Large-scale communication in the human brain is rhythmically modulated through alpha coherence

Summary

Recent evidence has suggested that coherent neuronal oscillations may serve as a gating mechanism for flexibly modulating communication between brain regions. For this to occur, such oscillations should be robust and coherent between brain regions that also demonstrate time-locked correlations, with time delays that match the phase delays of the coherent oscillations. Here, by analyzing functional connectivity in both the time and frequency domains, we demonstrate that alpha oscillations satisfy these constraints and are well suited for modulating communication over large spatial scales in the human brain. We examine intracranial EEG in the human temporal lobe and find robust alpha oscillations that are coherent between brain regions with center frequencies that are consistent within each individual participant. Regions demonstrating coherent narrowband oscillations also exhibit time-locked broadband correlations with a consistent time delay, a requirement for an efficient communication channel. The phase delays of the coherent alpha oscillations match the time delays of the correlated components, and importantly, both broadband correlations and neuronal spiking activity are modulated by the phase of the oscillations. These results are specific to the alpha band and build upon emerging evidence suggesting that alpha oscillations may play an active role in cortical function. Our data therefore provide evidence that large scale communication in the human brain may be rhythmically modulated by alpha oscillations.

In Brief

Chapeton et al. show that alpha oscillations in the human cortex satisfy several constraints that are necessary for using oscillatory coherence as a means of modulating large-scale cortical communication. The results are specific to the alpha band and supported by single unit spiking activity.

Introduction

Processing, transferring, and integrating information in the human brain requires effective communication, yet precisely how such communication occurs between different brain regions remains unknown. One possibility is that large-scale cortico-cortical communication is modulated by neural oscillations. Oscillations in neural activity are readily observed throughout the human brain at various spatiotemporal scales [1, 2, 3, 4], and have been linked with a variety of cognitive functions [5, 6, 7, 8, 9, 10, 11]. An emerging literature has suggested that synchronized oscillations could serve as a gating mechanism to quickly enable the selective routing of information between brain regions [12, 13, 14, 15, 16]. Given that a brain region may send and receive connections from multiple other regions, such a mechanism could allow for flexible communication between brain regions without requiring substantial changes to the underlying structural connectivity.

The alpha oscillation is the canonical example of rhythmic neural activity in the human brain. Alpha band activity can be distinguished from broadband activity by visual inspection alone and was the first rhythm observed and described in human recordings [1]. Historically, the alpha rhythm was first thought to reflect an idling state during which individuals are awake but at rest [17]. More recent studies, however, suggest that alpha oscillations play an active role in cortical functions that govern behaviors such as attention, memory, and even conscious perception [18, 19, 20, 21, 22, 23, 24, 25, 26, 8, 9, 27, 28]. A common thread among these studies is an emerging framework in which alpha oscillations reflect cortical inhibition, although the exact nature of this inhibition still remains a matter of debate [28, 9, 29, 27, 30]. One possibility is that these oscillations represent an overall inhibition of cortical processing, with higher alpha power leading to more inhibition. A different possibility that has gained support, however, is that alpha inhibition is rhythmic and consists of alternating states of inhibition and excitation (or disinhibition) [28, 30, 29, 27, 23]. In this scenario, the effects of alpha oscillations on cortical activity and function should be phase dependent.

If alpha oscillations provide rhythmic bouts of relative inhibition and excitation, then it may be possible to use coherent alpha oscillations to modulate communication between cortical regions. In this framework, the outputs of a sender brain region could consistently arrive at the excitatory (or dis-inhibited) phase of the alpha cycle in the receiver region if the two brain regions are oscillating with the right phase relationship. Adjusting the phase relationship between brain regions, or the extent to which their rhythms are coherent, can thereby modulate the effective connectivity between them. Such a mechanism is the core principle underlying the model of communication through coherence (CTC), in which coherent gamma oscillations are hypothesized to establish effective communication channels between local networks [31, 14, 32]. But while the original model of CTC focused on higher frequency gamma band coherence, several studies have suggested that the specific frequencies responsible for synchronizing activity between brain regions may depend on spatial scale, with lower frequencies being better suited for synchronizing large neuronal populations [33, 34, 35]. The alpha rhythm is particularly well suited for large-scale synchronization because it is robust in terms of power and frequency, and appears ubiquitously throughout the human cortex [36, 3]. Moreover, there is evidence in both humans and animals that conscious perception, which is thought to require co-activation of various cortical systems, may be discretized into ~100 ms bouts, approximately the duration of a single alpha cycle [37, 38, 39, 28].

