Sparse Bayesian learning of brain activity patterns
Swartz Center for Computational Neuroscience
Institute for Neural Computation
University of California,
San Diego
Although the use of anatomical location priors to spatially constrain the current density estimates to grey matter is highly beneficial, use of other priors including fixed source pdfs, and exact model order assumptions may lead to inaccurate solutions. The Bayesian evidence framework represents a clear departure from the classical objective of simply introducing a subjective prior to the inference process (i.e., MAP estimation). Alternative hypothesis corresponding to different priors are invoked for model comparison, and are objectively ranked by evaluating their Bayesian evidence, the probability of the data given the model (MacKay, 1992). Bayes rule automatically embodies Occam's razor to prefer simpler models that predict the data more strongly than overcomplex models. For regression and classification, this sparse Bayesian learning (SBL) methodology takes the form of the relevance vector machine (RVM) (Tipping, 2001), a Bayesian alternative to the support vector machine (SVM). Recently, SBL has been expanded to solve underdetermined inverse problems under sparsity constraints (Wipf and Rao, 2004). The SBL algorithm assumes a Gaussian likelihood model and a source distribution parameterized by a vector of hyperparameters that are learned from the data by integrating out the unknown source parameters (e.g., the current density) and performing evidence maximization (i.e., type-II maximum likelihood) using an Expectation Maximization (EM) algorithm or a fixed-point gradient update rule that minimizes the same cost, but more rapidly (Ramirez, 2005). Importantly, SBL finds the maximally sparse solution in cases where other algorithms, such as FOCUSS (Rao et al., 2003), converge to local minima. However, maximal sparsity is not always desired since sources have variable spatial extents.
Here, a new framework for neuroelectromagnetic source imaging is developed in which the fundamental data representation atoms are concatenated MEG/EEG gain vectors generated by locally distributed multi-resolution current density functions on cortical patches, instead of single dipoles. An overcomplete lead-field dictionary is constructed using these distributed neural bases at multiple scales. Cortical geodesic distances and orientation constraints are exploited to discriminate between gyri or sulci that are near in Euclidian but not geodesic space. A sparse solution to the multiscale-transformed inverse problem is computed using SBL. Although it is maximally sparse in terms of the transformed system, it represents a distributed current density estimate once it is back-transformed with the a priori neural basis matrix. Computer simulations demonstrate that we can reconstruct the spatial current density distribution of many simultaneously active sources in the presence of noise and modeling error. We analyze MEG data with SBL and infomax ICA to characterize the large-scale neuronal networks involved in cued visuospatial covert attention and target detection.