Estimation and interpretation of the cross-power spectrum to quantify brain connectivity from electrophysiological data

Magneto- and electro- encephalography (MEEG) are two neuroimaging tools capable of non invasively recording the magnetic field outside the head and the scalp potential produced by the electric currents that flow inside the brain. Their strong point is the extremely high temporal resolution which makes them suitable for the study of functional connectivity, i.e. the quantification of the statistical dependencies among the time courses that describe brain activity. Functional connectivity is usually estimated from MEEG data with a two-step process: (i) first an estimate of the time courses associated with brain activity is computed by means of a regularisation method; (ii) then, source space functional connectivity is computed from the estimated time courses. The most widely used regularisation method to face this problem is Tikhonov regularisation, also known as Minimum Norm Estimate (MNE), which requires the setting of a regularisation parameter. Such a parameter will be used throughout the process and will influence the final connectivity estimate. In the core part of this thesis we will show that the regularisation parameter providing the best estimate of brain activity does not provide the best possible connectivity estimate. Indeed, a smaller parameter should be set for the latter intent. This result will be supported by both analytical and numerical proofs. Our results highlight that the classic two-step approach plus Tikhonov regularisation presents two intrinsic drawbacks: (i) the propagation of the errors during the process, and (ii) a connectivity estimate which is over spread in space, due to Tikhonov regularisation. In this thesis we will present a one-step approach combined with the l1 regularisation, where source space functional connectivity is directly estimated from that at sensor space, without a prior estimation of source time courses. The proposed pipeline overcomes the problems of the classic approach and outperforms it, indeed, the one-step approach reduces the propagation of the errors and the l1 regularisation promotes sparsity on the connectivity estimate. Finally, we will present transfreq, a Python package for the automated computation of the theta-to-alpha transition frequency. A correct estimation of such a quantity is of utmost importance for the reliability of connectivity studies in both healthy subjects and patients.