Characteristics of Spontaneous Anterior–Posterior Oscillation-Frequency Convergences in the Alpha Band

Oscillation-frequency convergences are not mere coincidences

We first confirmed that anterior–posterior oscillation-frequency convergences occurred over and above what would be expected by random coincidences. We did so by comparing the total instances of frequency convergence of each size (one through eight) between the actual and time-shuffled frequency-convergence matrices. The instances of frequency convergence of four or more sites were consistently elevated in the actual data (red curves) relative to the time-shuffled controls (black curves) for all participants (Fig. 3A presenting the rest-with-eyes-closed condition as a representative example; see Extended Data Fig. 3-1 for the results from all behavioral conditions). Given that the total number of instances are plotted in a log scale, the elevations were substantial (by about an order of magnitude for convergence sizes of seven and eight). Thus, anterior–posterior oscillation-frequency convergences are coordinated rather than coincidental.

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Figure 3.

Instance, probability, and oscillatory power associated with oscillation-frequency convergence. Data for the rest-with-eyes-closed condition are presented as a representative example. A, Total instances of anterior–posterior oscillation-frequency convergence as a function of the size of frequency convergence, for the midline (left), left-hemisphere (middle), and right-hemisphere (right) routes. The actual data are shown in red, and the time-shuffled controls are shown in black. Each curve represents a participant. Note that the instances of frequency convergence of four or more sites were elevated in the actual data (red) relative to the time-shuffled controls (black) for all participants. Data for all behavioral conditions are presented in Extended Data Figure 3-1. B, The probability of k convergence (oscillation-frequency convergence across k sites) given k−1 convergence, that is, the probability of growing in convergence from k−1 to k, P(k|k–1) , as a function of k, for the midline (left), left-hemisphere (middle), and right-hemisphere (right) routes. P(k|k–1) for the actual data (red) are shown with the corresponding time-shuffled controls (black) as well as P(k|k–1) expected if oscillation-frequency convergences occurred randomly (gray). C, P(k|k–1) for the actual data minus that for the corresponding time-shuffled controls. The thin lines represent data from individual participants with the thick line representing the median. Note that the probability of incorporating another site into a frequency convergence increased as more sites frequency-converged. For (B) and (C), data for all behavioral conditions are presented in Extended Data Figure 3-2. D, Time-averaged oscillatory power as a function of the size of frequency convergence at each site along the anterior–posterior axis [color-coded from dark blue (posterior) to yellow (anterior)], for the midline (left), left-hemisphere (middle), and right-hemisphere (right) routes. The power values are relative to those obtained with the corresponding time-shuffled controls, normalized to the maximum actual power for each power-by-convergence curve (per site per participant). The values averaged across participants are plotted, with each open circle indicating Bonferroni-corrected statistical significance (α = 0.05, two-tailed) relative to zero (control levels). Note that most sites showed accelerated increases in power when they participated in frequency convergences of four or more sites. Data for all behavioral conditions are presented in Extended Data Figure 3-3.

To gain insights into the underlying mechanisms and functions of anterior–posterior oscillation-frequency convergences, we examined some of their general characteristics. While a variety of their spatial, spectral, and temporal features can be examined, we focused on the following questions. First, how does an oscillation-frequency convergence grow? For instance, does the probability of incorporating another site into a frequency convergence increase or decrease with the size of frequency convergence? Second, how are oscillation-frequency convergences related to oscillatory power? Third, are oscillation-frequency convergences accompanied by consistent phase relations? As discussed below, investigating these questions provided some insights into the potential mechanisms and functions of dynamic anterior–posterior oscillation-frequency convergences.

The probability of incorporating another site into an oscillation-frequency convergence increased as the size of frequency convergence increased

What mechanisms drive anterior–posterior oscillation-frequency convergence? One possibility was that multiple regions could be driven to oscillate at a matched frequency via inter-regional entrainment. Spreading inter-regional entrainment would reciprocally enhance within-region synchronization (Tewarie et al., 2019), which in turn would facilitate further inter-regional spreading of entrainment. This possibility would predict that (1) a frequency convergence across more regions (sites) would be associated with higher oscillatory power (a measure of within-region synchronization) at participating sites and (2) the probability of another site joining a frequency convergence would increase as more sites frequency-converge. We evaluated Prediction 2 here and Prediction 1 in the next section.

