(A-E) Power spectra and nested frequencies in spontaneous ECoG signals recorded from different arousal states in Patients #1-#5. Top: Raw power spectra for all electrodes plotted in log-log plots. Different colors plot different electrodes, the thick black trace plots the average across all electrodes. Middle: The fractal components of the power spectra, extracted by the CGSA method. The thick black trace plots the average across all electrodes. The low-frequency end (<0.1 Hz) and higher-frequency range (1-100 Hz) of the average power spectrum were each fit with a power-law function P(f) ∝ 1/f^β. The obtained exponents β are indicated in the graphs. The insets show the harmonic components extracted by the CGSA method. Results from all or a subset of electrodes are plotted on a linear scale; electrodes with pronounced harmonic activity were selected for presentation. Bottom: The percentage of electrodes with significant phase-amplitude cross-frequency coupling. Phase was extracted from 1-Hz-width bins with center frequencies from 1 to 20 Hz in 1-Hz steps. Amplitude was extracted from 5-Hz-width bins with center frequencies from 5 to 200 Hz in 5-Hz steps. The percentage of electrodes with significant MI Z-score (P < 0.05 after Bonferroni correction) is plotted as color for each frequency pair. (F) Electrode locations documented by plain X-ray pictures for each patient. R: electrodes over right hemisphere; L: electrodes over left hemisphere.

(A) Nested-frequency patterns from two example patients, #3 and #4. Phase was extracted from 1-Hz-width bands centered at 1, 6, 11, and 16 Hz. Amplitude was extracted from 5-Hz-width bands centered at 25, 50, 100, 150, and 200 Hz. For each frequency pair, the subplot shows a scatter-plot of all electrodes, each represented by one dot. The ordinate value plots the cross-frequency coupling strength as indexed by MI Z-score. The red horizontal line indicates significance level (P < 0.05 after Bonferroni correction). The abscissa value plots the preferred phase of the lower frequency, i.e., the phase of the lower-frequency fluctuation at which the amplitude of the higher frequency is largest. (B) For Patient #3, the preferred phase for each frequency pair in (A) and each electrode are plotted as color on a 2-D representation of the 8 × 8 electrode grid. The orientation and location of this grid on the cortical surface is shown in the top diagram. The six white cells in the grid are bad electrodes that have been eliminated from all analyses. The approximate locations of the central sulcus (CS) and Sylvian fissure (SF) are denoted on the bottom left grid. Squares in the right panel mark frequency pairs whose preferred-phase maps remained stable across waking and SWS (indexed by a significant spatial circular correlation; black: P < 0.002; red: P < 0.05, after Bonferroni correction for multiple comparisons).

Scale-free dynamics in spontaneous fMRI signals. Power spectrum of spontaneous fMRI signal was computed for 31 brain regions (10 pairs of homologous regions were each averaged together, for details see Table S2). (A) Normalized power spectrum (total power/variance = 1) for each brain region is plotted in a log-log plot. Line colors are grouped by cortical networks. Purple: default network; blue: attention network; orange: visual network; green: motor network; cyan: saliency network; red: non-neocortical group. (B) Each power spectrum in (A) was fit with a power-law function P(f) ∝1/f^β. The exponent β was entered into an ANOVA with brain network as the main factor. Error bars denote S.E.M. The effect of network was highly significant (F_5,15 = 5.05, P = 0.006). The two regions belonging to the saliency network, right frontoinsular cortex (R FI) and dorsal anterior cingulate cortex (dACC), were plotted separately for visualization, given the wide difference between their exponents. The non-neocortical group included the cerebellum, hippocampus and thalamus. Using Tukey/Kramer post-hoc test, significant differences were found between the default network and non-neocortical regions, and between the visual and non-neocortical regions. (C) Total fMRI signal variance and power-law exponent were correlated across brain regions.

