Advanced statistical techniques for brain functional connectivity: from network estimation to network comparisons

Magneto- and electro-encephalography (M/EEG) are two non-invasive neuroimaging techniques with high temporal resolution capable of recording the magnetic field outside the head and the scalp potential generated by the electric currents within the brain. From M/EEG recordings, it is possible to derive brain connectivity networks. In this thesis, we examine two main aspects concerning the study of brain networks. In the first part, we investigate the estimation of brain connectivity networks at the cortical level from M/EEG measurements. In detail, we first present a procedure based on $\ell_1$ regularisation that directly estimates the cross-power spectrum between brain regions while controlling the number of false positives. Then, we introduce an approach that exploits structural equation modelling (SEM) and statistical inference for estimating directed cortical networks aiming also to provide information on potential causal relantionships. In the second part, we investigate statistical tools to compare the estimated brain networks. In particular, we focus on the network-based statistic (NBS) tool for comparing groups of brain networks associated with different external factors. We provide a mathematical formulation of the NBS algorithm and present two alternatives to the F-test used when dealing with more than two groups: we compare the Tukey-Kramer test and a Bonferroni-corrected pairwise t-test, both capable of identifying which groups of networks differ significantly. Finally, we will apply the NBS to a dataset of subjects affected by Dementia with Lewy Bodies (DLB) to investigate the relationship between functional connectivity metrics and the core clinical feature characterising DLB patients.