C. C. Wood

Bayesian inference applied to the electromagnetic inverse problem.

We present a new approach to the electromagnetic inverse problem that explicitly addresses the ambiguity associated with its ill‐posed character. Rather than calculating a single “best” solution according to some criterion, our approach produces a large number of likely solutions that both fit the data and any prior information that is used. Whereas the range of the different likely results is representative of the ambiguity in the inverse problem even with prior information present, features that are common across a large number of the different solutions can be identified and are associated with a high degree of probability. This approach is implemented and quantified within the formalism of Bayesian inference, which combines prior information with that of measurement in a common framework using a single measure. To demonstrate this approach, a general neural activation model is constructed that includes a variable number of extended regions of activation and can incorporate a great deal of prior information on neural current such as information on location, orientation, strength, and spatial smoothness. Taken together, this activation model and the Bayesian inferential approach yield estimates of the probability distributions for the number, location, and extent of active regions. Both simulated MEG data and data from a visual evoked response experiment are used to demonstrate the capabilities of this approach. Hum. Brain Mapping 7:195–212, 1999 Published 1999 Wiley‐Liss, Inc. This article is a US government work and, as such, is in the public domain in the United States of America.

Spatiotemporal Bayesian inference dipole analysis for MEG neuroimaging data

Recently, we described a Bayesian inference approach to the MEG/EEG inverse problem that used numerical techniques to estimate the full posterior probability distributions of likely solutions upon which all inferences were based [Schmidt, D.M., George, J.S., Wood, C.C., 1999. Bayesian inference applied to the electromagnetic inverse problem. Human Brain Mapping 7, 195; Schmidt, D.M., George, J.S., Ranken, D.M., Wood, C.C., 2001. Spatial-temporal bayesian inference for MEG/EEG. In: Nenonen, J., Ilmoniemi, R. J., Katila, T. (Eds.), Biomag 2000: 12th International Conference on Biomagnetism. Espoo, Norway, p. 671]. Schmidt et al. (1999) focused on the analysis of data at a single point in time employing an extended region source model. They subsequently extended their work to a spatiotemporal Bayesian inference analysis of the full spatiotemporal MEG/EEG data set. Here, we formulate spatiotemporal Bayesian inference analysis using a multi-dipole model of neural activity. This approach is faster than the extended region model, does not require use of the subjects anatomical information, does not require prior determination of the number of dipoles, and yields quantitative probabilistic inferences. In addition, we have incorporated the ability to handle much more complex and realistic estimates of the background noise, which may be represented as a sum of Kronecker products of temporal and spatial noise covariance components. This reduces the effects of undermodeling noise. In order to reduce the rigidity of the multi-dipole formulation which commonly causes problems due to multiple local minima, we treat the given covariance of the background as uncertain and marginalize over it in the analysis. Markov Chain Monte Carlo (MCMC) was used to sample the many possible likely solutions. The spatiotemporal Bayesian dipole analysis is demonstrated using simulated and empirical whole-head MEG data.

Single vs. paired visual stimulation: superposition of early neuromagnetic responses and retinotopy in extrastriate cortex in humans

Neuromagnetic techniques were used in conjunction with magnetic resonance imaging (MRI) techniques to: (1) localize and characterize cortical sources evoked by visual stimuli presented at different locations in the lower right visual field; (2) examine the superposition of cortical responses by comparing the summation of responses to the presentation of single stimuli with responses to paired stimuli; and (3) examine the spatial resolution of magnetoencephalographic (MEG) techniques by comparing the identified source locations evoked by the presentation of single vs. paired stimuli. Using multi-dipole, non-linear minimization analyses, three sources were localized for each stimulus condition during the initial 80-170 ms poststimulus interval for all subjects. In addition to an occipital source, two extrastriate sources were identified: occipital-parietal and occipital-temporal. Each source evidenced a systematic shift in location associated with changes in stimulus placement parallel to the vertical meridian. To our knowledge, this is the first demonstration of retinotopic organization of extrastriate areas, using non-invasive neuromagnetic techniques. The paired presentation of stimuli reflected superposition of the responses evoked by single stimuli but only for early activity up to 150 ms poststimulus. Undersummation was evident after 150 ms. All sources identified for single stimuli were also identified in the paired-stimulus responses; but at the expense of larger errors for some of the estimated parameters.

Anatomical constraints for neuromagnetic source models

The localization of neural electromagnetic sources from measurements at the head surface requires the solution of an inverse problem; that is, the determination of the number, location, spatial configuration, strength, and time-course of the neuronal currents that give rise to the magnetic field or potential distribution. In most general form, the neuromagnetic and electrical inverse problems are ill-posed and have no unique solution; however, approximate solutions are possible if assumptions are made regarding the shape and conductivity of the head and the number and configuration of neuronal currents responsible for the surface distributions. To help resolve ambiguities and to reduce the number and range of free parameters required to model complex neuromagnetic sources, the authors are investigating strategies to constrain the locations of allowable sources, based on a knowledge of individual anatomy. The key assumption, justified by both physiological evidence and theoretical considerations, is that the dominant neuromagnetic sources which contribute to surface field distributions reside within the cortex. It is demonstrated that anatomically constrained source modeling strategies can produce significant improvements in source localization; however, the conclusion is that additional improvements in model fitting or source reconstruction procedures are required.