We computed the probability of growing in frequency convergence as a function of the number of frequency-converged sites. We first obtained the average base probability of oscillatory activity by computing the proportion of ones in the oscillation-frequency matrix from each site (containing ones where oscillations were identified, broadened by ±1 log unit; see Materials and Methods) and averaging the values across sites. We call this P(1) , the average probability of oscillatory activity. If oscillation-frequency convergences were due to random coincidences, the average probability of frequency convergence across k sites, P(k) , should be given by P(1)k .

We computed the probability of frequency convergences across k>1 sites as follows. For each combination of k sites, we stacked and summed their oscillation-frequency matrices and computed the proportion of k's in the resultant matrix to obtain the probability of k convergence. We then averaged the probabilities computed for all combinations of k sites to obtain the average probability of k convergence, P(k) . To estimate the probability of growing in frequency convergence at each level of convergence, we computed the conditional probability of obtaining k convergence given k–1 convergence, P(k|k–1) , by dividing P(k) by P(k–1) because k convergence was a subset of k–1 convergence. We computed P(k|k–1) (where 2≤k≤8 ) for each participant for each behavioral condition.

The probability of growing in convergence, P(k|k–1) , monotonically increased as a function of the number of frequency-converged sites, k (red curves; Fig. 3B presenting the rest-with-eyes-closed condition as a representative example; see Extended Data Fig. 3-2A for the results from all behavioral conditions). Slight increases were also observed in the time-shuffled controls (black curves), indicating that some general features of the oscillation-frequency matrices (e.g., major oscillations occurring in relatively narrow frequency ranges) slightly increased the probability of coincidental frequency convergences relative to the chance level (gray line). The data for large frequency convergences (for k=7 and especially for k=8 ) were somewhat noisy as their instances were relatively rare especially for the time-shuffled controls (Fig. 3A; Extended Data Fig. 3-1). Importantly, P(k|k–1) for the actual data increased substantially more steeply than that for the time-shuffled controls as a function of k (Fig. 3C presenting the rest-with-eyes-closed condition as a representative example; see Extended Data Fig. 3-2B for the results from all behavioral conditions). This indicates that the probability of growing in frequency convergence monotonically increased as a function of the size of frequency convergence, controlling for any potential effects of time-independent spectral features at the individual sites.

Larger oscillation-frequency convergences were associated with higher oscillatory power at participating sites

Here we evaluated the other prediction that if oscillation-frequency convergences were driven by inter-regional entrainment, frequency convergences across more regions (sites) would be associated with higher oscillatory power (a measure of within-region synchronization) at participating sites.

To evaluate this possibility, we temporally averaged the powers of all oscillations at each site, separately for each size of frequency convergence, k, in which it participated. For instance, we temporally averaged the powers of oscillations at AFz when their oscillation frequencies did not converge with any other sites (k=1) , when their oscillation frequencies converged with one other site (k=2) , their oscillation frequencies converged with two other sites (k=3) , and so on. This yielded the average oscillatory power at each site as a function of the size of frequency convergence in which the site participated. However, we may obtain a spurious association between higher-power and larger-frequency convergence due to time-independent spectral features. For example, if higher-power oscillations concentrated within a narrow frequency range while lower-power oscillations broadly distributed across frequencies, higher-power oscillations would coincidentally frequency-converge with a higher probability than lower-power oscillations. We thus applied the same analysis to the corresponding time-shuffled controls (see Materials and Methods).

For each site, we computed its average oscillatory power minus the value obtained from the corresponding time-shuffled control as a function of the size of frequency convergence in which the site participated. To facilitate comparisons of these power-by-convergence curves across sites, participants, and behavioral conditions, we normalized each curve (actual minus control) by dividing it by its maximum actual power (per site per participant per behavioral condition).

Figure 3D shows the (normalized) average oscillatory power at each site [color coded from dark blue (posterior) to yellow (anterior)] as a function of the size of frequency convergence in which the site participated (the rest-with-eyes-closed condition presented as a representative example; see Extended Data Fig. 3-3 for the results from all behavioral conditions). The open circles indicate Bonferroni-corrected (for eight convergence sizes) significant difference from the control level (i.e., zero) at α = 0.05 (two-tailed). The oscillatory power generally increased as a function of the size of frequency convergence at all sites, rising significantly above the control levels at frequency convergences of four or five sites and steeply increasing at larger convergences (except that the power did not rise significantly above the control level at the anterior-most site during the natural viewing conditions; see Extended Data Fig. 3-3, yellow curves).

