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

Leweke, Sarah; Michel, Volker; Fokas, A. S.

doi:10.1088/1361-6420/ab291f

Cited by 5 publications

(18 citation statements)

References 37 publications

(91 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, it has the disadvantage that, since the current is obtained from Ψ via the operation of the gradient, additional numerical errors in the reconstruction of the neuronal current are introduced. In the case of the spherical model, it is shown in [55] that it is possible to have an effective inversion by working directly with the current instead of its Helmholtz decomposition; the question of whether this approach can be extended to an arbitrary geometry remains open.…”

Section: Discussionmentioning

confidence: 99%

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

Hashemzadeh

Fokas

Schönlieb

2020

J. R. Soc. Interface.

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

Section: Discussionmentioning

confidence: 99%

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

Hashemzadeh

Fokas

Schönlieb

2020

J. R. Soc. Interface.

View full text Add to dashboard Cite

show abstract

“…In order to formulate the relation between the neuronal current and the measured quantities, we follow the path of optimize-then-discretize approaches in [15,21,22,37,38,44]. Via this ansatz, the problem is solved analytically as far as possible.…”

Section: Introductionmentioning

confidence: 99%

“…Within this framework, the problem of directly reconstructing the vector-valued neuronal current from magnetic flux density ν • B measurements by the magnetoencephalograph (MEG) (where ν denotes the normal vector of the MEG sensor surface) [13], and electric potential differences values on the scalp obtained via the electroencephalograph (EEG) [14], was addressed in [37,38]. Therein, singular value decompositions (SVD) of the corresponding operators are derived based on a novel set of vector-valued orthonormal basis functions.…”

Section: Introductionmentioning

confidence: 99%

Vector-valued spline method for the spherical multiple-shell electro-magnetoencephalography problem

2022

Self Cite

View full text Add to dashboard Cite

Human brain activity is based on electrochemical processes, which can only be measured invasively. Thus, quantities such as magnetic flux density (MEG) or electric potential differences (EEG) are measured non-invasively in medicine and research. The reconstruction of the neuronal current from the measurements is a severely ill-posed problem though the visualization of the cerebral activity is one of the main research tools in cognitive neuroscience. Here, using an isotropic multiple-shell model for the human head and a quasi-static approach for the electro-magnetic processes, we derive a novel vector-valued spline method based on reproducing kernel Hilbert spaces in order to reconstruct the current from the measurements. The presented method follows the path of former spline approaches and provides classical minimum norm properties. Besides, it minimizes the (infinite-dimensional) Tikhonov-Philips functional which handles the instability of the inverse problem. This optimization problem reduces to solving a finite-dimensional system of linear equations without loss of information, due to its construction. It results in a unique solution which takes into account that only the harmonic and solenoidal component of the neuronal current affects the measurements. Furthermore, we prove a convergence result: the solution achieved by the novel method converges to the generator of the data as the number of measurements increases. The vector splines are applied to the inversion of three synthetic test cases, where the irregularly distributed data situation could be handled very well. Combined with five parameter choice methods, numerical results are shown for synthetic test cases with and without additional Gaussian white noise. Former approaches based on scalar splines are outperformed by the novel vector splines results with respect to the normalized root mean square error. Finally, results for real data acquired during a visual stimulation task are demonstrated. They can be computed quickly and are reasonable with respect to physiological expectations.

show abstract

“…In the practical tests for diverse applications, very good approximations could be achieved not only for the downward continuation but also for other ill-posed inverse problems, for instance the (linear as well as the non-linear) inverse gravimetric problem or the inversion of MEG-and EEG-data, see, for instance, [6,7,19,21,23,24].…”

Section: Introductionmentioning

confidence: 99%

A first approach to learning a best basis for gravitational field modelling

Michel

Schneider

2020

Int J Geomath

Self Cite

View full text Add to dashboard Cite

Gravitational field modelling is an important tool for inferring past and present dynamic processes of the Earth. Functions on the sphere such as the gravitational potential are usually expanded in terms of either spherical harmonics or radial basis functions (RBFs). The (Regularized) Functional Matching Pursuit ((R)FMP) and its variants use an overcomplete dictionary of diverse trial functions to build a best basis as a sparse subset of the dictionary and compute a model, for instance, of the gravity field, in this best basis. Thus, one advantage is that the dictionary may contain spherical harmonics and RBFs. Moreover, these methods represent a possibility to obtain an approximative and stable solution of an ill-posed inverse problem, such as the downward continuation of gravitational data from the satellite orbit to the Earth's surface, but also other inverse problems in geomathematics and medical imaging. A remaining drawback is that in practice, the dictionary has to be finite and, so far, could only be chosen by rule of thumb or trial-and-error. In this paper, we develop a strategy for automatically choosing a dictionary by a novel learning approach. We utilize a nonlinear constrained optimization problem to determine best-fitting RBFs (Abel-Poisson kernels). For this, we use the Ipopt software package with an HSL subroutine. Details of the algorithm are explained and first numerical results are shown.

show abstract

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

Cited by 5 publications

References 37 publications

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

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

Vector-valued spline method for the spherical multiple-shell electro-magnetoencephalography problem

A first approach to learning a best basis for gravitational field modelling

Contact Info

Product

Resources

About