Several constraints must be satisfied for alpha oscillations to modulate large-scale cortical communication. These constraints arise out of theoretical considerations related to the implementation of communication schemes that depend on oscillatory entrainment or resonance [14, 15, 16], as well as in response to important concerns that have been raised specifically regarding the role of gamma oscillations in cortical function and communication [40, 41, 42]. First, narrowband alpha activity should have a consistent center frequency and be distinguishable from noise over large patches of cortex. This is because if oscillatory synchrony is to arise through mechanisms such as entrainment or resonance, these mechanisms are most effective when the amplitude of the input oscillations is large and when the difference in frequency between the oscillations at the sender and receiver is small [16, 43]. Second, any brain regions that are communicating should share both narrowband coherent oscillations as well as time-locked correlated activity over a wider frequency range. This constraint is motivated by the known relationship between a communication channel’s bandwidth and its information capacity [44, 45]. Transmitting a narrow band signal alone places severe limits on the rate at which information can be conveyed through any communication channel, and so an effective neural communication channel must be capable of transmitting frequency components outside of the alpha band. Third, the phase delay between the coherent oscillations should match the time delay between the correlated components so that inputs arrive during the duty cycle of the alpha oscillation. Finally, if particular phases of the alpha cycle correspond to time windows of excitation or inhibition, then both correlations in neural activity between brain regions, as well as neural activity within each region, should be modulated by the phase of the coherent oscillations. Together, these constraints can be used to directly test if rhythmic activity within the alpha band can be used to modulate large-scale communication in the human brain.

Here, we test the hypothesis that large-scale communication in the human brain is modulated by coherent oscillations in the alpha band by investigating human intracranial EEG (iEEG) captured from awake, freely behaving participants as they were being monitored with subdural electrodes for seizure activity. We were specifically interested in testing whether cortical activity satisfies the constraints identified above for modulating effective communication in large-scale cortical circuits. To this end, we examine functional connectivity between brain regions in both the time and frequency domains. In the time domain, we use a previously validated measure of directed functional connectivity, which we refer to as time-locked coupling, that is based on increases in correlated activity between brain regions that occur at a consistent time delay [46]. In the frequency domain, we quantify oscillatory synchronization between brain regions by computing the magnitude squared spectral coherence [47, 48], and extracting a metric which we call excess coherence. We analyze functional connectivity in both the time and frequency domains in order to distinguish broadband contributions from any frequency specific interactions underlying communication between brain regions. Moreover, we examine these measures over time scales ranging from minutes to days in order to assess the consistency of power, frequency, and the timing relationships in our data. In a subset of participants, we investigate whether single unit spiking activity in cortical regions is modulated by the same alpha rhythms that underlie spectral coherence between brain regions. Together, our results demonstrate that coherent alpha oscillations can provide a mechanism for the flexible transmission of information between cortical regions in the human brain.

Results

Neuronal spiking is rhythmically modulated by alpha oscillations

To determine the role of alpha oscillations in large-scale communication, we examined intracranial EEG (iEEG) recorded from subdural electrodes placed on the cortical surface in ten participants with drug resistant epilepsy. We collected iEEG data from each participant in 30-second blocks, during which time each participant was awake and behaving freely in the monitoring unit. In each participant, we analyzed 20 blocks of data, collected on two separate days of intracranial monitoring (10 blocks per session separated by 10.7 ± 0.2 minutes each; two sessions collected on separate days separated by 59 ± 17 hours; mean ± SEM). We also had the opportunity to analyze single unit and local field potential activity captured using implanted microelectrode arrays in the middle temporal gyrus in two of the participants during the same sessions used for the iEEG analyses.

Given that information in the brain is fundamentally transmitted via action potentials, we first examined whether alpha oscillations modulate neuronal spiking activity. Although previous evidence has suggested that the alpha rhythm plays an inhibitory role on cortical function [28, 9, 29, 27, 30] it is not clear if this inhibition is rhythmic or tonic. Rhythmic alpha inhibition could allow for selective processing during the periods of relative excitation, thereby providing a mechanism by which coherent alpha oscillations can establish flexible and effective communication channels between cortical regions (Figure 1A). We therefore examined the spike field coherence (SFC) between single units and the local field potential (LFP), recorded from the same microelectrode, in order to determine if spiking activity is rhythmically modulated by local alpha oscillations.

Figure 1. Alpha oscillations rhythmically modulate spiking activity.

