Multi-scale wavelet coherence

This paper develops a novel statistical approach to characterize temporally localised cross-oscillatory interactions between channels in a functional brain network. Brain signals are generally 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 multi-scale nature. Our approach formalises scale-specific subprocesses and cross-scale (CS) dependencies for a new class of multivariate locally stationary (MvLSW) wavelet processes that we refer to as CS-MvLSW. Under this model, we develop a novel spectral domain time-varying cross-scale dependence measure and its appropriate estimation. 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 modelling can deteriorate even for single-scale coherence when such complex structure is present in the process. The proposed cross-scale analysis is applied to electroencephalogram (EEG) data to study alterations in the functional connectivity structure in children diagnosed with attention deficit hyperactivity disorder (ADHD). Our approach identified novel, clinically pertinent cross-scale interactions in the functional brain network, differentiating brain connectivity between control and ADHD groups.