Computational NeuroscienceToward a proper estimation of phase–amplitude coupling in neural oscillations
Introduction
The mammalian brain is a complex system with a distributed organization of sensory, motor, and executive computation centers across large areas of the cortex. While the distributed organization allows for parallel and specialized processing of information, it requires a mechanism for binding information from different computations into a coherent, unitary mental experience (von der Malsburg, 1981, Engel and Singer, 2001). The multipurpose, functional organization of local brain circuits also requires a mechanism for achieving dynamic, context-dependent cognitive and attentional control, a mechanism for the efficient routing of information between different brain computation centers, as well as a robust and efficient mechanism for coding the information in the dynamics of neural discharge (Phillips and Singer, 1997, Kelemen and Fenton, 2010). Various forms of neural synchrony, the coordinated synchronized activation of same-function cells and the active desynchronization of different-function cells have been proposed as this fundamental mechanism of neural computation (von der Malsburg and Schneider, 1986, Buzsaki, 2010). In the recent decade, phase–amplitude synchrony of field potential oscillations, the coupling between the phase of a slow oscillation and the amplitude of a faster oscillation, has received significant attention as a candidate synchronizing mechanism and is the subject of the present work.
In phase–amplitude coupling (PAC), the amplitude of a fast signal (e.g. gamma 30–100 Hz) is modulated by the phase of a slow signal (e.g. theta 5–12 Hz). This interaction is sometimes called “nesting” because the fast oscillation is precisely fitted within the cycle of the slower oscillation (Lakatos et al., 2005). The term phase–amplitude cross-frequency coupling (CFC) has also been used for this phenomenon, because the interaction happens between two distinct oscillatory bands (Bragin et al., 1995). This particular property makes PAC principally different from other synchrony measures such as amplitude synchrony (assessed by cross-correlation) or phase synchrony (assessed by phase locking statistics) because it reflects the dynamical relationship between two oscillations that are generated by distinct neurophysiological mechanisms. As the oscillations have different biophysical origins, the consequent PAC is not easily attributed to the spurious occurrence of synchrony caused by volume conduction, selection of reference or synchronized noise. The concept of a cross-scale organization of neural activity (Jensen and Colgin, 2007, Le Van Quyen, 2011) offers a possible neural mechanism for integrating information between several functionally distinct networks, to accomplish perceptual binding, selective attention, cognitive control and the recruitment of computational and representational cell assemblies. Neural activity in macroscopic (slow oscillations), mesoscopic (high frequency oscillations) and microscopic (single neuron activity) scales are braided together such that a progressively faster activity occurs within a specific, short time window of a slower activity. Indeed, several conceptual and theoretical frameworks have been proposed for the computational role of PAC (Canolty and Knight, 2010). Given the growing interest, and the substantial value in PAC as a mechanism for neural computation it is important for the broader neuroscience community to understand how to accurately measure and interpret PAC, and appreciate the limitations of the current methods.
This paper is written in two parts. The first part is an analysis of the standard approach to computing PAC. We examine the assumptions and by parametric analyses, identify the filter and temporal requirements for estimating PAC accurately. The second part introduces a novel approach to estimate, measure and characterize PAC. It operates on two time scales, one is global, and like traditional methods it is only robust when applied to long time series of data, on the order of many seconds. The approach also allows the characterization of PAC on short time scales, as short as a single oscillation.
Section snippets
Materials and methods
Multiple algorithms for quantifying PAC have been proposed (Fig. 1). The common starting point of all algorithms is an extraction of the phase and the amplitude information from the sampled signal x(t). This can be accomplished by band-pass filtering the signal into the bands of interest, for example theta 7–9 Hz and fast gamma 62–100 Hz followed by the Hilbert transform (Fig. 1A–C). There are other methods for extraction of phase and amplitude information, for example convolution of the signal
Results
Part I
Properties of standard PAC algorithms
We investigated the boundary conditions for the appropriate estimation of phase–amplitude coupling in LFP recordings. We demonstrated the importance of filter properties and analysis window size for estimating PAC. Because these properties define the time–frequency resolution of the analyses, it is necessary to consider them carefully. Phase filters should be narrow band to obtain meaningful phase information but not excessively narrow to distort non-sinusoidal shapes of rhythms such as
Acknowledgements
Supported by NIMH Grants R01MH084038 and R01MH099128.
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