Mapping brain network topographies and topologies (BNT&T) via statistics of the MEEG sensor time series is a common fallacy. The alternative, mapping BNT&T by using 1) Inversion of the MEEG Lead Field to obtain source time series and the topography of networks 2) Determination of their connectivities or topology, would seem flawed due to effect of distortions (localization error and blurring) that originate from the ill-conditioning of the Lead Field, which affects both the estimated topographies and topologies. We recognize that precise determination of BNTT might achieved via Bayesian methods to perform 1) and 2) by placing structured sparse priors directly on the estimators of topographies or topologies. We present a toolbox: Brain Connectivity Variable Resolution Tomographic Analysis (BC-VARETA), designed with such Bayesian methods to map BNT&T that emerge from specific oscillatory activity (network spectral topographies and topologies). Two BC-VARETA libraries target the obtention of 1) Sparse network topographies, that are mathematically represented with the diagonal entries of the source cross-spectra (spectra) and estimated via the Spectral Structured Sparse Bayesian Learning method (sSSBL) [1]. sSSBL is an extension of the Sparse Bayesian Learning (SBL) method that leverages the hierarchical and complex valued Elastic Net (combined L1&L2 norms), as an a priori model of the source spectra. Statistical guaranties upon the sparse support of the network are obtained by means of the F-scores computed for the a posteriori mixed effect model of the spectra, which is estimated via maximum Bayesian evidence in sSSBL based on the Expectation-Maximization algorithm (EM). 2) Sparse topologies, which are computed upon but distinguish from the sparse network topography formerly estimated, that are mathematically represented with the precision matrix of the source cross-spectra and estimated via a Hidden Gaussian Graphical State-Space model (HIGGS) [2] with hermitian graphical LASSO prior (L1 norm). Statistical guaranties upon the sparse support of the topology are obtained by means of thresholds based on the Rayleigh distribution of the precision matrix unbiased statistics where the optimal threshold produces maximum Bayesian evidence in HIGGS based on the EM algorithm. Supplementary, grouped estimation of BNT&T within specific spectral (and not only specific frequency components) bands can be performed by BC-VARETA with the group Elastic Net or LASSO. In addition, to compensate the bias of sparsity in the estimated BNT&Ts, we leverage fused sparsity based upon multivariate reweighted L1&L2 or L1 norms, where the weight coefficients are specific designed to incorporate cortical priors. These priors include compensation of depth that is based upon cortical curvature, cortical Laplacian, and rotational invariance of cortical lead fields. This toolbox has produced BNT&Ts with unprecedent quality, in realistic simulations of EEG, which use a ground truth based on HCP MEG data, and comparison EEG vs ECoG. The comparison of MEG vs EEG in normative databases has proven the consistency of the BNT&Ts estimated with BC-VARETA.
References
[1] Gonzalez-Moreira, E., Paz-Linares, D., Areces-Gonzalez, A., Wang, Y., Li, M., Harmony, T. and Valdes-Sosa, P.A., 2020. Bottom-up control of leakage in spectral electrophysiological source imaging via structured sparse bayesian learning.
bioRxiv. https://www.biorxiv.org/content/10.1101/2020.02.25.964684v1
[2] Paz-Linares, D., Gonzalez-Moreira, E., Bosch-Bayard, J., Areces-Gonzalez, A., Bringas-Vega, M.L. and Valdes-Sosa, P.A., 2018. Neural Connectivity with Hidden Gaussian Graphical State-Model. arXiv:https://arxiv.org/abs/1810.01174