  Most biological signals are non-Gaussian, reflecting their origins in highly
nonlinear physiological systems. A versatile set of techniques for studying
non-Gaussian signals relies on the spectral representations of higher moments,
known as polyspectra, which describe forms of cross-frequency dependence that
do not arise in time-invariant Gaussian signals. The most commonly used of
these employ the bispectrum. Recently, other measures of cross-frequency
dependence have drawn interest in EEG literature, in particular those which
address phase-amplitude coupling (PAC). Here we demonstrate a close
relationship between the bispectrum and popular measures of PAC, which we
relate to smoothings of the signal bispectrum, making them fundamentally
bispectral estimators. Viewed this way, however, conventional PAC measures
exhibit some unfavorable qualities, including poor bias properties, lack of
correct symmetry and artificial constraints on the spectral range and
resolution of the estimate. Moreover, information obscured by smoothing in
measures of PAC, but preserved in standard bispectral estimators, may be
critical for distinguishing nested oscillations from transient signal features
and other non-oscillatory causes of "spurious" PAC. We propose guidelines for
gauging the nature and origin of cross-frequency coupling with bispectral
statistics. Beyond clarifying the relationship between PAC and the bispectrum,
the present work lays out a general framework for the interpretation of the
bispectrum, which extends to other higher-order spectra. In particular, this
framework holds promise for the detailed identification of signal features
related to both nested oscillations and transient phenomena. We conclude with a
discussion of some broader theoretical implications of this framework and
highlight promising directions for future development.
