Fig 1.
The use of the temporal derivative of EEG (EEGd) and DCT-phase-scrambled controls for investigating the dynamic control of synchronization of oscillatory activity.
A. An example of 1 sec EEG waveform (black) and its DCT-phase-scrambled control (blue) at FCz from one participant. B. The mean spectral-amplitude profiles of the full length (~5 min) versions of the same EEG data (black) and its DCT-phase-scrambled control (blue), with the fast Fourier transform (FFT) computed on each consecutive 5 sec waveform and then averaged, plotted in a log-log format. C. The temporal derivatives, which we call EEGd, of the example EEG waveform (black) and its DCT-phase-scrambled control (blue) shown in A. D. The mean spectral-amplitude profiles of the full length (~5 min) versions of the same EEGd data (black) and its DCT-phase-scrambled control (blue), with the fast Fourier transform (FFT) computed on each consecutive 5 sec waveform and then averaged, plotted in a semi-log format. For B and D, the shaded areas represent ±1 standard error of the mean based on the FFTs computed on multiple 5 sec waveforms. The units are arbitrary (a.u.).
Fig 2.
Probability distributions of EEG spectral power as deviations (in ratio) from exponential fits to phase-scrambled controls.
Spectral-power probability distributions (using 500 bins) are shown for the actual EEG data and their phase-scrambled controls for representative frequency bands, θ (6.0 Hz), α (10.5 Hz), β (14.6 Hz and 20.6 Hz) and γ (31.9 Hz, 40.0 Hz, and 50.1 Hz), color-coded from cooler to warmer. On the x-axis, spectral power is normalized to the median power per frequency per site per condition per participant. The probability distributions are averaged across sites and participants. The y-axis indicates spectral-power probability relative to the exponential fits to the corresponding phase-scrambled controls. For example, the value 1 indicates that the probability of a specific spectral-power value was as predicted by the exponential fit, 2 indicates that the probability was twice predicted by the exponential fit, etc. Note that all probability distributions for the phase-scrambled controls (thinner lines) were exponential, tightly conforming to the line of y = 1. The five panels show the probability distributions for the five conditions: ~5-min rest with the eyes closed (Rest EC), its replication (Rest EC rep), ~5-min rest with the eyes open in dark (Rest EO DK), and the earlier and later ~5-min viewing of a silent nature video (Nature video). The distributions for the actual EEG data (thicker lines) deviate from the exponential form in a characteristic U-shaped manner with elevated occurrences of the lowest and highest ranges of power. The shaded areas represent ±1 standard error of the mean with participants as the random effect.
Fig 3.
Relationship between log average spectral power ln(<SP>) and entropy Sobs for d = 3 sec intervals.
Each set of three panels shows the data for a specific condition: ~5-min rest with the eyes closed (Rest EC), its replication (Rest EC rep), ~5-min rest with the eyes open in dark (Rest EO DK), and the earlier and later ~5-min viewing of a silent nature video (Nature video). For each condition, the ln(<SP>)-Sobs pair was computed for each non-overlapping d = 3 sec interval per frequency per site per participant. Upper-left panels. 2D-density plots of all ln(<SP>)-Sobs pairs for the phase-scrambled controls. The linear fits (gray dashed oblique lines) indicate the line of maximum entropy indicative of spectral power fluctuations that maximize entropy for a given value of average spectral power (see text). Lower-left panels. 2D-density plots of all ln(<SP>)-Sobs pairs for the actual EEG data. Note that the distributions follow the line of maximum entropy (the gray dashed oblique lines) defined by the phase-scrambled controls. Main panels. Re-plotting of the 2D-density plots for both the phase-scrambled controls and the actual EEG data after aligning the phase-scrambled 2D-density plot for each frequency, site, and participant at its center at (0,0) and equivalently translating the corresponding actual-data density plots. The 2D-density plots for the phase-scrambled controls are shifted upward to avoid overlaps with those for the actual EEG data. The centering shows that the dynamic ranges of average spectral power (per d = 3 sec interval) were substantially extended along the line of maximum entropy (the gray dashed oblique lines) for the actual EEG data relative to their phase-scrambled controls in all conditions. This pattern was observed for all representative frequencies (Fig 4) and all participants (S1 and S2 Figs). Thus, on the timescale of up to about 3 sec, spectral power appears to be controlled in such a way that the dynamic ranges are substantially extended (relative to stochastic dynamics) while tightly conforming to the line of maximum entropy. All panels. Density is color-coded as percentile so that confidence intervals can be inferred. The extreme ranges of spectral power, ln(<SP>) < 0 and ln(<SP>) > 5.5 were excluded from the computations of the line of maximum entropy and the centered 2D-density plots (the main panels) to avoid the binning-related distortions (see text).
