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We study the performance of various beamformers for estimating a current dipole source at a known location using electroencephalography (EEG) and magnetoencephalography (MEG). We present our beamformers in the form of the generalized sidelobe canceler (GSC). Under this structure, the beamformer can be solved by finding a filter that achieves the minimum mean-squared error (MMSE) between the mainbeam response and filtered observed signal. We express the MMSE as a function of the filter's rank and use it as a criterion to evaluate the performance of the beamformers. We do not make any assumptions on the rank of the interference-plus-noise covariance matrix. Instead, we treat it as low-rank and derive a general expression for the MMSE. We present numerical examples to compare the MSE performance of beamformers commonly studied in the literature: principal components (PCs), cross-spectral metrics (CSMs), and eigencanceler (EIG) beamformers. Our results show that good estimates of the dipole source signals can be achieved using reduced-rank beamformers even for low signal-to-noise ratio (SNR) values. Index Terms--Beamforming, dipole source signal, electroencephalography, low-rank covariance matrix, magnetoencephalography, sensor array processing.

The effects of artifact repression by signal-space projection on dipole localization accuracy for magnetoencephalography measurements were investigated. Approximate analytical formulas, equivalent to the Cramer-Rao bound, were used and checked by Monte Carlo simulations which correlate the increase of localization error for individual coordinates to the similarity of the artifact field and respective quadrupole fields acquired by differentiating the dipole field with respect to its origin.

The electromagnetic field generated by a current dipole situated at an arbitrary position inside a conducting sphere is derived using the expansions of the spherical vector wave functions. The first few terms in a series expansion of this field with respect to the frequency are given for the normal magnetic field (used in magnetoencephalogram) and the tangential electric field (used in electroencephalogram) outside the conducting sphere at low frequency. It is shown that the first correction term to the static solution is linear in the frequency [Omega] (the second correction term is proportional to [[Omega].sup.3/2]) and, thus, the static solution can be used as a good approximation for the solution at a very low frequency. Index Terms - Conducting sphere, current dipole, electroencephalogram, expansion, frequency series, low-frequency, magnetoencephalogram, spherical vector wave functions.

Focal biomagnetic sources are described as pointlike current dipoles. The dipole parameters, position, and moment coordinates are commonly determined from biomagnetic data using iterative nonlinear optimization algorithms such as the Levenberg-Marquardt algorithm. However, even for single-dipole sources, mislocalizations can occur due to side minima of the cost function or due to a wrong choice of the start vector. This can be shown by introducing a cost function where the independent variables are only the position coordinates instead of position and moment coordinates. This dimensional reduction - which is also possible for multiple dipole sources - is achieved by calculating the cost function at each position with the position and data-dependent, optimum dipole moments. We call these dipoles with - in a least squares sense - optimum moments, locally optimal dipoles. The visualization of such a single-dipole cost function and of the iteration steps of the Levenberg-Marquardt algorithm show why mislocalizations cannot be avoided. Therefore, we propose an alternative noniterative localization algorithm for single-dipole sources without this drawback. It uses localization probabilities calculated by means of the locally optimal dipoles. Besides the determination of the dipole parameters, the proposed algorithm furnishes a reliable error for each localization. Its effectiveness is shown with simulated and real patient data.

We describe a statistical frequency domain approach to localizing equivalent dipole generators of human brain evoked potentials. The frequency domain representation allows considerable data reduction, constrains the magnitude function of the dipoles to be smooth, and accounts for the statistical properties of the background EEG. A previous paper described a restrictive model in which the dipole orientations were assumed to be fixed over time, and only one dipole was allowed. In this paper, we consider the more general model in which the orientation can vary over time, and which includes multiple dipole generators. The varying orientation model has the practical advantage of being more nearly linear and more flexible than the fixed orientation model, which facilitates convergence of the iterative fitting algorithm. We suggest a measure of goodness-of-fit that compares the likelihood of the dipole model with the likelihoods of saturated and null models. We report the results of fitting the model to recorded auditory and visual evoked potentials. A single dipole with fixed orientation seems to be an adequate model of the auditory midlatency response, while two dipoles with varying orientation are needed to fit the later P200 component. Analysis of the visual P100 response to unilateral stimulation localized a generator in the contralateral occipital cortex, as expected from anatomical considerations. A two-dipole model fit the visual P100 response of bilateral stimulations, and the locations of the two dipoles were similar to the locations obtained by single-dipole fits to the responses to left and right unilateral stimuli.

We discuss the use of magnetocardiography to detect reentry currents in cardiac flutter and fibrillation. The magnetic field produced by induced atrial flutter was measured in isolated rabbit hearts. A moving dipole model is proposed to treat the experimental data and to locate the reentry path.

The statistical properties of the EEG and the MEG are described mathematically as the result of randomly distributed dipoles. These dipoles represent the interactions of cortical neurons. For certain dipole distributions, the first- and second-order moments of the electric and magnetic fields are derived analytically. If the dipoles are in a spherical volume conductor and have no preference for any direction, the variance of a differentially measured EEG-signal is only a function of the electrode distance. In this paper, the theoretically derived variance function will be compared with EEG- and MEG-measurements. It is shown that a dipole with a fixed position and a randomly fluctuating amplitude is an adequate model for the [alpha]-rhythm. An expression for the covariance between the magnetic field and a differentially measured EEG-signal is derived. This covariance is considered as a function of the magnetometer position, and is compared with the measurements of Chapman et al. [23]. The theory can be used to obtain a (spatial) covariance matrix of the background noise, which occurs in evoked potential measurements. Such a covariance matrix can be used to obtain a maximum likelihood estimator of the dipole parameters in evoked potential studies, to evaluate the merits of the so-called 'Laplacian derivation,' and for the interpolation of electromagnetic data.

--A method is proposed to reduce the influence of external noise magnetic field on the accuracy of estimating current dipole parameters. It utilizes the spatial correlation of external noises, and is applied to data measured using a multichannel magnetometer. Computer simulation clearly demonstrates the effectiveness of the proposed method.

The image solution for the static magnetic field outside a conducting sphere with an internal current dipole is considered. The image current, which is a linear distribution of magnetic dipoles on the line segment between the dipole point and the center of the sphere, is derived by using the fact that the induced current does not have any contribution to the radial component of the magnetic field outside the sphere. The image is then used to obtain some explicit formulas for identifying the location and tangential moment of the primary current dipole. This explicit identification method is also tested with a real model for a patient's brain. Index Terms--Conducting sphere, current dipole, explicit identification, image, localization, magnetoencephalogram, magnetostatic.