EEG for Current With Two-Dimensional Support

Dassios, George; Fokas, A. S.; Hashemzadeh, Parham; Leahy, Richard M.

doi:10.1109/tbme.2017.2785342

Cited by 4 publications

(4 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the literature, the Helmholtz decomposition is the first choice decomposition for the inversion of the continuous EEG problem, see [6,9,[12][13][14]. The advantage of this approach is that the corresponding integral equation only depends on the scalar potential Ψ.…”

Section: Helmholtz Decomposition For Eegmentioning

confidence: 99%

“…This is a significant advantage since we do not have such a priori information about the neuronal current. This is to be contrasted with the approaches using the Hodge [6,13,15] or the Helmholtz [6,9,[12][13][14] decomposition, where further conditions on the neuronal current are required, such as boundary or smoothness conditions. The further insight into the structure of the inverse problem obtained via our approach, as well as the consistency of our results with the results of [12] are discussed in section 7.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Electro-magnetoencephalography for a spherical multiple-shell model: novel integral operators with singular-value decompositions

Leweke¹,

Michel²,

Fokas³

2020

Inverse Problems

View full text Add to dashboard Cite

The inverse magnetoencephalography and electroencephalography problems for spherical models have been extensively discussed in the literature. Using the spherical multiple-shell model, we derive novel vector-valued and singularity-free integral equations for both problems based on the quasi-static Maxwell's equations. These equations are solved via a Fourier series expansion. We call this procedure the Edmonds approach, since an orthonormal system based on the Edmonds-vector-spherical harmonics is used for the Fourier series. Employing the associated singular-value decomposition, we provide a complete answer to the non-uniqueness question of these two problems: only the harmonic part of the solenoidal component of the neuronal current is visible to the simultaneous use of magnetoencephalography and electroencephalography. The remaining components of the current are invisible to both techniques. We state Picard's condition for the existence of a solution and derive an explicit formula for the best-approximate solution of the neuronal current. In comparison to previous approaches, the Edmonds approach requires the fewest a-priori assumptions on the neuronal current. Finally, we show that the results obtained by means of the Edmonds approach are consistent with results derived earlier via the Helmholtz decomposition.

show abstract

Section: Helmholtz Decomposition For Eegmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Electro-magnetoencephalography for a spherical multiple-shell model: novel integral operators with singular-value decompositions

Leweke¹,

Michel²,

Fokas³

2020

Inverse Problems

View full text Add to dashboard Cite

show abstract

“…It was shown in [10][11][12] that, for the particular case of the spherical head model, v s (r, t) is given by v s (r, t) ¼…”

Section: Auxiliary Functions V J (R τ)mentioning

confidence: 99%

“…The medical significance of electroencephalography (EEG) is well established; see for example [1][2][3]. The estimation of the neuronal current from the measured electric potential (units volts) on the surface of the head provided via the EEG data can be formulated as a mathematical inverse problem [4][5][6][7][8][9][10][11][12][13]. The problem of computing the electric potential for a given head model and a given configuration of dipole sources is referred to as the forward problem [14][15][16][17].…”

Section: Introductionmentioning

confidence: 99%

A hybrid analytical–numerical algorithm for determining the neuronal current via electroencephalography

Hashemzadeh

Fokas

Schönlieb

2020

J. R. Soc. Interface.

Self Cite

View full text Add to dashboard Cite

In this study, the neuronal current in the brain is represented using Helmholtz decomposition. It was shown in earlier work that data obtained via electroencephalography (EEG) are affected only by the irrotational component of the current. The irrotational component is denoted by Ψ and has support in the cerebrum. This inverse problem is severely ill-posed and requires that additional constraints are imposed. Here, we impose the requirement of the minimization of the L 2 norm of the current (energy). The function Ψ is expanded in terms of inverse multiquadric radial basis functions (RBF) on a uniform Cartesian grid inside the cerebrum. The minimal energy constraint in conjunction with the RBF parametrization of Ψ results in a Tikhonov regularized solution of Ψ. The RBF shape parameter (regularization parameter), is computed by solving a 1-D non-linear maximization problem. Reconstructions are presented using synthetic data with a signal to noise ratio (SNR) of 20 dB. The root mean square error (RMSE) between the exact and the reconstructed Ψ is RMSE=0.1122. The proposed reconstruction algorithm is computationally efficient and can be vectorized in MATLAB.

show abstract

On the Geometric Sensitivity of the EEG Inversion Algorithm

Pasiou

Dassios

2022

La Matematica

View full text Add to dashboard Cite

EEG for Current With Two-Dimensional Support

Abstract: The resulting ability to restrict the source space greatly reduces the degree of ambiguity in the inverse solutions, offering the potential for more stable inverse solutions, since the auxiliary functions that define the mapping can be computed efficiently using standard numerical methods.

Cited by 4 publications

References 21 publications

Electro-magnetoencephalography for a spherical multiple-shell model: novel integral operators with singular-value decompositions

Electro-magnetoencephalography for a spherical multiple-shell model: novel integral operators with singular-value decompositions

A hybrid analytical–numerical algorithm for determining the neuronal current via electroencephalography

On the Geometric Sensitivity of the EEG Inversion Algorithm

Contact Info

Product

Resources

About