Fig 4.
The same as the main panels in Fig 3, but the centered relationships are shown separately for the representative frequencies (rows) and conditions (columns).
The phase-scrambled distributions are shifted upward to avoid overlaps with the actual-data distributions, and the gray dashed oblique lines indicate the line of maximum entropy. Note that the dynamic ranges of average spectral power were substantially extended along the line of maximum entropy for the actual EEG data relative to their phase-scrambled controls for all representative frequencies in all conditions.
Fig 5.
Probability distributions of entropy Sobs relative to the line of maximum entropy for the actual EEG data and their phase-scrambled controls as a function of interval duration d.
A. Probability distributions for the phase-scrambled controls (shaded areas) and actual EEG data (solid outlines), with the line of maximum entropy labeled as 0 on the x-axis and the negative tails of the actual-data distributions toward lower entropy shown as solid areas. The rows correspond to the five conditions and the columns correspond to the representative interval durations d (sec). The x-axis of each probability distribution has been normalized to the standard deviation of the corresponding phase-scrambled-control distribution. Note that up to about d = 3 sec (highlighted with a rectangle), the distributions for the actual EEG data and their phase-scrambled controls virtually overlap. B. Proportions of lower-entropy intervals (PrLEI) for the actual EEG data relative to their phase-scrambled controls (approximately the area proportion for the solid-colored negative tails shown in A) as a function of interval duration d (sec). This measure indicates the proportions of d (sec) intervals for which the actual EEG data yielded lower entropy than predicted by the line of maximum entropy. The circular symbols connected with thick lines indicate the median PrLEI values with the five conditions color-coded as in A (the black dotted lines indicating the replication of the rest-with-the-eyes-closed condition and the blue dotted lines indicating the later viewing of the nature-video condition) with the thin dotted lines showing the PrLEI values for the individual participants. Note that for the interval durations up to about d = 3 sec the actual EEG data closely followed the line of maximum entropy with less than ~5% deviations (in median PrLEI values) across all conditions, suggesting that neural dynamics on the spatial-scale of EEG current sources generally maintain maximum entropy up to the timescale of a few seconds (see text).
Fig 6.
Proportions of lower-entropy intervals (PrLEI) for d = 3 sec for the actual EEG data relative to their phase-scrambled controls as a function of frequency and site.
PrLEI indicates the proportion of d (sec) intervals for which the actual EEG data had lower entropy than predicted by the line of maximum entropy. A. PrLEI as a function of frequency. The thick lines indicate the median PrLEI values with the thin lines showing the values for the individual participants. The rows correspond to the five conditions. Note that the median PrLEI values were low regardless of frequency or condition. B. PrLEI as a function of site. The rows correspond to the five conditions as in A. The mean PrLEI values were globally low across all sites and conditions. The mid-central-posterior region (highlighted with dotted circles) yielded particularly low PrLEI values in the eyes-open conditions (the rest-with-the-eyes-open-in-dark and nature-video conditions) (the lower three rows). C. Same as B, but the data from each participant were z-transformed across sites to quantify the consistency of regional deviations of PrLEI values from the spatial average as t values (with |t|>3.95 for Bonferroni-corrected 2-tailed significance at α = 0.05). Cooler colors indicate regions with lower-than-average PrLEI values while warmer colors indicate regions with higher-than-average PrLEI values. The t-values confirm that the PrLEI values were consistently low in the mid-central-posterior region in the eyes-open conditions (see B). Further, consistent elevations in the PrLEI values (though still low with the means of less than 8.7% for all sites and conditions) occurred in areas surrounding the mid-central-posterior region, particularly in the right-lateral region in the eyes-closed conditions (the upper two rows).