These results and those presented in the preceding section are consistent with the idea that anterior–posterior oscillation-frequency convergences are driven by inter-regional entrainment that reciprocally enhances within-region synchronization (this section), which in turn facilitates further inter-regional spreading of entrainment (preceding section).

Anterior–posterior oscillation-frequency convergences were accompanied by approximately linear phase gradients

What functions might anterior–posterior oscillation-frequency convergences serve? When neural activities oscillate at a matched frequency across regions, communication between them can be controlled by adjusting phase lags. We thus examined how oscillation-frequency convergences influenced phase relations. For example, if frequency convergences were accompanied by linear phase gradients, one could infer that frequency convergences facilitate directional flows of information.

We computed sequential pairwise phase differences from posterior to anterior sites. Note that choosing this direction was arbitrary; the interpretation of the results would be the same regardless of the direction in which we computed phase differences. For example, for the midline route, we computed the phase of POz minus the phase of Oz, the phase of Pz minus the phase of POz, the phase of CPz minus the phase of Pz, the phase of Cz minus the phase of CPz, the phase of FCz minus the phase of Cz, the phase of Fz minus the phase of FCz, and the phase of AFz minus the phase of Fz. These sequential phase differences were computed at each oscillation frequency at each timepoint for each pair of sites that participated in a frequency convergence between two or more sites. Then, for each pair, their phase differences were averaged (as complex angles) across all instances, separately for each size of frequency convergence. Because of this temporal averaging, this analysis detected phase relations that were consistent over the ∼5 min period. The resultant time-averaged phase differences were averaged within each of the 19 partially overlapping frequency bins (1 Hz wide with 0.5 spacing).

To detect phase gradients (if any) in each frequency bin, we spatially integrated the corresponding sequential pairwise phase differences from the posterior to anterior sites (i.e., we cumulatively summed the sequential pairwise phase differences from the posterior to anterior sites) while arbitrarily setting the phase at the posterior-most site to zero. Overall, we observed two types of phase gradients (see below for details). One was negatively sloped (Fig. 4D, left) indicating that more anterior sites were more phase lagged, suggestive of a posterior-to-anterior flow of information, which we call the “posterior–anterior (P-A) gradient.” The other was positively sloped (Fig. 4D, right) indicating that more posterior sites were more phase lagged, suggestive of an anterior-to-posterior flow of information, which we call the “anterior–posterior (A-P) gradient.” It is clear from Figure 4D that both the P-A and A-P gradients developed, from barely sloped to substantially sloped, as the size of frequency convergence increased.

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Figure 4.

Characteristics of the anterior–posterior phase gradients accompanying oscillation-frequency convergence. A–C, Distribution of the linear trend (Pearson's r weighted by PLV) of the anterior–posterior phase gradient (averaged across frequency-convergence sizes) obtained across the 19 frequency bins (1 Hz wide with 0.5 Hz spacing, spanning 5–15 Hz), for the midline (A), left-hemisphere (B), and right-hemisphere (C) routes. Negative values indicate the P-A gradients (with more anterior sites more phase lagged; illustrated with a single red arrow when they were prevalent), whereas positive values indicate the A-P gradients (with more posterior sites more phase lagged; illustrated with a single blue arrow when they were prevalent). The line represents the distribution averaged across participants, and the shaded area represents the 95% confidence interval. The rows represent the five behavioral conditions. Note that the distributions are bimodal for all anterior–posterior routes in all behavioral conditions, indicating that the P-A and A-P gradients are distinct types. The asterisks (*p < 0.05; **p < 0.01; ***p < 0.001; ****p < 0.0001) indicate t test results (two-tailed) comparing the number of negative with positive r values, testing the asymmetry of each distribution. Note that the distributions for the minimal-visual-stimulation conditions are mostly asymmetric (top three rows), whereas those for the natural viewing conditions are mostly symmetric (bottom two rows). D, The P-A gradient (left; identified as the gradients with r less than −0.6; see the red-shaded regions in A–C) and the A-P gradient (right; identified as the gradients with r greater than 0.6; see the blue-shaded regions in A–C) averaged across the three anterior–posterior routes, participants, and behavioral conditions. Note that both gradients developed (i.e., became steeper) as more sites frequency-converged. E, Histograms of the correlation (Spearman's r, r_sp) between frequency-convergence size and phase-gradient slope for the P-A (left) and A-P (right) gradients (an entry per anterior–posterior route per participant per behavioral condition). As expected, the correlations were predominantly negative for the P-A gradients and predominantly positive for the A-P gradients, indicating that the phase gradients became steeper as more sites frequency-converged. Extended Data Figure 4-1 shows the correlation between the absolute slope of phase gradient (averaged across frequency-convergence sizes) and its average PLV across the 19 frequency bins. Extended Data Figure 4-2 shows the occurrences of the P-A and A-P gradients along each of the three anterior–posterior routes for each participant (per behavioral condition). Extended Data Figure 4-3 shows the distribution of the P-A and A-P gradients across the alpha range (5–15 Hz) along each of the three anterior–posterior routes (per behavioral condition). Extended Data Figure 4-4 shows the within-participant consistency of the alpha frequencies at which the P-A and A-P gradients formed during natural viewing (earlier vs later viewing of a silent nature video). Extended Data Figures 4-5–4-9 plot D and E separately for each of the three anterior–posterior routes and the five behavioral conditions. Extended Data Figure 4-10 plots D separately for three alpha subranges (5–8, 8–12, and 12–15 Hz). Extended Data Figure 4-11 is equivalent to D, but the phase gradients were computed without surface-Laplacian transforming the EEG data.

