Fig. 1: Directional phase estimation reveals information lost by conventional approaches.

(a) Directional ambiguity in cosine-based metrics. Two synthetic groups with mean phase offsets of +60° and −60° are indistinguishable using the cosine of phase difference (two-sample t, p = 0.74), but are clearly separable using full circular phase (Watson’s U², p = 0). (b) Population-level phase organization. Left: average complex-valued phase synchrony (CVPS) across the fBIRN cohort. Right: corresponding cosine (CRP) and sine (SRP) components of relative phase. The sine component, which encodes directional (lead–lag) relationships, is absent in CRP. (c) Empirical example (precuneus–thalamus): CRP shows no group difference (controls vs schizophrenia; p = 0.78), whereas CVPS reveals a significant separation (Watson’s test, p = 3.10 × 10⁻⁵). (d) Empirical example (hippocampus–superior parietal lobule): CRP indicates a difference (p = 6.36 × 10⁻³), but CVPS does not (p = 0.18), illustrating that cosine-only metrics can flag differences driven by magnitude while obscuring directional alignment. Together, these results establish that preserving directional phase information is critical for accurately characterizing interregional coordination. (e) Simulated non-stationary signal with slowly varying frequency (0.035–0.065 Hz; SNR = 10 dB). Left: time–frequency representation illustrating frequency modulation. Middle: phase tracking shows that the Gabor wavelet estimate closely follows the ground-truth trajectory, whereas the Hilbert transform deviates under nonstationarity. Right: RMSE across SNR levels (log scale) confirms improved accuracy of the Gabor approach (paired test; ***p<0.001). confirming that the Gabor wavelet tracks nonstationary phase more accurately than the Hilbert transform under BOLD-like conditions.