(A) Schematic demonstrating the principle of rhythmic communication. If neurons preferentially fire at a specific phase of an oscillation, and the oscillations at two regions are coherent with a phase delay matching the conduction delay between them, then spikes fired at the excitable phase in one region will arrive at the excitable phase in the downstream region. These rhythmic bouts of maximum gain can be used to set up an effective communication channel between two coherent regions. Conversely, when the phase does not match the delay then spikes will arrive away from the excitable phase and will be less likely to elicit firing. (B) Average spike field coherence spectrum (SFC) for the example participant. The average SFC spectrum across all units has a clear peak in the alpha band (f = [7,8] Hz). Inset: Individual spike waveforms (gray) and mean waveform (black) for a single example unit. Vertical scale bar = 10SD, Horizontal scale bar = 1.5ms. (C) Histogram of peak frequencies for individual units. For a large fraction of units (40%), the frequency of maximum SFC was also centered at f = [7,8] Hz. Inset: Distributions of correlations between LFP alpha power and instantaneous firing rate. On average, alpha power and spike rate are negatively correlated for alpha coherent units (orange) and uncorrelated for all other units (gray). (D) Spike triggered average for units with maximal SFC at f = [7,8] Hz. Alpha oscillations lasting several cycles are clearly visible. (E) Phase preference for the alpha coherent units. The preferred phase for spiking activity is near the trough of the alpha cycle. See also Figure S1.

In a single participant (participant 9), we found that the average SFC across all units exhibits a sharp peak in the high theta/low alpha range (Figure 1B), and that around 40% of the units are preferentially locked to this small range of frequencies (f = [7,8]Hz) (Figure 1C); in a second participant, we found peaks in the average SFC at 6 and 10 Hz, and a large fraction of units (16%) are preferentially locked to a tight range between 10 and 11 Hz; Figure S1A,B). For simplicity, we refer to these ranges as alpha, and define the units that are locked to these frequencies as alpha coherent units. The spike triggered average for all alpha coherent units shows clear oscillations that are sustained over several cycles (Figure 1D; Figure S1C). When we examined the instantaneous alpha phase of the LFP over all individual spike times, we found that there is an overwhelming preference for firing at or near the trough of the oscillation (Figure 1E; Figure S1D). These results demonstrate that the effects of alpha oscillations on spiking activity are indeed phase dependent.

To examine whether alpha oscillations have an inhibitory effect on neural activity, we computed the correlation between instantaneous alpha power of the LFP and the instantaneous firing rate of each unit. We found that periods of higher alpha power correspond to epochs of overall decreased firing, but this effect is limited to the alpha coherent units (Figure 1C inset and Figure S1B inset). We did not find a significant relationship between alpha power and firing rates for the remaining units (participant 9: t(28) = −3.31, p = 0.0025 for alpha coherent units, t(43) = −0.002, p = 0.998 for all others; participant 5: t(18) = −3.06, p = 0.007 for alpha coherent units, t(97) = −1.88, p = 0.063 for all others, t-test). These results demonstrate that the inhibitory effects alpha power on spiking activity are restricted to only those individual neurons that are rhythmically modulated by the alpha cycle.

Alpha activity and coherence in iEEG signals

Having established that neural activity depends on the phase of local alpha oscillations, we then tested the hypothesis that alpha oscillations mediate large scale cortico-cortical communication. For this to occur, alpha oscillations should be present over large areas of the cortex. For an example pair of electrodes in the same participant, (Figure 2A), we found strong narrowband power within the alpha band at both sites during a single block which persisted for the duration of the block (8 Hz; Figure 2B). We averaged the amplitude spectrum from all electrodes over all blocks in this participant, and also found a clear peak at 8 Hz, indicating that large amplitude alpha oscillations are a global phenomenon across time and space (Figure 2C).

Figure 2. Alpha activity and coherence.

(A) Subdural electrode grid for the example participant. (B) Time-frequency representation for the signals from two example electrodes (black circles in a) during a 30-second block. For both electrodes, there is consistently high power in a narrow band around 8 Hz. (C) Average power spectrum across all electrodes for the same participant (mean ± SEM over all blocks). There is clear peak at 8Hz, indicating that on average across all electrodes there is high narrowband alpha power above the f^−α background. (D) The frequency of maximum coherence for a large fraction of electrode pairs and blocks in this participant also occurred in the alpha band (f_c = [7,8] Hz; see inset for average coherence spectrum over all electrode pairs. See also Figure S2.

