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Edited by: Yan Mark Yufik, Virtual Structures Research, Inc., United States

Reviewed by: Adenauer Girardi Casali, Federal University of São Paulo, Brazil; Mehdi Adibi, University of New South Wales, Australia

*Correspondence: Aneta Stefanovska

†These authors have contributed equally to this work.

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Although neural interactions are usually characterized only by their coupling strength and directionality, there is often a need to go beyond this by establishing the functional mechanisms of the interaction. We introduce the use of dynamical Bayesian inference for estimation of the coupling functions of neural oscillations in the presence of noise. By grouping the partial functional contributions, the coupling is decomposed into its functional components and its most important characteristics—strength and form—are quantified. The method is applied to characterize the δ-to-α phase-to-phase neural coupling functions from electroencephalographic (EEG) data of the human resting state, and the differences that arise when the eyes are either open (EO) or closed (EC) are evaluated. The δ-to-α phase-to-phase coupling functions were reconstructed, quantified, compared, and followed as they evolved in time. Using phase-shuffled surrogates to test for significance, we show how the strength of the direct coupling, and the similarity and variability of the coupling functions, characterize the EO and EC states for different regions of the brain. We confirm an earlier observation that the direct coupling is stronger during EC, and we show for the first time that the coupling function is significantly less variable. Given the current understanding of the effects of e.g., aging and dementia on δ-waves, as well as the effect of cognitive and emotional tasks on α-waves, one may expect that new insights into the neural mechanisms underlying certain diseases will be obtained from studies of coupling functions. In principle, any pair of coupled oscillations could be studied in the same way as those shown here.

The complexity of the human brain makes its function exceptionally challenging to analyse and understand. Its electrophysiological activity emanates from the dynamics of large-scale cell ensembles (Traub et al.,

The different types of cross-frequency coupling (Jensen and Colgin,

Coupling functions describe in great detail the physical rule specifying how the interactions occur and manifest themselves. The coupling function as a whole can be described in terms of its strength and form. It is the functional form that has provided the new dimension and perspective on which we focus below. It probes directly the functional

Recent progress directed toward the extraction and reconstruction of the coupling functions between interacting oscillatory processes has led to a diversity of applications. These include chemical interactions (Kiss et al.,

We computed the wavelet transform (WT) (Kaiser, _{0} = 1. The power within each frequency interval was assessed by averaging the spectra over the corresponding frequency ranges.

Amplitude dynamics in living systems is often multidimensional, which can create complications in analysis. In contrast, the phase dynamics of a periodic process in such systems is describable in terms of a single-dimensional observable, which is usually much easier to detect and extract from data. It is well known that brain activity carries the signatures of several distinct neural oscillations that manifest themselves within characteristic frequency intervals (Buzsáki and Draguhn, _{i}(_{i}(_{i}(_{j}(τ)ξ_{k}(τ)…〉 = δ(_{ijk…}, where _{i, j, k..} gives the noise strength for the particular

The coupling functions _{i} act in such a way as to modify the natural frequency ω_{i}(_{i, 0} = 1 so that _{i, k}, scaled by _{i, k}(ϕ_{1}, ϕ_{2}, …, ϕ_{N}) act as base functions for the dynamical inference method.

Our aim is to reconstruct a dynamical model describing the interactions through the analysis of data, so that the model can then be used for extraction of the coupling functions. Our approach is based on the method of

Note that inference of cross-frequency couplings from the statistics of the coupled signals, e.g., through correlation, (bi-)coherence and Granger causality measures (Geweke,

In particular, coupling functions represent one type of dynamical mechanism and their inference yields the effective connectivity. More specifically, the form of the coupling function defines the functional law under which some input of the interactions (i.e., the mutual influence between the oscillations) is translated into an appropriate output. This is related, not only to the quantitative parameters of the net coupling strength i.e., net information flow, but also to how this information is functionally structured to give an effective mechanism. For example, as we will see below, the interactions can be such that the

A number of different techniques are available for estimating a model from data, based on different procedures and theories, and resulting in slightly different properties and characteristics. They include e.g., least-squares and kernel smoothing fits (Rosenblum and Pikovsky,

In what follows we use the dynamical Bayesian inference technique (Smelyanskiy et al.,

Using the inferred parameters we can calculate the coupling quantities and characteristics. The coupling functions _{i}(ϕ_{i}, ϕ_{j}, ϕ_{k}, …, ϕ_{N}) acting on the oscillator from each of the _{ϑ} generating the highest ρ carries dual information: the extent of the similarity (described by ρ itself) and the corresponding phase, given by ϑ. See Figure

The meaning of the polar similarity index. Two examples of coupling functions, plotted in blue, are compared with numerically-generated sinusoidal functions, plotted in red. The latter have been selected for being as similar as possible to the coupling functions: the only degree of freedom in the selection was the shift in phase (marked by the red dashed lines). The arrows in the polar planes in the top right corners have moduli equal to the similarity indices, and point to the corresponding phase values for:

