Modeling spatiotemporal covariance for magnetoencephalography or electroencephalography source analysis.

We propose a new model to approximate spatiotemporal noise covariance for use in neural electromagnetic source analysis, which better captures temporal variability in background activity. As with other existing formalisms, our model employs a Kronecker product of matrices representing temporal and spatial covariance. In our model, spatial components are allowed to have differing temporal covariances. Variability is represented as a series of Kronecker products of spatial component covariances and corresponding temporal covariances. Unlike previous attempts to model covariance through a sum of Kronecker products, our model is designed to have a computationally manageable inverse. Despite increased descriptive power, inversion of the model is fast, making it useful in source analysis. We have explored two versions of the model. One is estimated based on the assumption that spatial components of background noise have uncorrelated time courses. Another version, which gives closer approximation, is based on the assumption that time courses are statistically independent. The accuracy of the structural approximation is compared to an existing model, based on a single Kronecker product, using both Frobenius norm of the difference between spatiotemporal sample covariance and a model, and scatter plots. Performance of ours and previous models is compared in source analysis of a large number of single dipole problems with simulated time courses and with background from authentic magnetoencephalography data.

A generalized spatiotemporal covariance model for stationary background in analysis of MEG data.

Using a noise covariance model based on a single Kronecker product of spatial and temporal covariance in the spatiotemporal analysis of MEG data was demonstrated to provide improvement in the results over that of the commonly used diagonal noise covariance model. In this paper we present a model that is a generalization of all of the above models. It describes models based on a single Kronecker product of spatial and temporal covariance as well as more complicated multi-pair models together with any intermediate form expressed as a sum of Kronecker products of spatial component matrices of reduced rank and their corresponding temporal covariance matrices. The model provides a framework for controlling the tradeoff between the described complexity of the background and computational demand for the analysis using this model. Ways to estimate the value of the parameter controlling this tradeoff are also discussed.

MUSIC seeded multi-dipole MEG modeling using the Constrained Start Spatio-Temporal modeling procedure.

The Constrained Start Spatio-Temporal modeling program (CSST) is an objective multi-dipole, multi-start MEG/EEG analysis procedure that randomly selects from 100 to 100,000 initial dipole configurations, and runs a nonlinear simplex search on each of these configurations employing a reduced Chi-square statistic as the minimization criterion, to obtain a set of dipole configurations that best fit the data [Ranken, 2002]. A parallel version of CSST is implemented in IDL and MPI, making CSST usable on a single computer, or on a Linux cluster. We have now developed a multi-resolution version of MUSIC [Mosher, 1992] [Mosher, 1998] that provides an 80% or more reduction in the number of forward calculations needed to obtain results comparable to a 160,000 point MUSIC scan, on a 2 mm grid that defines a brain volume. The multi-resolution MUSIC scan provides an improved set of initial dipole estimates for the CSST analysis. In preliminary tests on real and simulated MEG data, with model orders ranging between 5 and 7 dipoles, the best performance improvements were obtained by mixing in 1 to 3 dipole locations randomly drawn from the best MUSIC locations, with randomly selected locations from the brain volume to complete the selected model order. We have also developed an improved method for sampling the brain volume for initial configurations. These improvements have led to a 75% reduction in the number of starting configurations required to obtain 5-10 best solutions with equal or lower reduced Chi-square values, when compared to the best solutions from the previous version of CSST.

A model for frequency dependence of conductivities of the live human skull.

A mathematical model (sigma(omega) approximately equal to A omega alpha, where, sigma is identical with conductivity, omega = 2 pi f is identical with applied frequency (Hz), A (amplitude) and alpha (unit less) is identical with search parameters) was used to fit the frequency dependence of electrical conductivities of compact, spongiosum, and bulk layers of the live and, subsequently, dead human skull samples. The results indicate that the fit of this model to the experimental data is excellent. The ranges of values of A and alpha were, spongiform (12.0-36.5, 0.0083-0.0549), the top compact (5.02-7.76, -0.137-0.0144), the lower compact (2.31-10.6, 0.0267-0.0452), and the bulk (7.46-10.6, 0.0133-0.0239). The respective values A and alpha for the respective layers of the dead skull samples were (40.1-89.7, -0.0017-0.0287), (5.53-14.5, -0.0296 - -0.0061), (4.58-15.9, -0.0226-0.0268), and (12.7-25.3, -0.0158-0.0132).