If alpha oscillations coordinate large scale communication, then we should also find that these oscillations are coherent between brain regions during periods when these regions are communicating. To test this, we examined spectral coherence between all electrode pairs in each participant. In the same example participant, we found that most pairs exhibit a peak in the coherence spectrum in the alpha band (f_c = [7,8] Hz; Figure 2D; see inset for the average coherence spectrum over all pairs), indicating that coherent alpha oscillations are also a global phenomenon. We found similar results in every participant. For each participant, there is a clear peak in coherence that is restricted to a small range of frequencies, although the specific frequency range f_c varies from one individual to the next (Figure S2). The range of coherent frequencies across participants includes frequencies that have been referred to as high theta or low beta (6 – 13 Hz range, 8.7 ± 0.5 Hz across participants) [49], but the fact that there is a clear peak in this frequency range for every participant suggests that this may be the same phenomenon. For simplicity, we refer to this as alpha coherence. The presence of coherent alpha oscillations across large areas of temporal cortex suggests that alpha oscillations may be well suited for synchronizing activity between cortical regions.

Electrode pairs that exhibit coherent alpha oscillations also exhibit correlated broadband activity

While coherence may demonstrate that two brain regions share an oscillatory rhythm, any brain regions that are communicating effectively should also share time-locked correlations over a broader frequency range [44, 45]. Time-locked correlated activity, which can be identified by a sharp dominant peak in the cross-correlation function, reflects a functional relationship between broadband activity of two brain regions at a consistent time delay, indicative of an effective communication channel. We quantified time-locked correlations between different cortical regions using a directed time domain measure of correlated activity that we refer to as time-locked coupling [46]. This measure, W(τ), provides a robust estimate of the correlation between two iEEG signals as a function of the time delay, τ, between them (see STAR Methods).

For each electrode pair in each participant, we computed the time-locked coupling during each block, and then the average over all blocks (Figure 3A). We identified the maximum value of coupling over all delays, W_max, which reflects the extent to which the correlation between two electrodes increases at a consistent time delay across all blocks. The time delay of maximum coupling, τ_w, defines the preferred time delay for that pair. We excluded electrode pairs for which τ_w = 0 from our analysis to mitigate the potential for spurious relationships due to volume conduction or re-referencing (see STAR Methods). We found that within each participant, the distribution of maximum time-locked coupling, W_max, across all electrode pairs exhibits a heavy tail, and often appears bimodal with a local minimum (for the same example participant, see Figure 3C top; Figure S2C for all participants). We fit these distributions with a mixture of two Gaussians (see STAR Methods) and used the posterior probability to assign pairs to each Gaussian component. We define pairs whose coupling value is assigned to the Gaussian with the larger mean, as having significant coupling. These distributions suggest that a subset of electrode pairs in each participant display strong time-locked correlations with a fixed time delay between them.

Figure 3. Relationship between alpha coherence and time-locked correlations.

(A) Time-locked coupling, a windowed and scaled measure of the time-lagged cross correlation (see STAR Methods), for the same electrode pair from Figure 2A. Across all blocks, the maximum coupling was large and occurred at a consistent time delay, τ_w. (B) Coherence spectra for the same example electrode pair in a. Across all blocks, there was a large and consistent peak in the coherence spectrum at f_c. The excess coherence, C_excess, captures the amount of coherence that is due only to narrowband activity (see STAR Methods). (C) The scatter plot shows all of the excess coherence and maximum coupling values for the example participant. By thresholding the distributions of these metrics (see STAR Methods) we can determine how many electrode pairs have both significant time-locked coupling and significant excess coherence (orange region). There are significantly more pairs than would be expected by chance (χ² test; see Table 1). (D) The Spearman correlations between excess coherence and average maximum coupling are positive and significant for all participants. (E) Correlations over time between alpha power and coupling for units with significant coupling and excess coherence are significant and positive. (F) Correlations between distance and coupling. (G) Correlations between distance and excess coherence. See also Figures S2, S3, S4A,B, and S5A,C.

Given that some electrode pairs in each participant exhibit strong coupling, and given that there is large alpha coherence between many electrode pairs, we were interested in whether these phenomena overlap and whether electrode pairs that exhibit large coherence also have strong time-locked coupling. We note, however, that a sharp peak at a single time delay in coupling, which reflects correlations over a broad range of frequencies, corresponds to an overall shift in the coherence spectrum across all frequencies where signal power exceeds noise power (see simulations of coupling and coherence in STAR Methods, Figure S3). We therefore separated frequency specific interactions from broadband correlations by computing a measure of excess coherence for each electrode pair, C_excess, that estimates the prominence of the peak coherence at f_c (see STAR Methods; Figure 3B). This measure captures the excess oscillatory coherence that is present between two brain regions at a specific frequency given the overall shift in the coherence spectrum that arises due to broadband correlated activity. Accounting for this shift by using the excess coherence is critical for analyzing any relationships between coupling and coherence, since the correlations between these measures will otherwise be trivially inflated (Figure S3E,F). We found that the distribution of excess coherence for each participant is also very heavy tailed and potentially bimodal (Figure 3C; Figure S2D). As with coupling, we fit these distributions with two Gaussians, and defined pairs that were assigned to the Gaussian with the larger mean as having significant excess coherence (see STAR Methods).