In neuroscience, the cross-frequency analyses reported to date have mostly focused on the _{i}(ϕ_{i}, ϕ_{j}) into its _{i}(ϕ_{i}, ϕ_{j}, ϕ_{k}) one can decompose the self, direct, and common components depending on either one or two phase variables. Additionally, one can have the direct component

The multichannel EEG recordings analyzed in this work were downloaded from the Neurophysiological Biomarker Toolbox (NBT) dataset (O'Gorman et al.,

The cross-frequency intervals were extracted by a standard (FIR and no-phase-shift) filtering procedure. The boundaries for the conventional frequency intervals were: delta δ = 0.8–4 Hz, theta θ = 4–7.5 Hz, alpha α = 7.5–14 Hz, beta β = 14–22 Hz, and gamma γ = 22–40 Hz. Special care was taken to minimize cardiac components and powerline interference (Lehnertz et al.,

The extensive changes that the simple closing of the eyes triggers in the brain caught the attention of the very first electroencephalographers (Berger,

The δ-to-α interaction reflects how δ activity, associated with deep dreamless sleep (Feinberg et al.,

Cross-frequency interactions are usually mediated by the slower oscillations modulating the faster ones (Brunel and Wang,

In the light of this, and because of the crucial role that the α oscillation (Klimesch et al.,

Moreover, the multichannel recordings allowed us to investigate couplings between δ and α oscillations extracted from different probes, and hence to create connectivity maps illustrating how the δ-to-α modulation differs in the EO and EC states. The coupling strength was first quantified. Note that, in earlier work (Musizza et al.,

When applying non-linear analysis techniques, one should bear in mind that the linear properties of the signals, like autocorrelation or spectral features, are likely to affect the measure. To discriminate the genuine results from the ones happened by chance, one can apply surrogate testing (Theiler et al.,

In practice, when inferring couplings even from very weakly-coupled (or completely uncoupled) systems, the methods always detect some non-zero values of apparent coupling strength. Surrogate testing can then be used to establish the “zero-level” of apparent coupling corresponding to uncoupled signals. In order not to bias the threshold with effects due to inter-subject or inter-probe variability, we applied the surrogate techniques to the same signals for which the coupling was to be measured, and we therefore define different thresholds for different subjects, pairs of probes and states.

We generated the necessary surrogates by use of the phase-shuffling (PS) method (Schreiber and Schmitz,

The surrogate populations were tested for normality with the Shapiro-Wilk test, with the null hypothesis that the data come from a normal distribution of unknown mean and variance. The test rejected the null hypothesis at the 5% significance level in only 3% of the surrogates, and we therefore accepted the assumption of a normal distribution. Hence, we could test the coupling from the original signal by comparison with the significance threshold.

The non-parametric Wilcoxon paired test was used to determine the significance of differences between the EO and EC distributions for each frequency within the power spectra, for the averaged power within each frequency interval, for the coupling strength and for the similarity of coupling functions.

Figure

Spectral comparison between signals recorded during the eyes open (EO, red) and eyes closed (EC, blue) conditions, for all the probes from all the subjects.

Figure

Strengths of the couplings for

In order to evaluate the spatial patterns of significant coupling, the dots shown in Figure

Spatial distribution of the validated coupling strengths. The color codes indicates the number of subjects with a higher direct-coupling strength than the corresponding surrogate threshold for

The color-scale in Figure

The figure shows how, for EO, two inter-hemispheric occipital-to-frontal δ-to-α couplings were exhibited by 5 subjects and one inter-hemispheric temporal long range connection, plus two intra-hemispheric, were detected in groups of 6 or 7 subjects. For EC, besides being in higher number, the significant connections were detected especially from temporal to occipital locations, and from temporal to the parietal Pz (for groups of 10–12 subjects). A clear pattern of temporal-to-frontal coupling was also detected, for smaller groups (8–9 subjects).

To complement the coupling strength analysis, we now focus on the coupling functions themselves and discuss their unique properties. The results are summarized in Figure _{δ}-axis, but is mostly constant along the ϕ_{α}-axis. This reveals the underlying functional mechanism i.e., shows that, when δ oscillations are between π and 2π, the sine-wave coupling function is higher and the δ activity accelerates the α oscillations; similarly, when the δ oscillations are between 0 and π, the coupling function is decreased and δ decelerates the α oscillations. The highest acceleration i.e., the ridge of the 3D function plot is around 3π/2. The form of the coupling function of Figure

Examples of inter-subject averages of coupling functions between particular pairs of probes. They have been selected for generating

These qualitative observations can be quantified and presented in terms of the polar similarity index. In Figure

Physiological systems and processes, including neural oscillations, do not exist in isolation. They can be affected by a variety of external influences making their dynamics, to a greater or lesser extent, time-varying. In such cases, one can use the dynamical Bayesian method to infer time-varying neural dynamics, as demonstrated in Figure

Time-evolution of the δ-to-α coupling functions in the resting state. Middle panel: Time-evolution of the similarity index ρ_{α}(δ, α) for the EO and EC states of a single representative subject. Top panel: The δ-to-α coupling functions for EC inferred at four particular moments in time, as indicated by the arrows. Bottom panel: The δ-to-α coupling functions for EO inferred at four particular moments in time. Complementary 2D color-contour plots of the coupling functions are given in the top right-hand corner of their respective panels.

