Hilbert transforms and wavelets are not equivalent in extracting parameters from electrophysiology signals for phase-amplitude coupling studies.

Abstract

In functional connectivity studies based on electrophysiological measurements, there are two popular approaches for extracting narrowband representations of brain activity time series and then computing their envelope amplitudes and instantaneous phases (which serve as inputs to subsequent data processing): Hilbert transforms (preceded by band-pass filtering), and wavelets. Since both approaches can be expressed as linear convolutions with the activity time series, they are mathematically equivalent; however, this equivalence is seldom observed in typical connectivity studies. In this work, we compared the approaches as they are actually implemented in the literature, focusing on the phenomenon of phase-amplitude coupling (PAC). 
Approach: The comparison of both approaches was carried out with simulated brain activity, from which we ran receiver operating characteristic (ROC) analyses, and experimental electrocorticography (ECoG) data from a finger movement study. 
Main results: The ROC analyses showed that the classification accuracy of wavelets was markedly better than that of Hilbert transforms in most scenarios tested - one significant exception was observed with interfering oscillations near the frequency bands of interest, in which case Hilbert transforms performed better. As for the ECoG data, wavelets allowed the identification of effects to which Hilbert transforms were insensitive, such as brain locations near the motor cortex with significant task-based changes in PAC. 
Significance: We demonstrated empirically that, contrary to what has been previously accepted, there can be a wide discrepancy between Hilbert transforms' and wavelets' performances, at least in the context of PAC estimations.