We then directly examined whether electrode pairs with significant time-locked coupling also exhibit significant excess coherence in two ways. First, based on the fraction of pairs that have significant coupling and the fraction that have significant excess coherence, we performed a χ² test. We found that for every participant, the number of electrode pairs that exhibit both significant coupling and excess coherence exceeds what would be expected by chance (p < 0.05, quadrant I in Figure 3C for the example participant; Table 1 for all participants). Second, we computed the Spearman correlation between time-locked coupling and excess coherence across all electrode pairs in each participant. We found a strong correlation between coherence and coupling that is consistent and significant within and across all participants (ρ ≥ 0.4, p ≤ 9.4 × 10⁻¹²; distribution of correlation coefficients t(9) = 23.02, p = 2.6 × 10⁻⁹ t-test; Figure 3C,D; Figure S4A). Together, these data demonstrate that in each participant there is a population of electrode pairs that share strong time-locked correlations and engage in coherent oscillations in the alpha band. For subsequent analyses, we refer to these pairs as coupled, and we define pairs whose coherence and coupling are both in the lower quartile of both distributions as uncoupled (see STAR Methods).

Table 1. Parameters and statistics for the number of electrode pairs with large excess coherence and large coupling for all participants.

See also Figures S4A and S5A.

Participant	# of pairs	fraction coupled	fraction coherent	χ2 Statistic	p-value
1	390	0.25	0.45	21.83	3.00E-06
2	199	0.53	0.56	6.85	0.0089
3	260	0.40	0.31	27.98	1.22E-07
4	256	0.38	0.25	18.99	1.33E-05
5	264	0.30	0.31	26.62	2.48E-7
6	265	0.30	0.49	14.16	1.70E-04
7	331	0.31	0.21	75.42	3.81E-18
8	353	0.31	0.25	82.55	1.03E-19
9	258	0.45	0.28	28.18	1.11E-7
10	334	0.43	0.24	31.56	1.93E-08

To ensure that these results are not being driven solely by the presence of strong alpha activity in both time series of an electrode pair we repeated our analyses by re-calculating the time-locked coupling after notch filtering and removing any alpha signals from the time series (see STAR Methods). Even when directly removing the alpha rhythm from the time series, we still find a significant number of electrode pairs which are both time-locked and coherent, and a strong correlation between time-locked coupling and excess coherence (Figure S5).

We also quantified the relationship between alpha power and time-locked coupling as follows. For every coupled electrode pair in each block, we calculated the average power at f_c as a function of time. We also constructed a time resolved version of the coupling metric (see STAR Methods), allowing us to compute the temporal correlations between these measures. On average, for coupled pairs there is a small but consistently positive correlation between alpha power at f_c and coupling (Figure 3E, t ≥ 5.3, p ≤ 2 × 10⁻⁶, t-test). This effect is substantially reduced or absent for uncoupled electrode pairs (Figure S4B, t ≥ 1.7, p ≤ 0.1, t-test). These data suggest that the strong time-locked correlations found between coupled electrode pairs are modulated by the power of the underlying alpha rhythm.

We examined the effect of inter-electrode distance on coupling as well as excess coherence for coupled electrode pairs and found negative correlations between distance and coupling as well as distance and excess coherence (Figure 3F,G), indicating that on average communication is more effective between nearby locations. This negative correlation is consistent and significant across participants (t(9) = −7.8, p = 2.8 × 10⁻⁵ for coupling and t(9) = −6.8, p = 7.6 × 10⁻⁹ for coherence, t-test), although the correlations are not statistically significant in each individual participant.

Coherent and correlated components have similar time delays

A critical requirement for oscillatory communication is that the phase delay between the oscillations in two brain regions can be tuned to match the conduction delay between them. Communication schemes based on entrainment or resonance suggest that phase relationships between oscillations can be modulated to satisfy this constraint [14, 15, 16]. We therefore examined whether the electrode pairs exhibiting strong coupling and alpha coherence in our data also satisfy this constraint. We estimated the latency of communication between two brain regions using the time delay of maximum time-locked coupling, τ_w, between them. For each coupled electrode pair in the same example participant, we found that the preferred time delay is relatively consistent across all 20 data blocks (Figure 4A). For each electrode pair, we also extracted the cross-spectrum phase difference, Δϕ, at the peak alpha coherence frequency, f_c, and used it to compute the phase delay between the alpha oscillations in the two regions (τ_ϕ = Δϕ ⁄ 2πf_c). We found that the average phase difference, and therefore the phase delay between each electrode pair, is consistent from block to block (Figure 4B inset). Moreover, the phase delay appears similar to the estimated time delay (Figure 4B; electrode pairs displayed in the same order as in Figure 4A).

