Neural Cross-Frequency Coupling Functions

Abstract

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.

Keywords: EEG; coupling function; cross-frequency coupling; dynamical Bayesian inference; effective connectivity; eyes-open; neural oscillations; resting brain.

Figure 1

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: (A) a coupling function with high similarity (ρ = 0.82) and (B) one with a low value (ρ = 0.23). A complementary 2D color-contour plot of the coupling function is given in the bottom right-hand corner of each panel.

Figure 2

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. (A) Paired statistical comparison between the inter-probe average power spectra from each subject in EO and EC, respectively. The lines show inter-subject medians, and the ranges of significance are shaded pink for p < 0.05, orange for p < 0.01 and yellow for p < 0.001. (B) Boxplots for the average power within the five frequency intervals. Diagonal lines symbolize statistical analyses pairing corresponding values for every probe and subject, and follow the changes in the medians. The p-value is indicated in each case. Note that the significance of the power in (A) corresponds closely to the boundaries of the α interval, and that the power in (B) increases significantly between EO and EC for every frequency band.

Figure 3

Strengths of the couplings for (A) EO (blue) and (B) EC (red) for all the subjects, shown as consecutive intervals on the x-axes. Only values higher than the corresponding PS surrogate threshold are shown. Couplings are selected when their strengths are higher than the mean+2STD of the corresponding surrogate distribution (gray shading). Horizontal lines indicate the average values of the surrogates and of the validated couplings (color-scheme as explained above).

Figure 4

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 (A) EO and (B) EC. Note the different scalings of the two color-bars, used for clarity.

Figure 5

Examples of inter-subject averages of coupling functions between particular pairs of probes. They have been selected for generating (A,C) the highest and (B,D) the lowest similarity indices, as shown. The arrows in the polar plots in the top right corners of each panel indicate the similarity indices for the averaged coupling functions, while the dots indicate the similarity indices for individuals. Note that in B and D the arrows are of negligible dimension. A complementary 2D color-contour plot of the coupling function is given in the bottom right-hand corner of each panel.

Figure 6

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.

Figure 7

Differences in the δ-to-α coupling strength above surrogates (A) and in the similarity of form of the δ-to-α coupling functions (B) for the two groups of subjects with EO and EC. The p-values indicated within each panel represent the statistical differences between the EO and EC states. Whiskers indicate ±2.7 standard deviations of the distribution.