To verify that these phase gradients accompanying oscillation-frequency convergences were distinctive types, we examined the distribution of the negative and positive phase gradients across the frequency bins. For each frequency bin, we computed the linear trend of the overall phase gradient (averaged across frequency-convergence sizes) by computing Pearson's r between the serial site index (1 through 8 from posterior to anterior) and the corresponding phase value. We weighted each r value by a measure of the temporal stability of the corresponding phase gradient, that is, by its average phase-locking value (PLV; Cohen, 2014; averaged across all sequential pairwise values within the gradient). We note that steeper phase gradients tended to have higher PLVs (especially in the minimal-visual-stimulation condition; Extended Data Fig. 4-1) indicating that steeper phase gradients tended to be temporally more stable. PLV's were corrected for sample-size–dependent biases by subtracting the expected chance levels (computed with the assumption of random sampling from a uniform angular distribution, well approximated by a power function of the sample size). As the corrected PLV's varied between close to 1, indicative of perfect phase locking, and near 0, indicative of no phase locking, the PLV-weighted r values varied between close to −1, indicative of a perfectly negative linear P-A gradient, to close to 1, indicative of a perfectly positive linear A-P gradient. For brevity, we will refer to the PLV-weighted r simply as r.

We made a histogram of the 19 r values (computed for the 19 frequency bins) for each participant. We then averaged the histograms across participants separately for the three anterior–posterior routes and the five behavioral conditions. The distributions were consistently bimodal (Fig. 4A–C), indicating that oscillation-frequency convergences were accompanied by two distinct types of phase gradients, the P-A gradients (with negative r values) and the A-P gradients (with positive r values).

Notably, in the minimal-visual-stimulation conditions (especially in the eyes-closed conditions), the P-A and A-P gradients were spatially segregated. Through the midline, oscillation-frequency convergences were predominantly accompanied by the P-A gradients, indicated by the dominant peak at a negative r value (Fig. 4A, top three rows). In contrast, through the left- and right-hemisphere, frequency convergences were predominantly accompanied by the A-P gradients, indicated by the dominant peak at a positive r value (Fig. 4B,C, top three rows). This spatial segregation disappeared during natural viewing. Frequency convergences were accompanied by about equal numbers of P-A and A-P gradients through the midline and each hemisphere, indicated by the symmetrically bimodal distributions of the r values (Fig. 4A–C, bottom two rows).

For nearly all participants for all anterior–posterior routes and behavioral conditions, oscillation-frequency convergences were accompanied by both the P-A and A-P gradients forming at different frequencies (Extended Data Fig. 4-2). The frequencies at which the two gradient types formed were not consistent across participants except that in the eyes-closed conditions either gradient type dominated (per anterior–posterior route) across a broad range of frequencies for most participants (Extended Data Fig. 4-3). Nevertheless, there was some within-participant consistency in the frequencies at which the two gradient types formed as the pattern of the negative and positive r values across frequency bins was significantly correlated between the earlier and later viewing of a silent nature video (Extended Data Fig. 4-4).