Nonlinear analysis of biological systems using short M-sequences and sparse-stimulation techniques

The m-sequence pseudorandom signal has been shown to be a more effective probing signal than traditional Gaussian white noise for studying nonlinear biological systems using cross-correlation techniques. The effectiveness is evidenced by the high signal-to-noise (S/N) ratio and speed of data acquisition. However, the “anomalies” that occur in the estimations of the cross-correlations represent an obstacle that prevents m-sequences from being more widely used for studying nonlinear systems. The sparse-stimulation method for measuring system kernels can help alleviate estimation errors caused by anomalies. In this paper, a “padded sparse-stimulation” method is evaluated, a modification of the “inserted sparse-stimulation” technique introduced by Sutter, along with a short m-sequence as a probing signal. Computer simulations show that both the “padded” and “inserted” methods can effectively eliminate the anomalies in the calculation of the second-order kernel, even when short m-sequences were used (length of 1023 for a binary m-sequence, and 728 for a ternary m-sequence). Preliminary experimental data from neuromagnetic studies of the human visual system are also presented, demonstrating that the system kernels can be measured with high signal-to-noise (S/N) ratios using short m-sequences.

Bayesian Inference for Neural Electromagnetic Source Localization: Analysis of MEG Visual Evoked Activity

We have developed a Bayesian approach to the analysis of neural electromagnetic (MEG/EEG) data that can incorporate or fuse information from other imaging modalities and addresses the ill-posed inverse problem by sampling the many different solutions which could have produced the given data. From these samples one can draw probabilistic inferences about regions of activation. Our source model assumes a variable number of variable size cortical regions of stimulus-correlated activity. An active region consists of locations on the cortical surface, within a sphere centered on some location in cortex. The number and radii of active regions can vary to defined maximum values. The goal of the analysis is to determine the posterior probability distribution for the set of parameters that govern the number, location, and extent of active regions. Markov Chain Monte Carlo is used to generate a large sample of sets of parameters distributed according to the posterior distribution. This sample is representative of the many different source distributions that could account for given data, and allows identification of probable (i.e. consistent) features across solutions. Examples of the use of this analysis technique with both simulated and empirical MEG data are presented.

Dynamic Neuroimaging by MEG, Constrained by MRI and fMRI

MEG is a direct measure of the electrical activity of populations of neurons, with excellent temporal resolution. However, the spatial characterization of sources depends on the solution of an ill-posed inverse problem. MRI provides high resolution volumetric data defining the anatomy of the head and brain; alternative MRI data acquisition strategies allow mapping of hemodynamic correlates of neural function. However, functional MRI (fMRI) and related techniques are limited by the slow timecourse of the hemodynamic response and the ill-defined relationship between neural activation and associated hemodynamic changes. Given such complementary strengths and weaknesses, the integration of multiple imaging technologies should provide dynamic functional neuroimaging capabilities with optimal spatial and temporal resolution. We are exploring a range of strategies for the integrated analysis of data from MRI, fMRI and MEG.

Bayesian analysis of MEG visual evoked responses

We have developed a method for analyzing neural electromagnetic data that allows probabilistic inferences to be drawn about regions of activation. The method involves the generation of a large number of possible solutions which both fit the data and prior expectations about the nature of probable solutions made explicit by a Bayesian formalism. In addition, we have introduced a model for the current distributions that produce MEG (and EEG) data that allows extended regions of activity, and can easily incorporate prior information such as anatomical constraints from MRI. To evaluate the feasibility and utility of the Bayesian approach with actual data, we analyzed MEG data from a visual evoked response experiment. We compared Bayesian analyses of MEG responses to visual stimuli in the left and right visual fields, in order to examine the sensitivity of the method to detect known features of human visual cortex organization. We also examined the changing pattern of cortical activation as a function of time.

Modeling Magno- and Parvo-Like Contribunons from Contrast-Response Functions Using Neuromagnetic Measures

In our recent study [1], we have measured the spatial frequency (SF) tuning functions as well as the contrast-response (CR) functions at different contrast levels (from near threshold to suprathreshold) and different eccentricities of the visual field in humans using noninvasive magnetoencephalography (MEG) techniques. In the current study, the estimated CR functions and SF tuning curves were further fitted by a two-stream (M & P) model for all 4 subjects at 3 different eccentricities. The 5 parameters in equation (2) for each of the M- and P-like systems were also estimated.

Spatial frequency tuning functions and contrast sensitivity at different eccentricities in the visual field

The human luminance spatial frequency contrast sensitivity function (CSF) has been well studied using psychophysical measurements by detecting spatial frequency (SF) grating patterns at threshold. Threshold CSFs at different eccentricities have proven to be quite useful in both basic and clinical vision research. However, near threshold, the CSF is measured at a linear area of the saturating contrast-response curve. In contrast, most of our everyday vision may be at suprathreshold levels, and thus may function most of the time at the nonlinear area of the contrast-response curve. In this study, in order to better characterize the CSF at normal contrast levels, we measured the SF tuning functions as well as the CR functions at different suprathreshold contrast levels and different eccentricities of the visual field using noninvasive MEG techniques. In this study, in addition to peak analysis, we have developed more reliable averaged power analysis methods where the average power can be calculated from the entire waveforms.