The bispectrum and its relationship to phase-amplitude coupling
Introduction
Many interesting properties of signals in nature relate to nonlinear, non-Gaussian and non-stationary dynamics, which are poorly indexed by second-order measures such as power and cross spectra. Because the spectrum of a stationary Gaussian process lacks any statistical dependence across frequencies, measures of frequency-domain dependence can be extremely useful for gauging the presence and nature of higher order dynamics (Sheremet et al., 2016). For stationary signals, higher order spectra, “polyspectra,” which capture such dependence, are the frequency-domain representations of higher moments (Nikias and Mendel, 1993), in which they mirror the relationship between the power spectrum of a signal and its autocorrelation. The bispectrum is the third-order polyspectrum, making it an obvious place to start in the approach towards higher-order dynamics, at least from a statistical standpoint. Bispectral analysis has proved its practical worth in applications to EEG (Dumermuth et al., 1971; Sigl and Chamoun, 1994), most notably in gauging the depth of anesthesia (Barnett et al., 1971; Gan et al., 1997; Kearse et al., 1998; Myles et al., 2004). Nevertheless, a significant drawback for the non-statistician remains its lack of an obvious connection to any simple physical interpretation (Fackrell et al., 1995). This point has been the source of some confusion in applied literature; for example, as recently reviewed by Hyafil (2015), various authors have suggested incorrectly that bicoherence relates to phase entrainment across pairs of frequencies. While it is true that bispectral measures cannot easily be reduced to any single interpretation, a goal of the present work is to show that specific forms of dependence leave easily recognized signatures in the bispectrum, making bispectral measures invaluable for distinguishing between a variety of phenomena related to cross-frequency coupling.
A separate body of work has recently emerged from EEG literature, which examines the role of another form of cross-frequency coupling, phase-amplitude coupling (PAC) (Canolty et al., 2006; Hyafil et al., 2015). PAC refers to dependence between analytic amplitude at one frequency and analytic phase at another. Rather than any statistical first principle, interest in PAC has been motivated by the empirical discovery of PAC in signals recorded from the brain, alongside emerging computational and physiological models of how it arises within populations of interacting neurons (Young and Eggermont, 2009; Jirsa and Müller, 2013; Akam and Kullmann, 2014; Hyafil et al., 2015). In this literature, the measurement of PAC is usually approached with second-order statistics, such as coherence, applied towards comparing analytic phase in one band with analytic amplitude extracted from another, treating the two as separate signals (Schack et al., 2002; Canolty et al., 2006; Tort et al., 2010). The situation with PAC is therefore reversed from bispectral measures: its physical meaning is more evident than its relationship to any general body of statistical theory.
The present work aims to bridge this gap and resolve ambiguities of meaning in both directions. The central result is that second-order measures of PAC may be fundamentally understood as estimators of the bispectrum. In the same way that windowed stationary estimates of the power spectrum can be equated to smoothings of the true signal spectrum, windowed bispectral estimators, which include those underlying measures of PAC, amount to different ways of smoothing the true signal bispectrum (Gerr, 1988; Nikias and Raghuveer, 1987; Swami, 1991; Hanssen and Scharf, 2003). In both cases, differences between estimators relate to properties of the respective smoothing kernels (Cohen, 1989). This observation demonstrates conclusively the meaning of PAC measures as they relate to the bispectrum and vice versa and establishes that second-order measures of PAC provide no unique information beyond what can be obtained from the bispectrum.
While PAC measures are fundamentally measures of the bispectrum, the reverse is not true; it is not correct to conclude that the bispectrum is principally a reflection of phase-amplitude coupling. Forms of “spurious” PAC, related, for example, to spectrally broad signal features, may be traced to the bispectral nature of PAC measures. One practical implication is that the superior resolution and lower bias of standard bispectral measures, in comparison to PAC measures, allows them to retain information critical for distinguishing between nested oscillations and other sources of apparent phase-amplitude coupling (Kramer et al., 2008). Following a brief review of bispectral and PAC estimation, we will observe how different regions of the bispectrum may be taken to reflect either phase-amplitude coupling or consistency of phase across a range of frequencies. It is shown that the bispectrum can be highly useful for ascertaining the presence and origin of phase-amplitude coupling. Properly applied and interpreted, bispectral statistics may overcome a number of recently highlighted limitations and ambiguities of existing PAC measures (Aru et al., 2015; Hyafil, 2015; Kramer et al., 2008; Lozano-Soldevilla et al., 2016; Scheffer-Teixeira and Tort, 2016; van Driel et al., 2015). For example, with conventional measures of PAC, the observable range of phase-providing frequencies is restricted by the bandwidth of the amplitude-providing band (Aru et al., 2015), yet no such limitation applies to bispectral estimates. In light of the relationship to the bispectrum, it becomes clear that this constraint is an artifact of the estimator rather than anything inherent to the quantity measured.
