The meaning of "coherent" and its quantification in coherent hemodynamics spectroscopy

We have recently introduced a new technique, coherent hemodynamics spectroscopy (CHS), which aims at characterizing a specific kind of tissue hemodynamics that feature a high level of covariation with a given physiological quantity. In this study, we carry out a detailed analysis of the significance of coherence and phase synchronization between oscillations of arterial blood pressure (ABP) and total hemoglobin concentration ([Hbt]), measured with near-infrared spectroscopy (NIRS) during a typical protocol for CHS, based on a cyclic thigh cuff occlusion and release. Even though CHS is based on a linear time invariant model between ABP (input) and NIRS measurands (outputs), for practical reasons in a typical CHS protocol, we induce finite "groups" of ABP oscillations, in which each group is characterized by a different frequency. For this reason, ABP (input) and NIRS measurands (output) are not stationary processes, and we have used wavelet coherence and phase synchronization index (PSI), as a metric of coherence and phase synchronization, respectively. PSI was calculated by using both the wavelet cross spectrum and the Hilbert transform. We have also used linear coherence (which requires stationary process) for comparison with wavelet coherence. The method of surrogate data is used to find critical values for the significance of covariation between ABP and [Hbt]. Because we have found similar critical values for wavelet coherence and PSI by using five of the most used methods of surrogate data, we propose to use the data-independent Gaussian random numbers (GRNs), for CHS. By using wavelet coherence and wavelet cross spectrum, and GRNs as surrogate data, we have found the same results for the significance of coherence and phase synchronization between ABP and [Hbt]: on a total set of 20 periods of cuff oscillations, we have found 17 coherent oscillations and 17 phase synchronous oscillations. Phase synchronization assessed with Hilbert transform yielded similar results with 14 phase synchronous oscillations. Linear coherence and wavelet coherence overall yielded similar number of significant values. We discuss possible reasons for this result. Despite the similarity of linear and wavelet coherence, we argue that wavelet coherence is preferable, especially if one wants to use baseline spontaneous oscillations, in which phase locking and coherence between signals might be only temporary.