Electro-magneto-encephalography for the three-shell model: numerical implementation via splines for distributed current in spherical geometry

Fokas, A. S.; Hauk, Olaf; Michel, Volker

doi:10.1088/0266-5611/28/3/035009

Cited by 27 publications

(38 citation statements)

References 28 publications

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“…(iii) The remark in (ii) regarding the 'minimum-norm solutions' is also valid for other regularization strategies used in the literature, such as the minimizations used in FOCUSS [25], RWMN [33] and LORETA [39]. (iv) In the case of independent EEG measurements, the results of [19] for the case of spherical and ellipsoidal geometries provide the explicit analytical solutions for the so-called ELEKTRA model [21] (this model has the advantage that it can be compared directly with intra-cranial recordings). (v) The question of implementing the reconstruction formulae of [19] supplemented with appropriate minimization constraints to real data and computing the relevant current with commercial software remains open.…”

Section: Discussionmentioning

confidence: 84%

The definite non-uniqueness results for deterministic EEG and MEG data

Dassios

Fokas

2013

Inverse Problems

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The solvability of the inverse problems of electroencephalography and magnetoencephalography has been studied extensively in the literature using a variety of models, including spherical and non-spherical geometries, homogeneous and inhomogeneous head models, and neuronal excitations involving the discrete and continuous distribution of dipoles. Among the important methods used are the methods based on spectral decompositions, physical arguments and integral representation techniques. Regarding the uniqueness of these inverse problems, a general result, independent of the geometry and the homogeneity of the conducting medium, has been obtained recently by the second author. This paper summarizes this result which appears to be mathematically definitive, in the sense that the geometry is arbitrary, the brain is surrounded by shells of varying conductivities, the neuronal current is arbitrary, the data are complete and the proofs are analytical. Furthermore, this paper includes a summary of the main steps of the proofs leading to the above result. In addition, it demonstrates the consistency of this general uniqueness result with all earlier results known in the literature.

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Section: Discussionmentioning

confidence: 84%

The definite non-uniqueness results for deterministic EEG and MEG data

Dassios

Fokas

2013

Inverse Problems

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“…Progress toward the analogous computational problems for the simpler case of spherical harmonics has been recently reported in . The question of extending the technique of to the ellipsoidal case remains open.…”

Section: Discussionmentioning

confidence: 99%

Electro‐magneto‐encephalography for the three‐shell model : a single dipole in ellipsoidal geometry

Dassios

Fokas

2012

Math Methods in App Sciences

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Analytic formulae are presented for determining the position and the moment of a single dipole from either Electroencephalography or Magnetoencephalography (MEG) measurements, under the assumption of the three-shell ellipsoidal model. It is remarkable that for this model all three components of the moment can be determined from the MEG measurements. This means that, in contrast to the spherical model, there exist no silent dipolar sources in MEG for the ellipsoidal model.

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“…This expression involves the inversion of a 7 × 7 matrix. It was observed in [13] that this matrix is ill conditioned and requires regularization. On the other hand, for the case of N spherical layers, with N arbitrary , an analytical expression for us(r,τ) is obtained in [14], [15].…”

Section: Computation Of Vs(rτ)mentioning

confidence: 99%