Many of the questions that arise in considering the relationship between PAC and the bispectrum prove to be of much more general relevance for understanding a range of signal properties that are neglected by traditional spectral measures, a topic whose importance is becoming increasingly clear (Cole and Voytek, 2017). In particular, we develop a model that uses the bispectrum to capture spectrally complex signal features, of which PAC is only one example. The concluding sections are devoted to a prospective review of the application of the bispectrum towards understanding the nature and functional significance of non-oscillatory and transient sources of cross-frequency coupling, beyond nested oscillations.
The overall aim of the current work is threefold; first, a general introductory background is provided to motivate applications of higher-order spectra in signal analysis, accompanied in the appendices by a more focused and technical review of the bispectrum and its estimation. The second aim is to describe a formal equivalence between bispectral estimators and measures of phase-amplitude coupling, details of which are presented in Appendix C. Finally, building on this formal relationship, a framework is developed to guide the interpretation of the bispectrum. In most places, more technical development has been left to the appendices, the results of which are summarized in the main text alongside some background explanation.
Section A primer on higher-order spectra in signal processing, in particular, focuses on introducing readers who have had little exposure to the ideas and applications of higher-order spectra to some general motivating principles, thus it deals with the subject very broadly. Section The bispectrum gives a short summary of the main relevant properties of the bispectrum described in more detail in Appendix A and Appendix B. Section Phase-amplitude coupling similarly introduces phase-amplitude coupling, which is given a more detailed treatment in Appendix C. Section Bispectral signatures of PAC develops a framework for understanding the relationship between PAC and the bispectrum, building on the proof provided in Appendix C. Section The bispectrum and phase-frequency coupling briefly considers an extension to phase-frequency coupling. The concluding sections set the framework within a broader theoretical context; section Tuning in to the bispectrum describes a model of signal encoding and transmission that exploits properties of the bispectrum, while section Future directions anticipates further development of the framework, especially in the direction of feature extraction and identification.
Section snippets
A primer on higher-order spectra in signal processing
Abstractly, signal processing is about making sense of sequential or otherwise meaningfully ordered quantities; it considers the question, what kinds of measures best capture the essential nature of any dependence among the quantities as it relates to their position in the ordering space (henceforth assumed to be time)? An endless variety of different measures can be conceived, the usefulness and interpretation of any of which naturally depends on very concrete particulars of the application.
The bispectrum
The previous section laid out some general context and motivation for appealing to higher-order spectra in signal processing. The remainder of the work is concerned with addressing the two stumbling blocks identified earlier in relation to third-order statistics: computation and interpretation. The appendices contain a more technically oriented overview of the bispectrum in which the question of how to design estimators is given particular attention. To address the problem of interpretation, we
Phase-amplitude coupling
Phase-amplitude coupling refers to dependence between the analytic envelope of an oscillatory signal component within one band and phase within another. For the envelope of the first component to fluctuate at the scale of the second, the bandwidth of the first component must be at least as great as the center frequency of the second and its center frequency correspondingly higher, for which reason the first component is a “fast oscillation” (FO), while the second is the “slow oscillation” (SO).
Bispectral signatures of PAC
Appendix C contains a proof that phase-power coherence is fundamentally a bispectral estimator, which differs from conventional bispectral estimators only in the shape of the associated smoothing kernel. While past authors have noted some similarities between bispectral and PAC measures (Kramer et al., 2008; Hyafil, 2015), this formal relationship appears not to have been previously described. The following sections extend this observation with a signal model that will offer some more practical
Multi-modal dependence
As mentioned in section Phase Amplitude Coupling, a variety of extensions to basic measures of PAC address more complicated forms of dependence between phase and amplitude, as when peaks of amplitude occur at two or more separate phases. It was also noted that similar extensions might be developed for the distribution of phase of within bispectral estimators. A major advantage of standard bispectral estimators is that their capacity to reveal phase-amplitude dependence is not inherently limited
Algorithm
General principles of estimation are reviewed in Appendix B. The following section briefly describes the implementation with discretely sampled data used in the present examples. The “direct technique” of bispectral estimation starts with a windowed Fourier time-frequency decomposition, summing bispectral products over time. Although the technique is typically described in its application to the short-time Fourier transform, any time-frequency decomposition, , that can be expressed as a
Discussion
With respect to the relationship between PAC and the bispectrum, we have shown:
- 1.
Common measures of PAC based on analytic phase and amplitude are in fact bispectral estimators, and as such provide no unique information beyond what is recovered by standard bispectral estimators.
- 2.
PAC measures are severely biased with respect to the symmetry of the bispectrum and introduce artificial constraints on the range and resolution of the estimator.
These limitations provide a clear rationale for favoring
Acknowledgements
We wish to thank Phil Gander, Alex Billig and Matthew A. Howard, III.
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