Power spectra and nested-frequency patterns of simulated scale-free dynamics. All simulated time series were set to 512-Hz sampling rate, and subjected to the same analyses as the ECoG data. In each panel, the left graph shows the power spectrum plotted in log-log scales; the right graph shows cross-frequency coupling strength (MI Z-score) as color in the 2-D frequency space (color range from Z = 3.84 to Z = 20, all P < 0.05 after Bonferroni correction). (A) A white-noise time series following Gaussian distribution from a pseudorandom number generator (mean = 0, variance = 10). The inset in the left graph shows the distribution of the values in the time series. No significant cross-frequency coupling was found. This white-noise time series was used as input to models in (B)-(E). (B) Spectrally generated scale-free time series. The white-noise time series in (A) was filtered in the frequency domain by P(f) ∝ 1/f^β (β = 1.8), without altering the phase, and then inverse-Fourier transformed. This time series does not have nested frequencies. (C) A first-order autoregressive (AR-1) process: x(t) = φ x(t-1) + ε(t), where φ = 0.9 and ε(t) is the same white-noise time series as in (A). (D) Aggregate of three AR-1 processes , where φ1 = 0.1, φ2 = 0.5, φ3 = 0.9, and ε_i(t) is the same white-noise time series as in (A). Neither C nor D has significant nested frequencies. (E) A random-walk model: x(t) = x(t-1) + ε(t), where ε(t) is the same white-noise input as above. This random-walk time series does have significant nested frequencies across many frequency pairs. The inset shows, for one example frequency pair, the higher-frequency amplitude averaged at different phases of the lower frequency. (F) A random-walk model: x(t) = x(t-1) + ε(t), where ε(t) is a white-noise time series following Gaussian distribution generated using random numbers from physical source (atmospheric noise). This random-walk model does not have nested frequencies.

Stability of the 1/f^β power spectrum and nested-frequency patterns. (A) Power spectra from three example electrodes in Patient #3. The left, middle and right panels are from the entire awake record (83 min), and two randomly selected 20-sec segments, respectively. Note the difference in scales between left vs. middle and right graphs. (B) Phase-amplitude cross-frequency coupling for each of the three electrodes computed from the entire awake record. MI Z-score is plotted as color for each frequency pair. Only significant values (P < 0.05 after Bonferroni correction) are shown. (C) The raw data records for the two 20-sec segments. For each segment, 0-10 sec is shown on the top and 10-20 sec shown on the bottom. (D) Nested-frequency patterns for selected frequency pairs in electrode #64. Amplitude of the higher frequencies (5-Hz-width bands centered at 25, 50, 100, 150, 200 Hz) was averaged at different phases of the lower frequencies (1-Hz-width bands centered at 1, 6, 11, 16 Hz). Phase ±π corresponds to the trough (surface negativity), and phase 0 to the peak (surface positivity) of the lower-frequency fluctuation. Nested-frequency patterns from the two 20-sec segments (middle and right) are very similar to that from the entire awake record (left).

Task modulation of scale-free brain activity. (A-C) Power-law exponent changes during task performance in electrodes over task-relevant brain areas. Cued: visual-cued button press condition. Selfpaced: self-paced button press condition. LH/RH: the button press was performed by the left or right index finger. Six example electrodes from three patients are shown. Five additional electrodes with significant alteration of the power-law exponent during task performance are shown in Fig. S4. Statistical significance of the difference of power-law exponent between rest and task conditions was assessed by t-tests, with P-values shown in the graphs. In Patient #5, SWS power spectrum (orange) is presented for comparison, but not used for statistical analysis. (D) Emergence of an oscillation from scale-free brain activity during task performance. Results are from electrode #64 in Patient #3, same as electrode #64 in Fig. 2. Left: Power spectra from the spontaneous awake state (black), SWS (orange), and four trial types of the task. An oscillatory peak at ~8 Hz emerges during all four task conditions (blue arrow). Top right: Randomly selected 20-sec raw data record during task. The presence of an 8-Hz oscillation is readily seen (arrows). Bottom right: Nested-frequency patterns for selected frequency pairs during task-performance (averaged across all task blocks), which are very similar to those during the spontaneous awake state (shown in Fig. 2D). The locations of these electrodes are indicated by arrows in the plain X-ray pictures (A&B), reconstructed image from anatomical MR and CT scans (C) and the clinical diagram (D).

Scale-free dynamics in earth seismic waves (left column) and stock market fluctuations (right column). (A) Power spectra plotted in log-log plots. For seismic data, frequency is in Hz (cycle/sec). For stock market data, frequency is in cycle/day. The power-law exponent β for seismic and stock market data was 1.99 and 1.95 respectively. (B) Top: Phase-amplitude cross-frequency coupling assessed by MI Z-score, plotted as color in the 2-D frequency space. Only significant values (P < 0.05 after Bonferroni correction) are shown. Bottom: Example nested-frequency patterns for selected frequency pairs. Amplitude of the higher frequency was averaged at different phases of the lower frequency and plotted.

A general picture emerging from the present results. Different mechanisms in a variety of systems – including the human brain, earth seismic activity, stock market fluctuations, and simulated time series – can all give rise to scale-free activity exhibiting a 1/f^β power spectrum, but these different dynamics have different nested-frequency patterns. Hence, different nested-frequency patterns might be indicative of different underlying generative mechanisms, even when the gross power spectrum is similar.