Figure 4. Time delays for broadband and narrowband components.

(A) Time delays, τ_w, for individual electrode pairs in a single participant during different data blocks estimated from the cross-correlation analysis. Each point represents the time delay during a data block for a single coupled pair. Colored clouds of points (rows) represent all time delays for a single coupled electrode pair. (B) Time delays, τ_ϕ, for individual electrode pairs during different data blocks estimated from the cross-spectrum phase difference at f_c. inset: Distribution of phase differences for a single pair. The phase differences appear to be tightly concentrated at approximately 315°. (C) Distribution of root mean squared errors (RMSE) for a single participant between: τ_w and τ_α for coupled pairs (orange), τ_w and τ_α for uncoupled pairs (blue), and τ_w and τ_δ for coupled pairs (purple) The errors for the coupled pairs using τ_α are typically less than 10 milliseconds, and are significantly smaller than the errors for uncoupled pairs and the errors for coupled pairs using τ_δ. (D) Same as c but pooled across all participants. (E) Correlations between distance and absolute time delay for every participant. (F) Correlations between distance and absolute phase lag. See also Figures S4C and S5B,D.

We directly quantified the extent to which the time delay and phase delay match one another by computing the root mean squared error (RMSE) between them across all blocks for each electrode pair. In the example participant, the majority of coupled electrode pairs have an RMS error between the time and phase delays of less than 10 ms (Figure 4C; median 8 ms; see Figure S4C for all individual participants). We found this correspondence in the majority of coupled electrode pairs pooled across all participants, with most RMS errors less than 10 ms (Figure 4D, median 9.6 ms). The size of these errors is about an order of magnitude smaller than the duration of an alpha cycle, suggesting that for coupled electrode pairs, the time delay of correlated activity matches well to the phase delay between the coherent alpha oscillations.

In contrast, the RMS errors between the time delay and phase delays are substantially higher for uncoupled electrode pairs both in individual participants and across all participants (example participant in Figure 4C, median 122.2 ms; pooled electrodes across participants, median 123.2 ms, Figure 4D). We compared the distribution of RMS errors between coupled and uncoupled electrodes, and found that the RMS errors are significantly smaller for coupled electrode pairs within (|z| ≥ 6.5, p ≤ 8.8 × 10⁻¹¹, Wilcoxon rank sum test) and across participants (t(9) = −25.2, p = 1.2 × 10⁻⁹, t-test on the distribution of median differences). We confirmed that this correspondence between phase differences and conduction delays is specific to the alpha band, and not just a general phenomenon for low frequencies with substantial power, by repeating our analysis using the phase differences extracted in the delta band (f_δ = 4 Hz). For coupled pairs, we found that the RMS error using the f_c specifically identified for each participant is significantly smaller than the RMS error using coherence in the delta band both in individual participants (|z| ≥ 9.9, p ≤ 9.9 × 10⁻⁵; Wilcoxon signed rank test) and across participants (t(9) = −7.3, p = 4.7 × 10⁻⁵, paired t-test on the distribution of median differences; Figure 4C,D). Even after notch filtering and removing any alpha components from the signals before computing the time delay, τ_w, we still find a good match between the time and phase delays (Figure S5B,D).

We quantified the relationship between distance and time delay as well as distance and phase delay for all coupled electrode pairs. On average across all participants, we found positive correlations between distance and time delay as well as between distance and phase delay (Figure 4E,F; t(9) = 3.6, p = 0.0058 for time delay, and t(9) = 5.2, p = 6.6 × 10⁻⁴ phase delay, t-test). Of note, some of the correlations between distance and time delay and distance and phase delay were small (or even negative in two cases) and not significant for individual participants. This is because some electrode pairs within each participant are separated by large distances and have relatively small, but non-zero, time delays for coupling. Given that other structural factors such as myelination and axon diameter, in addition to distance, also influence the time delay with which two brain regions communicate [50, 51], these short delays at large distances could arise in connections that are mediated by white matter tracts.