To investigate the quantitative statistics of each group of subjects we calculated the average values of the significant coupling strengths, with the corresponding surrogates' value subtracted, and the moduli of the polar similarity indices for the coupling functions of all the links for each subject. Then we compared statistically the distributions of these values for the two groups of subjects. To present the differences between the distributions visually, we use standard boxplots which refer to the descriptive statistics (median, quartiles, maximum and minimum).

The results in Figure

Differences in the δ-to-α coupling strength above surrogates

Much has already been done, mostly through fMRI and EEG analysis, to demonstrate the existence of resting state interactions, including the formation and dissolution of resting state functional network configurations around a stable anatomical connectivity (Berger,

As there were more significant couplings in the EC than in the EO state (Figure

The δ-to-α coupling functions had a specific shape, showing that the coupling is predominantly like a direct sine wave due to the δ influence, which accelerates and decelerates the α oscillations. Importantly, the form was similar for the EO and EC states (Figure

Because we reconstructed the form of the coupling functions, we were able to observe what they look like for both individual and averaged connections and subjects. Even though we found relatively similar forms of function, we also observed a certain degree of variability, both inter-subject variability (Figure

Finally, for the comparison of the EO and EC states (Barry et al.,

The assessment of neural coupling functions through the phase dynamics of interacting neural oscillations enables us to study their acceleration/deceleration, i.e., timing and coordination. The generalization to amplitude coupling functions is implicit. In such cases, one should be able to determine a plausible state model in relation to the dimensionality of the signals. Amplitude neural coupling functions can reveal the mechanism through which the strength and power of one neural oscillation are affected by the influence of the other oscillations.

Earlier effective connectivity methods for the inference of neural interactions have in principle contained coupling functions within their models of the interacting dynamical systems. The question we address here, in addition to presenting an efficient Bayesian method for determination of coupling functions, is that of how to

The pairwise investigation can further be generalized to higher degrees of network complexity (Kralemann et al.,

The time-varying form of the coupling functions (Figure

The limitations of the method should also be borne in mind. First, the whole analysis starts with the extraction of one-dimensional vectors of phases from data which probably have a non-trivial distribution of spectral content. Especially when the coupling mode is extracted from a single signal, the filtering must be done with extreme care: spillage between different frequency intervals, as well as splitting of one mode into two intervals, will result in an artificial “common” coupling. Whenever bandpass-filtering is involved, one should exclude the possibility of investigating high-to-low frequency coupling, because any modulation of the lower frequency due to the phase of the higher one will probably be erased from the filtered mode. In any case, these couplings will usually turn out to be insignificant compared to surrogates later in the analysis.

The windowed nature of dynamical Bayesian inference carries its own limitations, too, as the length of the window is fixed for every computation. This parameter must be chosen with care, and should be adjusted so as to include a sufficient number of periods of the lower frequency involved. We found that 6–10 periods is a reasonable lower limit for this number. Due to the uninformative flat prior used for the initial window, the resultant inference of the first window should be interpreted with care. Moreover, the signals' own particular features must also be taken into account: a high degree of time-variability would need a correspondingly shorter window for the dynamical inference to follow the evolution correctly. If the method is to be generalized for use other than with a phase dynamics model, one should be careful not to infer dynamics due to non-specific, non-stationary, processes instead of genuine coupling.

In conclusion, coupling functions bring a novel perspective to neuroscience that is unique in that it provides access to the functional

TS and VT did the analysis. TS and VT wrote the draft of the paper assisted by PM. AS planned and oversaw the entire enterprize. All authors edited the text and contributed ideas and content.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We are very grateful to Lars Michels and Simon-Shlomo Poil for their insightful comments on the results of our analyses of the eyes-open/eyes-closed data, and to them and the NBT research team for making the dataset available. Grateful thanks are also due to Klaus Lehnertz and Andreas Daffertshofer for valuable discussions. We are also grateful to Lall Hussain for pointing out the dataset, to Christopher Orrell for performing the initial analysis, and to Bastian Pietras and Federico Devalle for their useful comments on the manuscript.

The Supplementary Material for this article can be found online at: