Example of multiplicative frequency coupling. A, A “carrier” signal x̃(t, f1) = exp(i(2πf1t + φ1(t, f1))) at 80 Hz. The right side shows the time series of the real part of the carrier (thick line) and the instantaneous phase φ1(t, f1). The left side shows the associated power spectrum of the real part of the carrier signal. B, The 10 Hz real part of the signal component x̃(t, f2) = exp(i(2πf2t + φ2(t, f2))) with instantaneous phase φ2(t, f2) (right) and associated power spectrum (left). C, Real component of exp(i(2πf1t + φ1(t, f1)))exp(i(2πf2t + φ2(t, f2))), with a phase modulated by the components x̃(t, f1) and x̃(t, f2) [i.e., φ(t, f1 + f2) = φ1(t, f1) + φ2(t, f2) (left) and power spectrum (right), which peaks at 90 Hz]. D, Trial-wise computation of the bPLV. Left, The green arrows show the phases φ1(t, f1) and φ2(t, f2) in signal 1; the red arrows show the compound phase, φ1(t, f1) + φ2(t, f2), in signal 1; and the blue arrows show the phase at frequency f1 + f2 in signal 2. Right, The gray arrows show the trial-wise phase difference between the compound phase in signal 1 and the phase in signal two. The black arrow shows the mean phase difference across trials. The length of the black arrow is the bPLV.
Common phase modulation in the motor cortex across subjects. A, Electrode coverage. Blue, Subject 1; green, subject 2; red, subject 3; yellow, subject 4. Note that the positions for subject 1 have been projected from the right to the left hemisphere. The black markers indicate the seizure onset zone as identified by the clinician. B, Significant phase–phase interactions were found in all subjects (S1, light blue; S2, green; S3, red; S4, yellow) after clustering by spatial proximity. Brodmann areas are color coded as follows: BA1, brown; BA2, green; BA3, blue; BA4, pink; BA6, purple. Spatially consistent interactions across subjects form a cluster from premotor sources (BA6) to peri-Rolandic primary motor targets (BA4). Interactions are indicated by color-coded arrows, in which the point of the arrow indicates the direction of interaction.
Clustering scheme to find spatially consistent interactions across subjects. A, Initially, a large number of interactions between grid electrodes with uncorrected significances (p < 0.05) is found for each subject. B, Interactions are restricted to electrodes that lie in Brodmann areas 1, 2, 3, 4, and 6 (red circles). C, This procedure is applied to each subject. Exemplary, we show interactions from two subjects (black and green) overlaid on the same grid. Note that the individual grids can have different electrode numbering schemes, and thus electrodes with different indices can cover different Brodmann areas in different subjects. D, Interactions are grouped into clusters by spatial proximity. Interactions for which the beginning and end points are within a 2.8 cm range (the farthest distance between two electrodes in an 8-electrode neighborhood), indicated by the transparent blue circles, and which cover a distance >2.8 cm are grouped into the same cluster. E, The final cluster configuration.
Temporal evolution of the bPLV. Gray, Time intervals used as baseline for statistical testing. Green, Time interval used to compute the mean bPLV during movement. Blue, Smoothed temporal evolution of the bPLV. Red, Average thumb movement as recorded by the data glove. Black, Mean premovement bPLV. Numbers above the bar graphs indicate Bonferroni-corrected p values for movement-related increase in coupling.
Frequency–frequency maps of the mean bPLV over the movement segment [0 s −1 s] for channel interactions between premotor-to-motor channel pairs for all subjects. Note that there is a strong similarity for S2 and S3 in these maps. The common interaction frequency located at approximately (10, 80 Hz) is indicated by the white circles. The Bonferroni-corrected significance levels, in which we correct for the number of frequency bins, are indicated by the white (p ≤ 0.2) and black (p ≤ 0.05) contour lines.
Interaction cluster with revised position for the interaction in S1. The same color coding as in Figure 2 is used. The mean Talairach coordinates across all subjects were at x = −58, y = −14, z = 39 (source site), and x = −38, y = −31, z = 63 (target site).
Mean mutual information (over trials) for each subject between the phases φ1(t, f1) and φ2(t, f2) (white bars) in the premotor region during movement and mean mutual information between the phase in the motor region, φ(t, f1 + f2), and the sum of the phases in the premotor region (gray), φ1(t, f1) + φ2(t, f2), during movement. The error bars indicate the SEM. A shows the mutual information for the results from the initial analysis, and B, for the post hoc results.
Average integrated band power maps for S1–S4 for the interaction frequencies (10, 80) to 90 Hz. Values for each electrode have been smoothed by a Gaussian kernel to produce spatially smooth maps (Miller et al., 2007a). The band power for each band has been integrated over the movement time period [0 s −1 s] and normalized with respect to the baseline period [−1.5 s to −0.5 s]. The green arrows indicate the nonlinear interaction for premotor-to-motor electrode pairs in S1, S2, S3, and S4. The frequency bands were selected based on the interacting cross-frequencies as identified by the pBLV increase during movement.