Investigations of dipole localization accuracy in MEG using the bootstrap
Introduction
The Equivalent Current Dipole (ECD) is a widely used model for event-related neuronal activity. The location, orientation, and time series of ECDs can be estimated from noninvasive surface measurements of the associated magnetic fields and electric potentials generated by the human brain. The ECD model yields a fixed number of source locations and source time series from each data set. These dipole source positions can be estimated to an arbitrary precision by means of nonlinear optimization (Mosher et al., 1992, Scherg, 1990). There are many factors that can affect the accuracy of the estimated locations: brain and environmental noise, subject motion during the experiment, errors in the head and sensor models used in the inverse procedure, and trial to trial variations in the brain's response to the specific stimulus or task. Furthermore, since the localization methods are nonlinear, the accuracy of localization is also dependent on the number of sources localized.
The arbitrary precision of localization results can be misleading, especially in studies where differences in source localizations due to different experimental conditions are analyzed. Some studies report dipole localization differences on the order of 2 mm (Baumgartner et al., 1998, Biermann et al., 1998, Buchner et al., 1995, Buchner et al., 1999, Yamazaki et al., 2000, Virtanen et al., 1998). The significance of these differences depends on all sources of error in localization and statistical tests that take these factors into account should be used to assess significance. Independent repetitions in a single subject can be used for this purpose (Buchner et al., 1995, Waberski et al., 2003). However, the large number of repetitions required to achieve statistical significance is impractical for most studies; for example, the study by Waberski et al. (2003) uses 4500 repetitions per digit in a somatosensory study. Furthermore, these studies are used to test for significant differences between digits but do not give confidence intervals for individual sources. Another approach to establishing statistical significance of differences in dipole locations is to use multiple subjects in a common stereotactic coordinate system (Schoonhoven et al., 2003, Thoma et al., 2003). Dipole reconstructions are performed over a number of subjects and average positions and standard deviations for these positions are computed in a common coordinate system. Again, while this approach can detect significant separations between sources among subjects, it does not provide any direct measure of uncertainty in individuals. In contrast, our goal here is to assess the uncertainty in the localization of multiple dipoles within a single subject from a set of repeated trials.
The accuracy of dipole localization using EEG and magnetoencephalography (MEG) has been widely studied in theoretical, simulation, and phantom studies. Lower bounds on parameter variances using the Cramer–Rao inequality were described for this problem in Mosher et al. (1993), Muravchik and Nehorai (2001), and Radich and Buckley (1995). While the bounds were shown to be reasonably tight in simulation studies, they assume stationary Gaussian noise and a deterministic time series for each source. In practice, these assumptions may not hold. Furthermore, this model is based on the assumption of a fixed number of sources. A more detailed review of other theoretical analyses of localization can be found in Mosher et al. (1993). Phantom studies of dipoles implanted in a human skull filled with conducting gelatin (Baillet et al., 2001b, Greenblatt and Robinson, 1994, Leahy et al., 1998) or a cadaver head (Barth et al., 1986) have also been used to determine the expected accuracy of EEG and MEG source localizations. These phantom studies have the advantage over a theoretical analysis that they include a more realistic forward mapping through the head and the effects of environmental noise and realistic sensors. For our 32-dipole skull phantom study we found average localization errors on the order of 7.6 mm for EEG and 3.4 mm for MEG. However, these can probably be viewed as a best case analysis since they do not include the effects of additional background brain activity, the effects of anisotropic and inhomogeneous conductivity in the brain, or the effects of trial to trial variation in the subjects response or head motion.
We now turn our attention to methods to assess localization accuracy directly from experimental data. Braun et al. (1997) review and compare a number of approaches to estimation of confidence intervals, including linear perturbation analysis, Monte Carlo simulation, and an iterative confidence interval estimation (ICE) scheme. Simulation studies revealed that the perturbation analysis and ICE did not perform well with correlated noise and that Monte Carlo simulation was the preferred approach. The Monte Carlo approach samples from a Gaussian distribution with covariance computed from an estimate of the experimental noise correlation matrix, which can be computed from prestimulus data. These noise samples are added to the forward field calculated for the estimated source location and used to estimate a new source location. Repetition of this procedure results in a cluster of source locations from which a confidence volume can be computed as an ellipsoid based on an eigendecomposition of the spatial covariance of the cluster. This method is essentially a parametric bootstrap, which assumes that the source is invariant from trial to trial and that the only source of variability is additive noise at the sensors. Thus, the effects of trial to trial variations, subject motion, and errors in the forward model on source localization accuracy are not considered. Here we attempt to also include these factors by working directly with individual epochs and using a nonparametric bootstrap approach. The above Monte Carlo method can also make use of the residual error (Hamalainen et al., 1993) rather than the prestimulus data to compute a confidence interval. This can potentially lead to underestimation of the true confidence regions since the residual variance decreases, and hence the confidence intervals shrink, as the number of dipolar sources is increased. The true number of sources is unknown and typically estimated based on a priori assumptions about the experiment (Asada et al., 1999, Forss and Jousmaeki, 1998); overestimation will lead to unrealistically small confidence volumes.
