Dynamic cross-scale wavelet coherence

This paper develops a novel statistical approach that allows for the {\em first time} the {\em cross}-oscillatory characterisation of temporally localised interactions between channels in a functional brain network. Brain signals are often nonstationary and the proposed framework uses wavelets as an effective tool for capturing (i) single-scale channel transient features, due to their adaptiveness to the dynamic signal properties, and (ii) cross-scale channel interactions, due to their multiscale nature. Our approach introduces scale-specific {\em subprocesses} and {\em cross-scale (CS) dependencies} for a new class of multivariate locally stationary (MvLSW) wavelet processes that we refer to as CS-MvLSW. Under this new model, we develop two consistent estimation procedures for the {\em localised} single- and cross-scale channel dependence. Extensive simulation studies demonstrate that the theoretically established properties hold in practice. The proposed CS-MvLSW framework remains accurate under pronounced cross-scale dependence, whereas existing MvLSW coherence estimates dramatically deteriorate even for single-scales when such complex structure is present. The proposed approach was used for electroencephalogram (EEG) data to study alterations in the functional connectivity structure in children diagnosed with attention deficit hyperactivity disorder (ADHD), and identified novel clinically pertinent cross-scale interactions in the functional brain network across the left and right hemispheres, differentiating brain connectivity between control and ADHD groups.