverse problems has a unique solution was known to Helmholtz in 1853. Nevertheless, complete quantitative results on the non-uniqueness of the inverse MEG problem were obtained only recently, in [9], [10] for the spherical model …”

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confidence: 99%

Electro-magneto-encephalography and fundamental solutions

Dassios

Fokas

2009

Quart. Appl. Math.

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Abstract. Electroencephalography (EEG) and Magnetoencephalography (MEG) are two important methods for the functional imaging of the brain. For the case of the spherical homogeneous model, we elucidate the mathematical relations of these two methods. In particular, we derive and analyse three different representations for the electric and magnetic potentials, as well as the corresponding electric and magnetic induction fields, namely: integral representations involving the Green and the Neumann kernels, representations in terms of eigenfunction expansions, and closed form expressions. We show that the parts of the EEG and MEG fields in the interior of the brain that are due to the induction current are related via Kelvin's inversion transformation. We also derive closed form expressions for the interior and exterior vector potentials of the corresponding magnetic induction fields. Introduction.Electrochemically generated neuronal currents in the brain give rise to a magnetic field, which in turn excites an induction current within the conductive brain tissue. This electromagnetic activity of the brain is recorded by measuring the electric potential on the scalp (EEG) and the magnetic induction field at distances 4-6 cm from the head (MEG). There exists an extensive literature on how to utilise the EEG and MEG recordings [13], [17] in order to determine the electric potential and the magnetic induction field outside the head. From the mathematical point of view, the two basic problems for EEG and MEG are the forward problem, namely find the interior and exterior fields in terms of the neuronal current and the inverse problem, namely find the neuronal current in terms of the electric potential or in terms of the magnetic field. The fact that neither of these inverse problems has a unique solution was known to Helmholtz in 1853. Nevertheless, complete quantitative results on the non-uniqueness of the inverse MEG problem were obtained only recently, in [9], [10] for the spherical model

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“…Since then, many efforts have been made to produce analytic solutions for the related direct problem where the field generated by a given source is sought. We mention the works of Ilmoniemi, Hämäläinen and Knuutila [10], Sarvas [15] and Fokas, Kurylev and Marinakis [5] for the spherical brain model, the works of Cuffin and Cohen [1] and de Munck [6] for the spheroidal brain model and the work of Nolte, Fieseler and Curio [14] for perturbative models of the brain. But, as anatomy indicates, the actual geometry of the human brain is best approximated by a triaxial ellipsoid [16], a geometrical shape far more complicated than the sphere or even the spheroid.…”

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confidence: 99%

Untitled

2005

Quart. Appl. Math.

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Abstract. The forward problem of Magnetoencephalography for an ellipsoidal inhomogeneous shell-model of the brain is considered. The inhomogeneity enters through a confocal ellipsoidal shell exhibiting different conductivity than the one of the brain tissue. It is shown that, as far as the leading quadrupolic moment of the exterior magnetic field is concerned, the complicated expression associated with the field itself is the same as in the homogeneous case, while the effect of the shell is focused on the form of the generalized dipole moment. In contrast to the spherical case, where no shell inhomogeneities are "readable" outside the skull, the ellipsoidal shells establish their existence on the exterior magnetic induction field in a way that depends not only on the geometry but also on the conductivity of the shell. The degenerated spherical results are fully recovered.1. Introduction. The mathematical theory of Electroencephalography (EEG) and Magnetoencephalography (MEG) was founded in the late 60s, mainly on the basis of the works of Geselowitz [7,8]. Since then, many efforts have been made to produce analytic solutions for the related direct problem where the field generated by a given source is sought. We mention the works of Ilmoniemi, Hämäläinen and Knuutila [10], Sarvas [15] and Fokas, Kurylev and Marinakis [5] for the spherical brain model, the works of Cuffin and Cohen [1] and de Munck [6] for the spheroidal brain model and the work of Nolte, Fieseler and Curio [14] for perturbative models of the brain. But, as anatomy indicates, the actual geometry of the human brain is best approximated by a triaxial ellipsoid [16], a geometrical shape far more complicated than the sphere or even the spheroid. Intense efforts towards a complete analytic solution for EEG and MEG problems in ellipsoidal geometry led to results included in [3,4,11,12]. The present work aims in obtaining an analytic expression of the leading quadrupolic term for the exterior magnetic field in the

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The unique determination of neuronal currents in the brain via magnetoencephalography

Cited by 99 publications

References 16 publications

Electro–magneto-encephalography for a three-shell model: distributed current in arbitrary, spherical and ellipsoidal geometries

Electro–magneto-encephalography for a three-shell model: distributed current in arbitrary, spherical and ellipsoidal geometries

Electro-magneto-encephalography and fundamental solutions

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