Electric and Magnetic Activity of the Brain in Spherical and Ellipsoidal Geometry

Dassios, George

doi:10.1007/978-3-642-03444-2_4

Cited by 42 publications

(58 citation statements)

References 31 publications

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“…These terms are then rearranged to form an expression similar to Eq. (22). Comparing the resulting relation with Eq.…”

Section: Forward and Inverse Problem For A Single Dipole And Multiplementioning

confidence: 97%

“…1 Note that the compatibility condition (13) Employing analytic techniques, the interested reader can find all details in Ref. [22]; it is not hard to show that the solution regarding Eqs. (15) and (16) evaluated at the surface is given as…”

Section: Forward and Inverse Problem For A Single Dipole And Multiplementioning

confidence: 99%

“…Expression (17) can be simplified using a summation formula [22,24] yielding the following closed form:…”

Section: Forward and Inverse Problem For A Single Dipole And Multiplementioning

confidence: 99%

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Mathematical Foundation of Electroencephalography

Doschoris¹,

Kariotou²

2017

Electroencephalography

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Electroencephalography (EEG) has evolved over the years to be one of the primary diagnostic technologies providing information concerning the dynamics of spontaneous and stimulated electrical brain activity. The core question of EEG is to acquire the precise location and strength of the sources inside the human brain by knowledge of an electrical potential measured on the scalp. But in what way is the source recovered? Leaving aside the biological mechanisms on the cellular level responsible for the recorded EEG signals, we pay attention to the mathematical aspects of the narrative. Our goal is to provide a brief and concise introduction of the mathematical terminology associated with the modality of EEG. We start from the very beginning, presenting step by step the mathematical formulation behind EEG in a simple and clear manner, keeping the mathematical notation to a minimum. Whilst we serve only the key relations for the described problems, we focus specifically on the limitations of each modelling approach. In this fashion, the reader can appreciate the beauty of the formulas presented and discover every single piece of information encoded within these formulas.

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“…These terms are then rearranged to form an expression similar to Eq. (22). Comparing the resulting relation with Eq.…”

Section: Forward and Inverse Problem For A Single Dipole And Multiplementioning

confidence: 97%

Section: Forward and Inverse Problem For A Single Dipole And Multiplementioning

confidence: 99%

See 1 more Smart Citation

Mathematical Foundation of Electroencephalography

Doschoris¹,

Kariotou²

2017

Electroencephalography

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“…Straightforward calculations, using either eigenfunction expansions or images [4], lead to the following solutions:…”

Section: Electroencephalographymentioning

confidence: 99%

“…Simple analytic algorithms for the solution of the inverse problem for both EEG and MEG corresponding to a single dipole, a finite set of dipoles, and a current localised within a small sphere, are presented in [5]. For the more realistic ellipsoidal brain-model, the direct problems are discussed in [3], [4].…”

mentioning

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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Third‐order harmonic expansion of the magnetoencephalography forward and inverse problems in an ellipsoidal brain model

Alcocer-Sosa

Gueorguiev

2016

Numer Methods Biomed Eng

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We present a forward modeling solution in the form of an array response kernel for magnetoencephalography. We consider the case when the brain's anatomy is approximated by an ellipsoid and an equivalent current dipole model is used to approximate brain sources. The proposed solution includes the contributions up to the third-order ellipsoidal harmonic terms; hence, we compare this new approximation against the previously available one that only considered up to second-order harmonics. We evaluated the proposed solution when used in the inverse problem of estimating physiologically feasible visual evoked responses from magnetoencephalography data. Our results showed that the contribution of the third-order harmonic terms provides a more realistic representation of the magnetic fields (closer to those generated with a numerical approximation based on the boundary element method) and, subsecuently, the estimated equivalent current dipoles are a better fit to those observed in practice (e.g., in visual evoked potentials). Copyright © 2016 John Wiley & Sons, Ltd.

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Electric and Magnetic Activity of the Brain in Spherical and Ellipsoidal Geometry

Cited by 42 publications

References 31 publications

Mathematical Foundation of Electroencephalography

Mathematical Foundation of Electroencephalography

Electro-magneto-encephalography and fundamental solutions

Third‐order harmonic expansion of the magnetoencephalography forward and inverse problems in an ellipsoidal brain model

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