Time-locked correlations are enhanced during specific phases of alpha oscillations

The constraint that the conduction delay between two brain regions should match the phase delay of the coherent oscillations ensures that any action potentials sent during a particular phase of an oscillation in one region will arrive during the same phase in the other. If additionally, the duty cycle for neural activation is limited to a specific excitable phase of the coherent rhythm, then action potentials sent at the excitable phase will more efficiently elicit neuronal firing in coherent downstream regions, since they will also arrive during the excitable phase (Figure 1A). In this case, correlations between the neural activity of two brain regions which are engaged in coherent oscillations should depend on the phase of the underlying rhythm.

To test this possibility, we examined coupled pairs of electrodes and extracted the instantaneous phase of the oscillatory rhythm, f_c. We then segmented each time series into periods when the phase of the alpha oscillation was within one of four phase bins, corresponding to the rise, peak, fall, and trough, and concatenated segments belonging to the same phase bin. For each coupled electrode pair, we aligned the corresponding time series according to the preferred time delay and computed the absolute value of the correlation using only segments where the instantaneous phase of both time series belonged to the same phase bin (see STAR Methods). In this manner, we computed the extent to which two brain regions exhibit correlated neural activity at the preferred delay as a function of the phase of the coherent rhythm. To ensure that there is no bias for any particular phase, we also constructed a surrogate distribution for this measure (see STAR Methods).

For the example participant, there are more electrode pairs with maximal correlations at the trough of the alpha cycle than at any other phase, and there are significantly more pairs with maximum correlation at the trough, and to a lesser degree at the peak, than what would be expected if there was no phase dependence (Figure 5A; t(20.56) = 5.46, p = 2.2 × 10⁻⁵ for the trough; t(20.41) = 2.22, p = 0.038 for the peak; two sample t-test against the surrogate distribution shown in gray assuming unequal variances). We found that this was generally consistent across participants, where the fraction of pairs with maximal correlation at the trough is significantly larger than what would be expected by chance (Figure 5B; t(9.02) = 3.4, p = 0.0089). The fraction of pairs with maximal correlation at the peak is larger than 25%, although not significantly different than what would be expected by chance (t(9.1) = 1.75, p = 0.11; see Figure S6 for all individual participants).

Figure 5. Phase preference for large-scale correlations.

(A) Single participant. There are more electrode pairs with maximal correlations at the trough of the alpha cycle than at any other phase bin, and the fractions of pairs with maximal correlations at the peak and trough are larger than what would be expected if there was no phase dependence (gray bars). (B) Same as a but pooled across all participants. See also Figure S6.

Discussion

This study provides several lines of evidence supporting the hypothesis that coherent alpha oscillations modulate large-scale communication in the human brain. By investigating single unit spiking activity, we first show that cortical firing rates are preferentially enhanced at the trough of a local alpha oscillation, providing direct support to the hypothesized role of alpha in rhythmically modulating periods of excitation and inhibition of neural activity. At larger spatial scales, we find that multiple sites across the temporal cortex engage in coherent alpha oscillations at the same frequency that modulates spiking activity, and that the center frequency of these oscillations is consistent across time and cortical regions within each participant. Importantly, we find that the regions that exhibit strong alpha coherence also display time-locked broadband correlations that are rhythmically modulated by the same alpha frequency, and whose time delays match the phase lags of the narrowband coherent oscillations. These results demonstrate that the alpha rhythm satisfies several constraints required for effectively modulating communication in the human cortex.

Our results build upon accumulating evidence that alpha oscillations in the human cortex play an active role in cortical processing [28, 9, 29, 27, 30]. Indeed, alpha oscillations may reflect rhythmic cortical inhibition, leading to alternating states of inhibition and excitation within cortical regions. Synchronized bouts of excitation and inhibition between two brain regions would therefore allow them to communicate more effectively, or in a more selective and temporally precise manner. Our data are consistent with this possibility. The physiological mechanisms responsible for the generation of the alpha rhythm are not yet fully understood, making it unclear if alpha oscillations represent a modulation of inhibition, excitation, or both [30]. This specific distinction is not relevant for our results that only require periods of relative excitation and inhibition for modulating both local neuronal activity and communication between brain regions. However, a better understanding of the physiological basis for alpha oscillations would be critical for testing models such as pulsed-inhibition and rhythmic pulsing [27, 29], whose implementation depends on the specific dynamics of the alpha cycle, as well as for constructing biologically plausible models of the effects of alpha oscillations on networks of spiking neurons.

