There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Gyri-precise head model of transcranial direct current stimulation: Improved spatial focality using a ring electrode versus conventional rectangular pad

The equation for the magnetic lead field for a given magnetoencephalography (MEG) channel is well known for arbitrary frequencies omega but is not directly applicable to MEG in the quasi-static approximation. In this paper we derive an equation for omega = 0 starting from the very definition of the lead field instead of using Helmholtz's reciprocity theorems. The results are (a) the transpose of the conductivity times the lead field is divergence-free, and (b) the lead field differs from the one in any other volume conductor by a gradient of a scalar function. Consequently, for a piecewise homogeneous and isotropic volume conductor, the lead field is always tangential at the outermost surface. Based on this theoretical result, we formulated a simple and fast method for the MEG forward calculation for one shell of arbitrary shape: we correct the corresponding lead field for a spherical volume conductor by a superposition of basis functions, gradients of harmonic functions constructed here from spherical harmonics, with coefficients fitted to the boundary conditions. The algorithm was tested for a prolate spheroid of realistic shape for which the analytical solution is known. For high order in the expansion, we found the solutions to be essentially exact and for reasonable accuracies much fewer multiplications are needed than in typical implementations of the boundary element methods. The generalization to more shells is straightforward.

The magnetic lead field theorem in the quasi-static approximation and its use for magnetoencephalography forward calculation in realistic volume conductors.

A solution of the forward problem is an important component of any method for computing the spatio-temporal activity of the neural sources of magnetoencephalography (MEG) and electroencephalography (EEG) data. The forward problem involves computing the scalp potentials or external magnetic field at a finite set of sensor locations for a putative source configuration. We present a unified treatment of analytical and numerical solutions of the forward problem in a form suitable for use in inverse methods. This formulation is achieved through factorization of the lead field into the product of the moment of the elemental current dipole source with a "kernel matrix" that depends on the head geometry and source and sensor locations, and a "sensor matrix" that models sensor orientation and gradiometer effects in MEG and differential measurements in EEG. Using this formulation and a recently developed approximation formula for EEG, based on the "Berg parameters", we present novel reformulations of the basic EEG and MEG kernels that dispel the myth that EEG is inherently more complicated to calculate than MEG. We also present novel investigations of different boundary element methods (BEMs) and present evidence that improvements over currently published BEM methods can be realized using alternative error-weighting methods. Explicit expressions for the matrix kernels for MEG and EEG for spherical and realistic head geometries are included.

/pdf/eeg-and-meg-forward-solutions-for-inverse-methods-5f9oygs4a9.pdf

EEG and MEG: forward solutions for inverse methods

An Introduction to Magnetohydrodynamics

Modeling in magnetoencephalography (MEG) and electroencephalography (EEG) requires knowledge of the in vivo tissue resistivities of the head. The aim of this paper is to examine the influence of tissue resistivity changes on the neuromagnetic field and the electric scalp potential. A high-resolution finite element method (FEM) model (452162 elements, 2-mm resolution) of the human head with 13 different tissue types is employed for this purpose. Our main finding was that the magnetic fields are sensitive to changes in the tissue resistivity in the vicinity of the source. In comparison, the electric surface potentials are sensitive to changes in the tissue resistivity in the vicinity of the source and in the vicinity of the position of the electrodes. The magnitude (strength) of magnetic fields and electric surface potentials is strongly influenced by tissue resistivity changes, while the topography is not as strongly influenced. Therefore, an accurate modeling of magnetic field and electric potential strength requires accurate knowledge of tissue resistivities, while for source localization procedures this knowledge might not be a necessity.

Influence of tissue resistivities on neuromagnetic fields and electric potentials studied with a finite element model of the head

The problem of determining defects in structures using eddy current methods was investigated. The goal of this work is to demonstrate that the forces generated by the eddy currents and acting back on the magnet system can be used to detect defects in the object. Numerical simulations and experimental investigations have been performed. This novel technique has been found to be sensitive enough to detect even deep defects in an Aluminium bar moving relative to the field-generating magnet system.

/pdf/eddy-current-testing-of-metallic-sheets-with-defects-using-4px0d2i8d9.pdf

Eddy Current Testing of Metallic Sheets with Defects Using Force Measurements

We present the logical expressions (LE) approach that allows fast computation of three-dimensional eddy current problems, including parts in motion. The approach applies time-dependent logical expressions to describe moving parts of the model on a fixed computational grid. The study is motivated by a novel nondestructive testing technique called Lorentz force eddy current testing (LET), which enables the detection of defects lying deep inside a conducting material. Depending on the definition of the frame of reference, we present two different implementations of the LE approach referred to as 1) moving magnet approach, and 2) moving defect approach. In order to demonstrate the advantages of the LE approach, we compare its results with the sliding mesh technique. The validation of the obtained results with experiments is also presented.

Finite Element Analysis of Nondestructive Testing Eddy Current Problems With Moving Parts

We report an investigation of the motion of a free-falling permanent magnet in an electrically conducting pipe containing an idealized defect. This problem represents a highly simplified yet enlightening version of a method called Lorentz force eddy current testing which is a modification of the traditional eddy current testing technique. Our investigation is a combination of analytical theory, numerical simulation and experimental validation. The analytical theory allows a rigorous prediction about the relation between the size of the defect and the change in falling time which represents the central result of the present work. The numerical simulation allows to overcome limitations inherent in the analytical theory. We test our predictions by performing a series of experiments. We conclude that our theory properly captures the essence of Lorentz force eddy current testing although a refinement of the experiment is necessary to reduce the discrepancy to the predictions. In spite of its apparent simplicity the present system can serve as a prototype and benchmark for future research on Lorentz force eddy current testing.

Lorentz Force Eddy Current Testing: a Prototype Model

Modelling in magnetoencephalography (MEG) and electroencephalography (EEG) is increasingly based on the boundary element method (BEM). We quantify the influence of boundary element discretization on the neuromagnetic and neuroelectric forward and inverse problem for different dipole depths, brain regions and the quasispherical correction. In particular we derive standards for the general use of BEM models in MEG/EEG source localization. For this purpose simulation with single current dipoles, and source reconstructions from somatosensory evoked potentials and magnetic fields were employed. It was found that both local and global discretization influence source reconstruction. Only at a minimum triangle side length of 10 mm was it possible to achieve stable results for MEG and EEG. In order to obtain acceptable errors within the stable region, the ratio of dipole depth to triangle side length must not be less than 0.5. The results obtained from a comparison of the different brain regions indicate that the similarity to spherical geometry might well have an influence on the estimated dipole location, but not so much on its strength. Source reconstruction employing quasispherical correction was found to be the most stable, in particular in the case of coarse BEM discretization.

Hartmut Brauer

Papers

Influence of tissue resistivities on neuromagnetic fields and electric potentials studied with a finite element model of the head

Eddy Current Testing of Metallic Sheets with Defects Using Force Measurements

Finite Element Analysis of Nondestructive Testing Eddy Current Problems With Moving Parts

Lorentz Force Eddy Current Testing: a Prototype Model

Der Einfluss der Randelementediskretisierung auf die Vorwartsrechnung und das inverse Problem in Elektroencephalographie und Magnetoencephalographie