Figure 7.
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/fβ (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 εi(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.