Original Research
Quantifying phase–amplitude coupling in neuronal network oscillations

https://doi.org/10.1016/j.pbiomolbio.2010.09.007Get rights and content

Abstract

Neuroscience time series data from a range of techniques and species reveal complex, non-linear interactions between different frequencies of neuronal network oscillations within and across brain regions. Here, we briefly review the evidence that these nested, cross-frequency interactions act in concert with linearly covariant (within-frequency) activity to dynamically coordinate functionally related neuronal ensembles during behaviour. Such studies depend upon reliable quantification of cross-frequency coordination, and we compare the properties of three techniques used to measure phase–amplitude coupling (PAC) – Envelope-to-Signal Correlation (ESC), the Modulation Index (MI) and Cross-Frequency Coherence (CFC) – by standardizing their filtering algorithms and systematically assessing their robustness to noise and signal amplitude using artificial signals. Importantly, we also introduce a freely-downloadable method for estimating statistical significance of PAC, a step overlooked in the majority of published studies. We find that varying data length and noise levels leads to the three measures differentially detecting false positives or correctly identifying frequency bands of interaction; these conditions should therefore be taken into careful consideration when selecting PAC analyses. Finally, we demonstrate the utility of the three measures in quantifying PAC in local field potential data simultaneously recorded from rat hippocampus and prefrontal cortex, revealing a novel finding of prefrontal cortical theta phase modulating hippocampal gamma power. Future adaptations that allow detection of time-variant PAC should prove essential in deciphering the roles of cross-frequency coupling in mediating or reflecting nervous system function.

Introduction

Oscillatory activity is a pervasive feature of biological systems in general and nervous systems in particular. Neuronal oscillations reflect interdependencies between the relative timing (phase) and power (amplitude) of rhythmic activity in individual components of neurons, networks and systems. Oscillation cycle lengths range from milliseconds (e.g. 100–200 Hz hippocampal ripples), to seconds (e.g. cardio-respiratory rhythms in brainstem), to hours (e.g. circadian modulation of cortical excitability), with oscillations at these distinct frequencies arising from distinct cellular, synaptic and neuromodulatory processes (Buzsaki, 2006, Young and Eggermont, 2009). The powers of these diverse oscillatory frequencies can be dynamically modulated over a similar range of timescales, as can the coherence of a given frequency of oscillation between networks of neurons across numerous brain regions. This within-frequency coordination reflects and mediates functional connectivity, allowing specialized structures to both encode information independently and to interact selectively according to behavioural demands (Fries, 2009, Varela et al., 2001). However, rather than constituting independent communication channels analogous to AM radio signals, different frequencies of neuronal activity simultaneously interact with one another in nested, multiplexed signals. This cross-frequency coupling may reflect an important component of the synchronized neuronal activity believed to underlie brain function, and deciphering its mechanisms and roles necessitates increasingly sensitive and complex analyses and models.

Cross-frequency coupling of neuronal activity is evident in at least two forms: (1) phase synchrony, during which a consistent number of higher-frequency cycles occur within single cycles of a lower-frequency rhythm (Tass et al., 1998) and (2) phase–amplitude coupling (PAC), during which the phase of a lower-frequency rhythm modulates the amplitude of a higher-frequency oscillation. Although the extent to which phase synchrony and PAC reflect similar mechanisms and roles remains to be established, both types of coupling are evident in a range of EEG, electrocorticogram (ECoG), magnetoencephalogram (MEG) and local field potential (LFP) data recorded from a range of brain regions and species. The majority of phase synchrony examples to date stem from studies of human neocortex (Darvas et al., 2009a, Darvas et al., 2009b, Palva et al., 2005, Palva and Palva, 2007); in contrast, PAC is prevalent in both human and rodent neocortical, allocortical and subcortical regions, and currently represents a more experimentally tractable model of cross-frequency coupling (see Jensen and Colgin, 2007).

The archetypal example of neuronal PAC was first uncovered in the CA1 subfield of the hippocampus, where LFP recordings reveal a consistent, cyclic variation of gamma-frequency (30–100 Hz) power with concurrent theta-frequency (5–10 Hz) phase (Bragin et al., 1995; see Fig. 1 for example). Since the connectivity and activity patterns of hippocampal excitatory and inhibitory principal neurons and interneurons are increasingly well understood (Klausberger and Somogyi, 2008), the hippocampus therefore presents a valuable model network in which to dissect PAC’s mechanisms and roles. For example, Wulff et al., 2009 used a genetically altered mouse line lacking functional GABA-A receptors on a subset of interneurons to suggest a role for rapid, synaptic feedback inhibition in shaping hippocampal PAC.

Models have attempted to link hippocampal theta–gamma PAC with CA1-dependent memory processing, whereby subsets of hippocampal units co-active during individual gamma cycles are recruited in consistent, sequential order dependent upon theta cycle phase (Lisman and Buzsaki, 2008, Lisman and Idiart, 1995; see also Fuentemilla et al., 2010). There is also evidence to suggest that hippocampal pyramidal cells differentiate in the preferred phase at which they fire in relation to theta–gamma PAC activity. This could allow for the simultaneous implementation of multiple coding schemes for memory items, stored in both a sequential and non-sequential context (Senior et al., 2008). Recent data from LFP recordings from both rat and human hippocampus provide evidence for a functional role of theta–gamma PAC in mnemonic processing (Axmacher et al., 2010, Shirvalkar et al., 2010, Tort et al., 2009), but these hippocampal models and examples also reflect a broader hypothesis of PAC function: hierarchies of inter-locked oscillatory frequencies allow ensembles of anatomically localized neurons co-active on short timescales (i.e. within higher-frequency cycles, e.g. Siegel et al., 2009) to be temporally aligned (‘bound’) across longer timescales and anatomical distances by lower-frequency modulation (see Sarnthein et al., 1998). VanRullen and Koch (2003) posited a PAC-based model of alpha–gamma-frequency interactions mediating perception, and a recent example of data supporting a functional role for hippocampal PAC showed that separable bands of the gamma-frequency range coincide with different phases of the theta rhythm in CA1. It was suggested that PAC of gamma at these different frequencies reflects the dynamic influence of afferent inputs from CA3 and entorhinal cortical regions to CA1 during different phases of the theta cycle (Colgin et al., 2009); as such, similar phenomena may reflect interactions in other systems of connected brain regions.

