Bioelectromagnetic forward problem: isolated source approach revis(it)ed.

Electro- and magnetoencephalography (EEG and MEG) are non-invasive modalities for studying the electrical activity of the brain by measuring voltages on the scalp and magnetic fields outside the head. In the forward problem of EEG and MEG, the relationship between the neural sources and resulting signals is characterized using electromagnetic field theory. This forward problem is commonly solved with the boundary-element method (BEM). The EEG forward problem is numerically challenging due to the low relative conductivity of the skull. In this work, we revise the isolated source approach (ISA) that enables the accurate, computationally efficient BEM solution of this problem. The ISA is formulated for generic basis and weight functions that enable the use of Galerkin weighting. The implementation of the ISA-formulated linear Galerkin BEM (LGISA) is first verified in spherical geometry. Then, the LGISA is compared with conventional Galerkin and symmetric BEM approaches in a realistic 3-shell EEG/MEG model. The results show that the LGISA is a state-of-the-art method for EEG/MEG forward modeling: the ISA formulation increases the accuracy and decreases the computational load. Contrary to some earlier studies, the results show that the ISA increases the accuracy also in the computation of magnetic fields.

Realistic conductivity geometry model of the human head for interpretation of neuromagnetic data.

In this paper, the computational and practical aspects of a realistically-shaped multilayer model for the conductivity geometry of the human head are discussed. A novel way to handle the numerical difficulties caused by the presence of the poorly conducting skull is presented. Using our method, both the potential on the surface of the head and the magnetic field outside the head can be computed accurately. The procedure was tested with the multilayer sphere model, for which analytical expressions are available. The method is then applied to a realistically-shaped head model, and it is numerically shown that for the computation of B, produced by cerebral current sources, it is sufficient to consider a brain-shaped homogeneous conductor only since the secondary currents on the outer interfaces give only a negligible contribution to the magnetic field outside the head. Comparisons with the sphere model are also included to pinpoint areas where the homogeneous conductor model provides essential improvements in the calculation of the magnetic field outside the head.

Spatial resolution of neuromagnetic records: theoretical calculations in a spherical model.

Spatial resolution of magnetoencephalography (MEG) was studied by computer simulations using a spherical conductor model for the head. The accuracy obtainable in the absolute location of a dipole was found by calculating the confidence limits for source location in 3 dimensions. The accuracy in determining the relative locations of two sources was estimated by calculating the smallest shift in source location that could be detected with statistical significance. The results were used to illustrate the dependence of spatial resolution on several factors including noise, source depth, source strength, flux transformer configuration and the choice of the measurement locations. Under optimal conditions, separations of a couple of millimeters in superficial non-simultaneous sources can be detected, whereas for deeper sources the resolution is worse.

Basic mathematical and electromagnetic concepts of the biomagnetic inverse problem.

In this paper basic mathematical and physical concepts of the biomagnetic inverse problem are reviewed with some new approaches. The forward problem is discussed for both homogeneous and inhomogeneous media. Geselowitz' formulae and a surface integral equation are presented to handle a piecewise homogeneous conductor. The special cases of a spherically symmetric conductor and a horizontally layered medium are discussed in detail. The non-uniqueness of the solution of the magnetic inverse problem is discussed and the difficulty caused by the contribution of the electric potential to the magnetic field outside the conductor is studied. As practical methods of solving the inverse problem, a weighted least-squares search with confidence limits and the method of minimum norm estimate are discussed.

Feasibility of the homogeneous head model in the interpretation of neuromagnetic fields.

During the past few years it has been demonstrated that active areas in the human brain can be located by measuring the magnetic fields arising from the electric currents in the neurons. An established model for the conductivity geometry of the head in these studies is the layerwise homogeneous sphere. If, however, the measurement grid is too large or the local radius of curvature of the head is changing rapidly in the measurement area, this simple model may become inadequate. In this paper we investigate the feasibility of replacing the conducting sphere by a homogeneous body having the shape of the brain. We show by a semi-quantitative argument that the homogeneous body model is justified. A numerical procedure for the calculation of the magnetic field is presented with examples of the accuracy that can be obtained. An example of significant differences between the predictions of the homogeneous and sphere models is discussed.

Mixed and sensory nerve stimulations activate different cytoarchitectonic areas in the human primary somatosensory cortex SI. Neuromagnetic recordings and statistical considerations.

Magnetic responses evoked by stimulation of the mixed median nerve at the wrist and its cutaneous branches on the glabrous skin of the index and middle fingers were studied. The first responses to mixed nerve stimulation peaked at 19-24 ms, and those to cutaneous nerve stimulation about 4 ms later. The responses, up to a latency of 150 ms, reversed in polarity between the upper and lower parts of the rolandic fissure. Equivalent dipoles for the mixed nerve stimulation were stronger and they lay statistically significantly deeper from the scalp than those activated by the cutaneous nerve stimulation. It is suggested that mixed nerve stimulation activates areas 3a and 3b whereas cutaneous stimulation activates mainly area 3b at the human primary somatosensory cortex. Statistical procedures were developed for comparison of different field patterns and for determining confidence limits of source model parameters. For these purposes the quality and quantity of the noise were studied. The error caused by inaccuracies in the positioning of the magnetometer was found to be minimal in comparison with the signal noise which was estimated from the standard deviation of the averaged response.