Fig 1.
Examples of CFC in eight simulated signals.
The plots on the left of each column depict a trace from the original (background) signal used for the simulation (black), the added Gaussian train (red), and the compound signal after adding the potentials (blue). The blue trace is therefore the sum of the black and the red ones. Note the temporal jitter in the Gaussian periodicity (i.e. variability of the inter-peak interval). The image on the right in each column shows the comodulogram for the compound signal (blue to red color scale corresponds to 0 to 1 in all panels). Clusters of significant CFC in the comodulograms are marked with a black outline (p<0.01). Note that values in the bottom triangle of each plot are not shown as CFC is invalid for frequencies-for-amplitude lower than the frequency-for-phase. The left column shows results for EEG-based simulations with a 100-ms mean inter-peak interval; the right column shows results for pink-noise-based simulations with a 166-ms mean inter-peak interval. It can be easily seen that the periodicity determines the frequency-for-phase of the CFC (10Hz and 6Hz, respectively), as well as its harmonics. A, D, the raw background traces with no added spikes. B, F, added spikes with an amplitude of 3 STDs and a width of 10 ms (full-width at half-maximum). C, G, Same as B,F but with an amplitude of 1.5 STDs. Note the diminished magnitude of the effect. D, H, Same as B,F but with a width of 20 ms. Note the lower frequency-for-amplitude range for which CFC occurs.
Fig 2.
Preferred frequency-for-amplitude as a function of Gaussian spike width.
Shaded area reflects one standard deviation of the preferred frequency across 100 iterations (see Materials and Methods).
Fig 3.
Phase-shift of the background EEG signal after the addition of periodic potentials.
The signal was generated by adding a Gaussian spike train with 100-ms mean inter-peak interval, 15-ms width and amplitude of 1.5 STDs to the background EEG trace. A, Distribution of 10Hz phase values at the time points where Gaussian peaks occurred. Blue: 10Hz phase extracted from the background EEG signal before adding the Gaussians train; red: phase extracted from the compound signal. It can be seen that while the spikes were inserted at random points of the 10Hz cycle of the background signal, the same time points show a prominent phase-concentration around the cycle peak (phase 0) once the Gaussians were integrated with the signal. That is, the 10Hz cycle tends to align with the transient periodicity such that a phase-amplitude relation emerges. B, Example of alpha phase change of the EEG background signal in a few consecutive cycles, before and after the introduction of the spike train. Blue trace represents the original signal, red trace represents the compound signal. Note that as the Gaussian peak falls away from the original alpha peak, the new alpha cycle will shift farther in phase (compare the two arrows). C, This rearrangement of the alpha phase can occur without any significant change to the power spectrum.
Fig 4.
CFC for periodic vs. non-periodic spike trains.
A-C. phase-amplitude comodulograms for semi-periodic sharp waveforms (100 ms mean interval, 10 ms jitter, 15 ms Gaussian spike width). The spike amplitude is set to 1, 2, or 5 standard deviation of the background signal from the left (A) to the right (C) panel. Clusters of significant CFC in the comodulograms are marked with a black outline (p<0.01). D-F. The same comodulograms for a non-periodic signal. It can be seen that in the absence of periodicity, the CFC effect occurs on a wider range of frequencies-for-phase, and the CFC magnitude is smaller.
Fig 5.
Effect of periodicity and mean interval duration on CFC.
A. The magnitude of the CFC diminishes with decreasing periodicity (increased randomness). The level of CFC was defined as the maximal CFC magnitude for a 10Hz frequency-for-phase across all frequencies-for-amplitude in a 1-100Hz range. Randomness = 1: fully random; randomness = 0: completely periodic. B. CFC level depends on mean interval in a random sequence. Note that the optimal interval and exact shape of the curve will depend on the spike width (15ms Gaussian width in this example).
Fig 6.
Waveform-dependent CFC in empiric signals.
Left column: example traces of each analyzed signal. The blue trace is the raw trace, and the red is the low-frequency band-pass filtered component. Black trace in the bottom two rows shows the artificial data added to induce CFC. Middle column: comodulogram showing the PLV value for each frequency-pair. Right column: Phase-phase dependency: histogram of high-frequency phase values (in radians) occurring at the preferred phase of the low-frequency cycle (i.e. the phase for which high-frequency amplitude is maximal) for the basic rhythm frequency (10Hz for A,C,D, 17Hz for B) with its first harmonic (20Hz for A,C,D, 34Hz for B). The histogram is repeated twice along its horizontal axis to improve readability. A, Intracranial recording of mu oscillations (~10Hz). B, Intracranial recording of beta oscillations (~17Hz). Note the non-sinusoidal single sharp peaks driving the oscillatory component in A and B, as well as the harmonics in the comodulogram and the non-uniform phase-phase relation between the low- and high-frequency components. C, CFC induced by superimposing a periodic Gaussian train on a background EEG trace, featuring no CFC initially. D, "Coupled-sources" CFC induced by modulating the amplitude of the 30-100Hz band by the instantaneous 10Hz phase. Note the similar behavior of A, B compared to C and not to D.
Fig 7.
Statistical significance of phase-amplitude and phase-phase coupling for low-amplitude transients.
A, Each pixel in the image represents the average p-value, across 100 repetitions, for a phase-amplitude CFC effect driven by adding a Gaussian spike train (inter-spike interval: 100 ms) of the given amplitude and width, to a background EEG signal featuring no initial CFC. The color scale corresponds not to the actual p-value, but to n = 0.05/p, i.e. the number of Bonferroni multiple-comparisons the effect “survives”, resulting in a simple linear scale of significance. The white outline indicates the cluster of significant pixels (n> = 1). Importantly, note that significant CFC occurs even for very low-amplitude transients (under 1 standard deviation). It can also be seen that the magnitude of CFC depends on the width of the transient waveform: very narrow transients do not contain enough energy to drive the phase of the slow-frequency component, while wide transients contain a weaker high-frequency component. B, Significance of phase-phase coupling between 10Hz and 20Hz is shown for the same range of transient amplitudes and widths. Since a parametric significance test was used, p-value tended to be very small and so the logarithmic scale n = log10(0.05/p) was used (such that n = 0 corresponds to p = 0.05, n = 1 to p = 0.005, etc.). White border indicates area where p<0.05. Note that the area indicated as significant in this analysis does not fully overlap that of the phase-amplitude coupling analysis (Fig 7A).