We next examined how the P-A and A-P gradients changed as a function of the size of frequency convergence. To consider robust cases of the P-A and A-P gradients (which occurred in specific frequency bins; Extended Data Figs. 4-2, 4-3), we selected the gradients with r values less than −0.6 (occupying mostly below the lower bimodal peak; Fig. 4A–C, highlighted with red-shaded regions) as representative P-A gradients and those with r values greater than 0.6 (occupying mostly above the upper bimodal peak; Fig. 4A–C, highlighted with blue-shaded regions) as representative A-P gradients. We then separately averaged these samples of the P-A and A-P gradients (taken from frequency bins in which they were robust) across the three anterior–posterior routes, participants, and behavioral conditions. As noted above, it is clear that both the P-A and A-P gradients developed, that is, changed from barely sloped to substantially sloped, as more sites frequency-converged (Fig. 4D). To confirm this observation, we computed the correlation (Spearman's r, r_sp) between frequency-convergence size and phase-gradient slope (the slope of the linear fit going through the origin) separately for the P-A and A-P gradients per anterior–posterior route per participant per behavioral condition. For the P-A gradients, the r_sp values were predominantly negative (Fig. 4E, left) indicating that the negative slopes increased with the increasing frequency-convergence size. For the A-P gradients, the r_sp values were predominantly positive (Fig. 4E, right) indicating that the positive slopes increased with the increasing frequency-convergence size. This confirms that both the P-A and A-P gradients became steeper, that is, both phase gradients developed, as more sites frequency-converged. We further note that the phase gradients were approximately linear when all anterior–posterior sites frequency-converged (Fig. 4D, the darkest line), implying traveling waves propagating in the anterior and posterior directions (see Discussion below). Similar patterns of results were obtained separately in each behavioral condition (Extended Data Figs. 4-5–4-9).

Are the observed oscillation-frequency convergences an artifact of source mixing (primarily volume conduction) effects?

In scalp-recorded EEG, source mixing (primarily volume conduction) effects are extensive (extending >5 cm; Schaworonkow and Nikulin, 2022). We used the surface-Laplacian transform to reduce source mixing effects to 1–3 cm (Nunez et al., 1994; Tenke and Kayser, 2012; Cohen, 2014). For the current dataset, we verified that volume conduction effects attenuated within ∼1.2 times the average interelectrode distance (Menceloglu et al., 2024; Fig. 4). Thus, the observed frequency convergences across four or more sites are unlikely to have been generated by volume conduction effects from a single oscillatory source. Importantly, a theoretical consideration and the characteristics of the phase gradients reasonably rule out the possibility that the observed oscillation-frequency convergences could have been spuriously generated by source mixing effects.

Given that source mixing effects are virtually instantaneous, any volume conducted oscillatory activity from a source detected elsewhere has a phase lag of zero. The phase gradients we observed therefore could not have been generated by source mixing effects. If anything, given the likely presence of at least some source mixing effects (especially between adjacent sites), the actual anterior–posterior phase gradients would have been steeper than what we observed. The reason is that any source mixing effects would reduce observed phase differences. Suppose oscillation at site2 was ahead of oscillation at site1 by Δθ , that is, site2=A2sin(2πft+Δθ) while site1=A1sin(2πft) , where A2 and A1 are oscillation amplitudes at site2 and site1 , f is the oscillation frequency, and t is time. Adding the volume conducted oscillation from site1 , A1VCsin(2πft) , to oscillation at site2 , A2sin(2πft+Δθ) , yields A′sin(2πft+Δθ′) , where:Δθ′=atan[A2sin(Δθ)/(A1VC+A2cos(Δθ))]. Given that Δθ′<Δθ (as A1VC is in the denominator), source mixing effects would always reduce observed phase differences relative to the actual phase differences. Indeed, the slopes of the P-A and A-P gradients decrease by a factor of ∼10 if we do not reduce source mixing effects by applying the surface-Laplacian transform (compare Fig. 4D with Extended Data Fig. 4-11).

Furthermore, if oscillation-frequency convergences were driven by source mixing effects from a prominent source, frequency convergencies across more sites would be driven by more extensive source mixing effects from stronger oscillation at the source. Stronger source mixing effects (e.g., larger A1VC in the above example) would have made the phase gradients less steep (see Eq. 1, having A1VC in the denominator). To the contrary, we observed that the phase gradients became steeper as more sites frequency-converged (Fig. 4D,E).