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A critical note on the definition of phase–amplitude cross-frequency coupling

https://doi.org/10.1016/j.jneumeth.2011.08.014Get rights and content

Abstract

Recent studies have observed the ubiquity of phase–amplitude coupling (PAC) phenomenon in human and animal brain recordings. While various methods were performed to quantify it, a rigorous analytical definition of PAC is lacking. This paper yields an analytical definition and accordingly offers theoretical insights into some of the current methods. A direct PAC estimator based on the given definition is presented and shown theoretically to be superior to some of the previous methods such as general linear model (GLM) estimator. It is also shown that the proposed PAC estimator is equivalent to GLM estimator when a constant term is removed from its formulation. The validity of the derivations is demonstrated with simulated data of varying noise levels and local field potentials recorded from the subthalamic nucleus of a Parkinson's disease patient.

Highlights

► We give an analytical definition of phase–amplitude coupling. ► The coupling definition leads to a direct estimator. ► Other known estimates such as general linear model measure are evaluated. ► Derivations are validated on data acquired from a Parkinson's disease patient.

Introduction

Phase–amplitude coupling (PAC) between low and high frequency components of electrophysiological signals has received great interest in neuroscience (Jensen and Colgin, 2007, Jerbi and Bertrand, 2009, Canolty and Knight, 2010). Many diverse estimation methods were suggested and utilized to measure this phenomenon (Bruns and Eckhorn, 2004, Mormann et al., 2005, Canolty et al., 2006, Tort et al., 2008, Lakatos et al., 2008, Osipova et al., 2008, Cohen, 2008). Some of these methods were also evaluated and their performances were numerically compared in various studies (Penny et al., 2008, Tort et al., 2010, Onslow et al., 2011). However, a formal analytical definition of PAC is still lacking.

These methods were designed intuitively to capture and measure PAC, but we believe the usual strategy hitherto is standing on its head by asserting the methods without a proper prior definition of PAC itself as a universal phenomenon. A recent study by He et al. (2010) already gives clues of this universality as it addresses PAC in other man-made and natural processes such as Dow–Jones index and seismic waves.

From a signal processing point of view, the natural way would be first to define a universal PAC function and then consider appropriate methods that would capture the most accurate estimate possible. In this second step, the estimation method would mainly depend on the statistical properties and type of the data set being analyzed. Hence, our approach in this study aims to invert the research path taken so far by providing a rigorous analytical definition of PAC and then proceeding to consider the appropriate estimators for the quantification of it.

In this respect, we present a direct PAC estimator stemming from this definition and relate it theoretically to general linear model (GLM) estimator (Penny et al., 2008), modulation index (MI) with and without statistics (Canolty et al., 2006). We also discuss some other widely used estimators such as envelope-to-signal correlation (ESC) (Bruns and Eckhorn, 2004) and cross-frequency coherence (CFC) (Osipova et al., 2008) following the same context.

Throughout this paper estimates (not the true values) are denoted with a triangular hat while E, *, superscript T and ≔ stand for expectation operator, convolution operator, transpose and symbol of “defined as”, respectively.

Section snippets

Definition of PAC function

Let aH(n) be the amplitudes of a narrowband random vector zH(n) and let φL(n) be the phases of narrowband random vector zL(n), where zH(n) and zL(n) are bandpass filtered complex analytic representations from a common random signal or two separate signals such thatzL(n)=|zL(n)|eiφL(n)zH(n)=|zH(n)|eiφH(n)aL(n)|zL(n)|,aH(n)|zH(n)|wherezL(n)z(n)*hL(n)+iH{z(n)*hL(n)}zH(n)z(n)*hH(n)+iH{z(n)*hH(n)}HL(ω)1,ωLΔωLωωL+ΔωL0,otherwiseHH(ω)1,ωHΔωHωωH+ΔωH0,otherwiseωH>ωL+ΔωL+ΔωHwith hL(n)

Parkinsonian data example

We demonstrate an example with local field potential (LFP) data acquired from a Parkinsonian patient to illustrate the appropriateness of the given framework of aforementioned PAC estimation methods. There have been recent works (López-Azcárate et al., 2010, Özkurt et al., 2011) reporting an existent PAC between the phases of beta band (13–30 Hz) and the amplitudes of HFO (200–300 Hz) on the LFPs acquired from subthalamic nucleus (STN) of akinetic-rigid type Parkinsonian patients under no

Simulations

The Parkinsonian data example in the previous section validated our theoretical results. While patient data provide a realistic scenario including brain nonstationarity and nonlinearity, it also has the inherent limitation of unbeknownst ground truth such as precise coupling frequencies. Here we employ simulated data in order to verify these results and point to noise sensitivities of the methods in a controlled setting.

We would like to note that there are various ways of producing data

Conclusion

This study has attempted to present a different approach to the problem of PAC. Instead of jumping to the estimation of PAC measures without a formal description of the PAC function like the previous studies, we have provided an analytically rigorous definition and evaluated some estimators in the recent literature in this respect. Our approach has proven to give a clear insight into the nature of these estimators and relation of them to the true PAC function itself. Patient data example and

Acknowledgements

We thank Dr. Zhi-Hong Mao of University of Pittsburgh for his valuable suggestions and comments on the manuscript. We would like to thank Dr. Tayfun Akgül of Istanbul Technical University for his stimulating thoughts through a private communication. We are also thankful to the anonymous reviewers for their helpful and constructive comments. This work was supported by the ERANET-Neuron Grant “PhysiolDBS” (Neuron-48-013). The first author is supported by a grant from Scientific and Technological

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