To achieve acceptable estimates, dipole reconstruction is typically performed on data sets averaged over tens or hundreds of repetitions of the same experiment. Each averaged data set produces a point estimate of the ECD locations and time series. The bootstrap method (Efron, 1979, Efron and Tibshirani, 1986) provides a nonparametric method for assessing the reliability of the estimated sources. Bootstrap resamples can be generated by sampling with replacement from the set of repeated trials. The advantage of the nonparametric bootstrap approach is that no specific assumptions are made regarding the distribution of the noise, the dipole time series, or the number of dipoles. A similar approach to analysis of event-related data is described in DiNocera and Ferlazzo (2000), but the analysis was applied only to the scalp data rather than brain sources estimated from these data.
We applied the bootstrap method to MEG data from a somatotopic experiment in order to estimate the accuracy and reliability of estimated ECDs. Each of the four measurements, which consisted of an electric stimulation of the thumb, index, middle, and small finger, was resampled 5000 times from about 400 trials per finger. We reconstructed sources from these resamples using the Recursively Applied and Projected (RAP)-MUSIC algorithm (Mosher and Leahy, 1999). The size of the subspace (rank 19) for this reconstruction was considerably larger than the number of possible sources, thus allowing us to capture all significant sources that might be involved in generating the magnetic field pattern. We also performed simulations with two dipolar sources with overlapping time series. We simulated 400 trials for these sources and added noise from the prestimulus interval in the somatosensory data. The strength of the simulated sources was adjusted to match the signal-to-noise ratio (SNR) of the real data in order to be able to compare the bootstrapped simulation with results from real data.
Since more than one dipolar source was reconstructed per bootstrap resample of the data and there is no inherent order in the dipoles themselves, the resulting dipoles were clustered using their normalized time series and topography. Using this clustering scheme, spurious sources can be identified easily, as their time series, topography, and location will vary considerably over the number of resamples, which will lead to clusters with few members and highly variable time series, topographies, and location. Conversely, statistically significant sources can be expected to populate spatial tight clusters with well-defined time series and topographies.
Section snippets
MEG data acquisition
Somatosensory measurements were performed on one healthy right-handed male. The somatosensory stimulation was an electrical square-wave pulse delivered separately to four fingers of each hand: thumb, index, middle, and little finger. The stimulation was applied between the middle and distal phalanxes of each finger. The stimulation order was randomized. The pulse duration was 0.2 ms and the amplitude was set to twice the perceptual threshold. The interstimulus interval (ISI) was varied randomly
Reconstruction from the average data
The mapping of the sources in contralateral S1 primary somatosensory cortex from the average data for right and left hand stimulation is shown in Fig. 3. Their associated time series are shown in Fig. 4. Since we found additional sources that do not belong to the S1 activation, we plot here only those sources that had the strongest signal power in the interval from 30 to 50 ms. The location of these sources follows the homuncular cortical representation of the fingers as described in Nakamura
Discussion
The purpose of this study was to investigate the utility of the nonparametric bootstrap in determining the uncertainty in source locations estimated from MEG averages taken over multiple repetitions of the stimulus. In particular, we were interested in assessing the contribution to uncertainty that arises from trial to trial variability in brain response as well as the effects of background brain activity, environmental and sensor noise, subject motion, and modeling errors.
The simulation
Conclusion
We have described the use of the nonparametric bootstrap for estimating uncertainty in dipole locations and time series estimated from multiple trial MEG data. A comparison of experimental with simulated data indicated larger uncertainty than would be predicted based on additive noise considerations alone. Consequently, it is important to consider the effects of trial to trial variability in the data when determining confidence regions for dipole locations. The results for a somatotopic study
Acknowledgments
This work was supported by NIBIB under Grant R01 EB002010 and by Los Alamos National Laboratory, operated by the University of California for the United States Department of Energy, under Contract W-7405-ENG-36. The authors thank Tom Nichols of the University of Michigan and Stephane Bahrami of the CNRS, Paris, for helpful discussions on issues related to use of the bootstrap. They also thank Sabine Meunier of the Physiology and Physiopathology of Human Motricity Lab, La Salpetriere Hospital,
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