Beyond the hippocampal formation, PAC phenomena have been reported in sensory, frontal and parietal human neocortex during a range of auditory, linguistic and working memory tasks (Canolty et al., 2006, Osipova et al., 2008, Sauseng et al., 2008), plus in monkey auditory and visual cortices (Lakatos et al., 2007, Lakatos et al., 2008) and rodent olfactory bulb (see Rojas-Libano and Kay, 2008). The oscillation frequencies demonstrating PAC in these various systems and behaviours are by no means restricted to theta–gamma cross-frequency interactions, but also encompass delta (1–4 Hz) and alpha (8–12 Hz) rhythms, though variable definitions of frequency band labels can confound comparisons across species. Importantly, some studies suggest a continuous hierarchy of frequencies, with delta modulating theta which in turn modulates gamma (Lakatos et al., 2005); simultaneous PAC across such a range of timescales raises important questions about the underlying network structure giving rise to the phenomenon, as well as the relative functional contributions of oscillations at distinct frequencies.

Importantly, PAC does not occur only within functionally specialized brain regions, but also across functionally related brain regions. For example, hippocampal–striatal PAC is dynamically modulated alongside behavioural task demands in rat (Tort et al., 2008), and hippocampal theta phase can also modulate neocortical gamma power (Sirota et al., 2008). Like within-frequency coherence, PAC is therefore well placed to underpin or reflect the temporal coordination of neuronal networks across distributed brain regions, though the basic features of excitatory and inhibitory network connectivity that give rise to PAC of different frequencies and in different anatomical regions have not yet been established.

Fig. 1 shows 1 s of LFP data recorded from hippocampal CA1 of a freely behaving rat; whilst PAC is clearly evident upon visual inspection, these data demonstrate some central challenges that arise when attempting to quantify its extent and nature. These include, but are not limited to:

  • i.

    a variable signal-to-noise ratio between the amplitudes of both phase modulating and amplitude-modulated signal;

  • ii.

    estimating the statistical significance of any PAC present, given that PAC may arise by chance in signals simultaneously containing power in low and high frequencies;

  • iii.

    quantifying the time-variant dynamics of PAC, which is non-stationary and may come and go from one lower-frequency cycle to the next;

  • iv.

    limits imposed by the length of available data series, for example precluding analyses of low frequency signals;

  • v.

    establishing whether PAC applies to all frequencies present, or is restricted to specific pairs of modulating and modulated oscillations;

  • vi.

    determining whether amplitude-modulated power varies continuously with modulating phase, or whether step-like changes in amplitude underlie PAC.

Robust, sensitive analysis methods with sufficient temporal resolution and statistical power are therefore essential for the study of PAC, particularly in limited and/or noisy neurobiological data. A number of methods have been published in recent years, some of which have been compared and reviewed in different combinations elsewhere (Cohen, 2008, Penny et al., 2008, Tort et al., 2010). Here, we briefly review three of the available PAC analysis methods; we have standardized the algorithms for their implementation to enable an objective, quantifiable comparison of their advantages and limitations, which are demonstrated and discussed in relation to both simulated and real LFP data.

Section snippets

Notation

Throughout the following sections we denote the raw signals as Xph(t) and Xamp(t), corresponding to the signal assumed to contain the lower, modulating frequency and the signal assumed to contain the higher, modulated frequency respectively. In analyses used to investigate the phase of a slower oscillation modulating the amplitude of a faster oscillation within the same signal, Xph=Xamp. Each of the PAC detection measures relies on filtering one or both of these signals for particular

Results and discussion

We first tested if the methods detect statistically significant PAC at a fixed combination of frequencies (4 Hz modulating 60 Hz was used throughout simulations, modeling the theta–gamma PAC reported in a range of neural data) in artificial data that did not contain PAC. As we expected all methods detected significant PAC on average on 5% of simulated signals, and there were no statistically significant differences between the methods.

Unsurprisingly, consistent detection of statistically

Conclusions

We present adaptations to ESC, MI and CFC methods that allow estimates of statistical significance of PAC, thus addressing important concerns that the phenomenon may reflect passive spectral properties of mixed-frequency signals, rather than underlying neurophysiology. These statistical methods are critical in consistently quantifying PAC levels, particularly when comparing different experimental, physiological and pathological conditions. For example, quantifying the impact of pharmacological

Acknowledgements

Our thanks to Ole Jensen of the F.C. Donders Centre for Cognitive Neuroimaging for generous sharing of CFC Matlab code. Related tools are freely available at: http://megcommunity.org/index.php?option=com_content&view=article&id=30&Itemid=36.

LFP data were kindly provided by Hannah Chandler (University of Bristol, Department of Physiology and Pharmacology); MWJ thanks the MRC, BBSRC and The Wellcome Trust for financial support of related experimental work. ACEO is supported by the EPSRC grant

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