Two major problems exist in dipole fitting procedures. First, due to non-linear optimization there are problems of local minima when more than two dipole parameters are estimated and this is usually manifested by sensitivity to initialization (Huang et al. 1998). Brute-force search methods have a huge computational burden—exponential in the number of parameters (Mosher et al. 1992, 1993). A second, more difficult problem in parametric methods is that often these methods require a priori knowledge of the number of dipoles. Often, such information about model order is not known a priori, especially for complex brain mapping conditions, and the resulting localization of higher-order cortical functions can sometimes be unreliable. Although information theoretic or Bayesian estimation criteria have been proposed to address this problem, the success of these approaches is less clear as these are not widely used (Campi et al. 2011; Kiebel et al. 2008; Sorrentino et al. 2009; Wolters et al. 1999). Nevertheless, many basic neuroscience and clinical studies to date have successfully used dipole-fitting procedures to gain important insights (Aine et al. 2010; Salmelin et al. 1994; Susac et al. 2009).

Non-parametric whole brain imaging is an alternative approach to estimate the inverse problem. The relevant localization problem can be posed as follows. The measured signal is a d _b  × n matrix B, where d _b equals the number of sensors and n is the number of time points at which measurements are made and the unknown sources are given by a d _s  × n matrix S which is the (discretized) amplitude of the source activity at d _s candidate locations obtained from the forward model calculations. In this case, B and S are related by the generative model
where L is the composite forward-field matrix that captures the relationship between unit sources all over the brain and the expected pattern of magnetic field measurement on the sensor array. The number of candidate source locations is much larger than the number of sensors (d _s  >> d _b ). Therefore, the problem reduces to estimation of the activity in each source regions, which are reflected by the non-zero rows of the source estimate matrix Ŝ. E is a noise or interference term discussed earlier.

Many whole-brain imaging algorithms impose constraints on source locations i.e. the candidate locations for sources based on anatomical and functional information obtained from other brain imaging modalities. Such constraints within a Bayesian framework are embedded in a prior distribution p(S) either implicitly or explicitly. If under a given experimental or clinical paradigm this p(S) were somehow known exactly, then the posterior distribution can be computed via Bayes rule:

This distribution contains all possible information about the unknown S conditioned on the observed data B. Two fundamental problems prevent using p(S|B) for source localization. First, for most priors p(S), the normalization distribution p(B) given by
cannot be computed analytically. If only a point estimate for S is desired, rather than a full distribution, then this normalizing distribution may not be needed. For example, a popular estimator is the minimum-norm estimator which involves finding the value of S by assuming that prior p(S) has a Gaussian distribution with a single scalar variance term. This variance is related to the regularization constant in many implementations of the minimum-norm estimator and can be obtained by maximizing the posterior distribution (a.k.a. the MAP estimate) of p(S|B) which is invariant to p(B). Second, and more importantly, we do not actually know the prior p(S) and so some appropriate distribution must be assumed, perhaps based on neurophysiological constraints or computational considerations. In fact, it is this choice, whether implicitly or explicitly specified, that differentiates a wide variety of localization methods (Phillips et al. 1997; Wipf and Nagarajan 2009).

While seemingly quite different in many respects, we recently presented a generalized framework that encompasses different whole-brain imaging methods for source localization and points to intimate connections between algorithms. We showed that many seemingly disparate algorithms for source imaging can be unified using a hierarchical Bayesian modeling framework with a general form of prior distribution, called Gaussian scale mixture, with flexible covariance components, and two different types of inferential procedures. The wide variety of Bayesian source localization methods that fall under this framework can be differentiated by the following factors: (1) selection of covariance component regularization terms; (2) choice of initial covariance component set; (3) optimization method/update rules; and (4) approximation to the lower bound on the marginal likelihood of the data. Bayesian source localization methods demonstrate a number of surprising similarities or out-right equivalences between what might otherwise appear to be very different algorithms. Specifically, from the vantage point of a simple Gaussian scale mixture model with flexible covariance components, our initial work in this area analyzed and extended several broad categories of Bayesian inference directly applicable to source localization including empirical Bayesian approaches, standard MAP estimation, and variational Bayesian (VB) approximations. This perspective leads to explicit connections between many established algorithms and suggests natural extensions for handling unknown dipole orientations, extended source configurations, correlated sources, temporal smoothness, and computational expediency. Specific imaging methods elucidated under this paradigm include weighted minimum L2-norm, FOCUSS, minimum-L1 norm (also called minimum-current estimation (MCE)), VESTAL, sLORETA, ReML and covariance component estimation, beamforming, Variational Bayes, and Automatic relevance determination (ARD) with multiple sparse priors (MSP). Perhaps surprisingly, all of these methods can be formulated as particular cases of covariance component estimation using different concave regularization terms and optimization rules, making general theoretical analyses and algorithmic extensions/improvements particularly relevant.

These ideas help to bring an insightful perspective to Bayesian source imaging methods, reduce confusion about how different techniques relate to one another, and expand the range of feasible applications. Additionally, there are numerous promising directions for future research, including time-frequency extensions, alternative covariance component parameterizations, and integration with robust interference suppression. These insights allow for continued development of novel algorithms for whole-brain imaging in relation to prior efforts in this enterprise. Figure 2 shows performance in simulations using one such novel algorithm, called Champagne, as well as reconstructions from popular benchmark algorithms for comparisons that highlight their poorer spatial resolution and sensitivity to correlated sources and noise (Owen et al. 2012; Wipf et al. 2010). When compared to ground-truth it can be seen that Champagne is the algorithm that is able to reconstruct the source configuration. Figure 3 shows source reconstructions of auditory evoked responses using called Champagne, and benchmarks algorithms. Auditory evoked responses are challenging datasets because of high degree of correlations between bilateral auditory cortices. In these real datasets from three different subjects, it can also be seen that Champagne is the only algorithm able to reliably reconstruct bilateral auditory cortical activity.

Instead of simultaneous estimation of all sources a popular alternative is to scan the brain and estimate source amplitude at each source location independently. It can be shown that such scanning methods are closely related to whole-brain imaging methods, and the most popular scanning algorithms are adaptive spatial filtering techniques, more commonly referred to as “adaptive beamformers” or just “beamformers” (Sekihara and Nagarajan 2008). Adaptive beamformers have been shown to be quite simple to implement and are powerful techniques for characterizing cortical oscillations and are closely related to other whole-brain imaging methods. However, one major problem with adaptive beamformers is that they are extremely sensitive to the presence of strongly correlated sources. Although they are robust to moderate correlations, in the case of auditory studies, since auditory cortices are largely synchronous in their activity across the two hemisphere, these algorithms tend to perform poor for auditory evoked datasets without workarounds), and many modifications have been proposed for reducing the influence of correlated sources (Dalal et al. 2006). The simplest such workaround is to use half the sensors corresponding to each hemisphere separately, and this approach works surprisingly well for cross-hemispheric interactions. Other modifications to the original algorithms have been proposed in the literature that require some knowledge about the location of the correlated source region (Dalal et al. 2006; Quraan and Cheyne 2010). Recently, we have shown that significant improvements in performance can be achieved by modern Bayesian inference algorithms that are closely related to minimum-variance adaptative beamformers and these extensions allow for accurate reconstructions of a large number of sources from typical configurations of MEG sensors (Wipf et al. 2010; Zumer et al. 2007, 2008).