Construction of invariant features for time-domain EEG/MEG signals using Grassmann manifolds

A challenge in interpreting features derived from source-space electroencephalography (EEG) and magnetoencephalography (MEG) signals is residual mixing of the true source signals. A common approach is to use features that are invariant under linear and instantaneous mixing. In the context of this approach, it is of interest to know which invariant features can be constructed from a given set of source-projected EEG/MEG signals. We address this question by exploiting the fact that invariant features can be viewed as functions on the Grassmann manifold. By embedding the Grassmann manifold in a vector space, coordinates are obtained that serve as building blocks for invariant features, in the sense that all invariant features can be constructed from them. We illustrate this approach by constructing several new bivariate, higher-order, and multidimensional functional connectivity measures for static and time-resolved analysis of time-domain EEG/MEG signals. Lastly, we apply such an invariant feature derived from the Grassmann manifold to EEG data from comatose survivors of cardiac arrest and show its superior sensitivity to identify changes in functional connectivity. Author SummaryElectroencephalography (EEG) and magnetoencephalography (MEG) are techniques to non-invasively measure brain activity in human subjects. This works by measuring the electric potentials on the scalp (EEG) or the magnetic fluxes surrounding the head (MEG) that are induced by currents flowing in the brains grey matter (the "brain activity"). However, reconstruction of brain activity from EEG/MEG sensor signals is an ill-posed inverse problem and, consequently, the reconstructed brain signals are linear superpositions of the true brain signals. This fact complicates the interpretation of the reconstructed brain activity. A common approach is to only use features of the reconstructed activity that are invariant under linear superpositions. In this study we show that all invariant features of reconstructed brain signals can be obtained by taking combinations of a finite set of fundamental features. The fundamental features are parametrized by a high-dimensional space known as the Grass-mann manifold, which has a rich geometric structure that can be exploited to construct new invariant features. Our study advances the systematic study of invariant properties of EEG/MEG data and can be used as a framework to systematize and interrelate existing results. We use the theory to construct a new invariant connectivity measure and apply it to EEG data from comatose survivors of cardiac arrest. We find that this measure enables superior identification of affected brain regions.