Conductivities of three-layer live human skull.

Electrical conductivities of compact, spongiosum, and bulk layers of the live human skull were determined at varying frequencies and electric fields at room temperature using the four-electrode method. Current, at higher densities that occur in human cranium, was applied and withdrawn over the top and bottom surfaces of each sample and potential drop across different layers was measured. We used a model that considers variations in skull thicknesses to determine the conductivity of the tri-layer skull and its individual anatomical structures. The results indicate that the conductivities of the spongiform (16.2-41.1 milliS/m), the top compact (5.4-7.2 milliS/m) and lower compact (2.8-10.2 milliS/m) layers of the skull have significantly different and inhomogeneous conductivities. The conductivities of the skull layers are frequency dependent in the 10-90 Hz region and are non-ohmic in the 0.45-2.07 A/m2 region. These current densities are much higher than those occurring in human brain.

Conductivities of three-layer human skull.

In this study, electrical conductivities of compact, spongiosum, and bulk layers of cadaver skull were determined at varying electric fields at room temperature. Current was applied and withdrawn over the top and bottom surfaces of each sample and potential drop across different layers was measured using the four-electrode method. We developed a model, which considers of variations in skull thicknesses, to determine the conductivity of the tri-layer skull and its individual anatomical structures. The results indicate that the spongiform and the two compact layers of the skull have significantly different and inhomogeneous conductivities ranging from 0.76 +/- .14 to 11.5 +/- 1.8 milliS/m.

Mapping function in the human brain with magnetoencephalography, anatomical magnetic resonance imaging, and functional magnetic resonance imaging.

Integrated analyses of human anatomical and functional measurements offer a powerful paradigm for human brain mapping. Magnetoencephalography (MEG) and EEG provide excellent temporal resolution of neural population dynamics as well as capabilities for source localization. Anatomical magnetic resonance imaging (MRI) provides excellent spatial resolution of head and brain anatomy, whereas functional MRI (fMRI) techniques provide an alternative measure of neural activation based on associated hemodynamic changes. These methodologies constrain and complement each other and can thereby improve our interpretation of functional neural organization. We have developed a number of computational tools and techniques for the visualization, comparison, and integrated analysis of multiple neuroimaging techniques. Construction of geometric anatomical models from volumetric MRI data allows improved models of the head volume conductor and can provide powerful constraints for neural electromagnetic source modeling. These approaches, coupled to enhanced algorithmic strategies for the inverse problem, can significantly enhance the accuracy of source-localization procedures. We have begun to apply these techniques for studies of the functional organization of the human visual system. Such studies have demonstrated multiple, functionally distinct visual areas that can be resolved on the basis of their locations, temporal dynamics, and differential sensitivity to stimulus parameters. Our studies have also produced evidence of internal retinotopic organization in both striate and extrastriate visual areas but have disclosed organizational departures from classical models. Comparative studies of MEG and fMRI suggest a reasonable but imperfect correlation between electrophysiological and hemodynamic responses. We have demonstrated a method for the integrated analysis of fMRI and MEG, and we outline strategies for improvement of these methods. By combining multiple measurement techniques, we can exploit the complementary strengths and transcend the limitations of the individual neuro-imaging methods.

Evaluation of boundary element methods for the EEG forward problem: effect of linear interpolation.

We implement the approach for solving the boundary integral equation for the electroencephalography (EEG) forward problem proposed by de Munck [1], in which the electric potential varies linearly across each plane triangle of the mesh. Previous solutions have assumed the potential is constant across an element. We calculate the electric potential and systematically investigate the effect of different mesh choices and dipole locations by using a three concentric sphere head model for which there is an analytic solution. Implementing the linear interpolation approximation results in errors that are approximately half those of the same mesh when the potential is assumed to be constant, and provides a reliable method for solving the problem.