Distributed source imaging
Source imaging approaches have developed in parallel to the other techniques discussed above. Imaging source models consist of distributions of elementary sources, generally with fixed locations and orientations, which amplitudes are estimated at once. MEG/EEG source images represent estimations of the global neural current intensity maps, distributed within the entire brain volume or constrained at the cortical surface.
Source image supports consist of either a 3D lattice of voxels or of the nodes of the triangulation of the cortical surface. These latter may be based on a template, or preferably obtained from the subject’s individual MRI and confined to a mask of the grey matter. Multiple academic software packages perform the necessary segmentation and tessellation processes from high-contrast T1-weighted MR image volumes.
The cortical surface, tessellated at two resolutions, using: (top row) 10,034 vertices (20,026 triangles with 10 mm2 average surface area) and (bottom row) 79,124 vertices (158,456 triangles with 1.3 mm2 average surface area).
As discussed elsewhere in these pages, the cortically-constrained image model derives from the assumption that MEG/EEG data originates essentially from large cortical assemblies of pyramidal cells, with currents generated from post-synaptic potentials flowing orthogonally to the local cortical surface. This orientation constraint can either be strict (Dale & Sereno, 1993) or relaxed by authorizing some controlled deviation from the surface normal (Lin, Belliveau, Dale, & Hamalainen, 2006).
In both cases, reasonable spatial sampling of the image space requires several thousands (typically ~10000) of elementary sources. Consequently, though the imaging inverse problem consists in estimating only linear parameters, it is dramatically underdetermined.
Just like in the context of source localization where e.g., the number of sources is a restrictive prior as a remedy to ill-posedness, imaging models need to be complemented by a priori information. This is properly formulated with the mathematics of regularization as we shall now briefly review.
Adding priors to the imaging model can be adequately formalized in the context of Bayesian inference where solutions to inverse modeling satisfy both the fit to observations – given some probabilistic model of the nuisances – and additional priors. From a parameter estimation perspective, the maximum of the a posteriori probability distribution of source intensity, given the observations could be considered as the ‘best possible model’. This maximum a posteriori (MAP) estimate has been extremely successful in the digital image restoration and reconstruction communities. (Geman & Geman, 1984) is a masterpiece reference of the genre. The MAP is obtained in Bayesian statistics through the optimization of the mixture of the likelihood of the noisy data – i.e., of the predictive power of a given source model – with the a priori probability of a given source model.
We do not want to detail the mathematics of Bayesian inference any further here as this would reach outside the objectives of these pages. Specific recommended further reading includes (Demoment, 1989), for a Bayesian discussion on regularization and (Baillet, Mosher, & Leahy, 2001), for an introduction to MEG/EEG imaging methods, also in the Bayesian framework.
From a practical standpoint, the priors on the source image models may take multiple faces: promote current distributions with high spatial and temporal smoothness, penalize models with currents of unrealistic, non-physiologically plausible amplitudes, favor the adequation with an fMRI activation maps, or prefer source image models made of piecewise homogeneous active regions, etc. An appealing benefit from well-chosen priors is that it may ensure the uniqueness of the optimal solution to the imaging inverse problem, despite its original underdeterminacy.
Because relevant priors for MEG/EEG imaging models are plethoric, it is important to understand that the associated source estimation methods usually belong to the same technical background. Also, the selection of image priors can be seen as arbitrary and subjective an issue as the selection of dipoles in the source localization techniques we have reviewed previously. Comprehensive solutions for this model selection issue are now emerging and will be briefly reviewed further below.
The free parameters of the imaging model are the amplitudes of the elementary source currents distributed on the brain’s geometry. The non-linear parameters (e.g., the elementary source locations) now become fixed priors as provided by anatomical information. The model estimation procedure and the very existence of a unique solution strongly depend on the mathematical nature of the image prior.
A widely-used prior in the field of image reconstruction considers that the expected source amplitudes be as small as possible on average. This is the well-described minimum-norm (MN) model. Technically speaking, we are referring to the L2-norm; the objective cost function ruling the model estimation is quadratic in the source amplitudes, with a unique analytical solution (Tarantola, 2004). The computational simplicity and uniqueness of the MN model has been very attractive in MEG/EEG early on (Wang et al.., 1992).
The basic MN estimate is problematic though as it tends to favor the most superficial brain regions (e.g., the gyral crowns) and underestimate contributions from deeper source areas (such as sulcal fundi) (Fuchs, Wagner, Köhler, & Wischmann, 1999).
As a remedy, a slight alteration of the basic MN estimator consists in weighting each elementary source amplitude by the inverse of the norm of its contribution to sensors. Such depth weighting yields a weighted MN (WMN) estimate, which still benefits from uniqueness and linearity in the observations as the basic MN (Lin, Witzel, et al.., 2006).
Despite their robustness to noise and simple computation, it is relevant to question the neurophysiological validity of MN priors. Indeed – though reasonably intuitive – there is no evidence that neural currents would systematically match the principle of minimal energy. Some authors have speculated that a more physiologically relevant prior would be that the norm of spatial derivatives (e.g., surface or volume gradient or Laplacian) of the current map be minimized (see LORETA method in (Pascual-Marqui, Michel, & Lehmann, 1994)). As a general rule of thumb however, all MN-based source imaging approaches overestimate the smoothness of the spatial distribution of neural currents. Quantitative and qualitative empirical evidence however demonstrate spatial discrimination of reasonable range at the sub-lobar brain scale (Darvas, Pantazis, Kucukaltun-Yildirim, & Leahy, 2004, Sergent et al.., 2005).
Most of the recent literature in regularized imaging models for MEG/EEG consists in struggling to improve the spatial resolution of the MN-based models (see (Baillet, Mosher, & Leahy, 2001) for a review) or to reduce the degree of arbitrariness involved in selected a generic source model a priori (Mattout, Phillips, Penny, Rugg, & Friston, 2006, Stephan, Penny, Daunizeau, Moran, & Friston, 2009). This results in notable improvements in theoretical performances, though with higher computational demands and practical optimization issues.
As a general principle, we are facing the dilemma of knowing that all priors about the source images are certainly abusive, hence that the inverse model is approximative, while hoping it is just not too approximative. This discussion is recurrent in the general context of estimation theory and model selection as we shall discuss in the next section.
Distributed source imaging of the [120,300] ms time interval following the presentation of the target face object in the visual RSVP oddball paradigm described before. The images show a slightly smoothed version of one participant’s cortical surface. Colors encode the contrast of MEG source amplitudes between responses to target versus control faces. Visual responses are detected by 120ms and rapidly propagate anteriorly. By 250 ms onwards, strong anterior mesial responses are detected in the cingular cortex. These latter are the main contributors of the brain response to target detection.
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