Although our results suggest that coherent alpha band oscillations modulate large-scale communication, substantial theoretical and experimental work has focused on higher frequency gamma rhythms [4, 13]. Given that each iEEG electrode captures activity from approximately 10⁵ neurons, representing mesoscopic scale activity [52], one interesting possibility is that the specific oscillatory rhythm involved in communication is related to the spatial scale over which communication occurs. Prior studies have suggested that slower rhythms are better suited for synchronizing neuronal activity over larger spatial scales, whereas faster rhythms such as gamma may be responsible for synchronizing activity in local circuits [33, 34, 35]. Although some of the precise timing offered by the gamma rhythm may be lost if large-scale synchronization is mediated by alpha oscillations, this may be a necessary trade-off in order to robustly coordinate activity between large patches of cortex. The range and variability of conduction delays for cortico-cortical white matter tracts introduces substantial temporal dispersion on the arrival of inputs from one area to another [53, 54]. This may be overcome at the cost of temporal precision by allowing for the larger windows of integration, or longer duty cycles, that alpha oscillations provide.

Our data are also consistent with recent demonstrations that cross-frequency interactions are ubiquitous in human neural recordings [55, 56]. A common feature of these studies is that the phase of low frequency activity modulates higher frequency broadband activity (phase amplitude coupling; PAC). Given that high frequency broadband activity is often taken as a surrogate for cortical activation [57], then the presence of PAC is consistent with the hypothesis that lower frequency rhythms such as alpha provide bouts of excitation and inhibition for neural activity. Our results on the modulation of spiking activity by the phase of the alpha rhythm directly demonstrate this, as do our results demonstrating that correlations between brain regions are also modulated by the phase of the alpha rhythm. Moreover, if alpha and gamma activity do in fact reflect communication at different levels of a spatial hierarchy, then an additional possibility would be that the interaction between communication processes at these spatial scales could also be reflected in these observed cross-frequency interactions.

We find that the precise frequency of the coherent oscillations can vary between individuals, but each participant has their own dominant rhythm, f_c, that was consistent throughout all of that participant’s recordings and regions. For simplicity, we refer to the dominant frequency within each participant as being in the alpha band. However, we note that these frequencies did extend as low as 6 Hz in one participant and as high as 13 Hz in another, and could also be labeled as high theta or low beta activity. Nonetheless, in each case, we found that the identified dominant frequency exhibited peak prominences and phase differences that aligned well with the maximum broadband coupling and time delays identified using time-lagged correlation. Such consistency suggests that this dominant 6–13 Hz rhythm, despite its precise label, is the rhythm that mediates large-scale communication in each participant. Moreover, we found that these features are specific to this frequency, f_c, and not a general feature of low frequency activity. When we examined delta coherence, we found that the phase delays between delta rhythms do not appear to be well suited for large scale communication through coherence. Some discrepancies between previous studies examining the role of oscillations may be due to the use of predefined frequency bands. A potential solution may be to identify the dominant coherent rhythm and define relevant frequency bands for each individual participant separately.

Large amplitude alpha oscillations are a common phenomenon in visual cortices and can be strongly modulated during resting state studies. However, it is important to note that our experimental paradigm is not a resting state study in which participants may modulate alpha activity by opening or closing their eyes. Instead, participants were free to engage in spontaneous behaviors such as reading, watching television, or conversing with family or staff as they were being monitored for seizures. These are complex behaviors whose underlying neural activity is likely to reflect typical brain function. Nevertheless, a common and persistent feature in our recordings is the presence of strong coherent alpha oscillations across large patches of temporal cortex.

Together, our results demonstrate that coherent alpha oscillations in the human brain satisfy important and necessary constraints for mediating communication between brain regions. Using complementary measures of functional connectivity in the time and frequency domain, we specifically demonstrate that large scale effective connectivity is rhythmically modulated by alpha oscillations. Whether alpha oscillations modulate the rate or timing of action potentials, or whether these oscillations are a byproduct of intrinsically rhythmic or externally modulated firing of large neuronal populations remains unknown. Similarly, whether there is a specific population of interneurons that generates and adjusts these oscillations to modulate communication between assemblies of principal cells, or whether there is an external source such as the thalamus that imposes this rhythmicity on all cortical regions is also unclear. Answering such questions will require performing experiments in which the oscillations in the field potentials and the spiking activity can be modulated independently in order to understand the physiological mechanisms behind alpha oscillations and to test if the location of the spikes relative to the alpha rhythm affects behavior. Nonetheless, our data demonstrate that correlated neural activity between brain regions, and local spiking activity within brain regions, are rhythmically modulated by alpha oscillations, providing a plausible substrate for how communication through large scale cortical networks can be flexibly modulated by neural oscillations.

STAR Methods

Lead contact and materials availability

Further information and requests for resources should be directed to and will be fulfilled by the Lead Contact, Kareem A. Zaghloul (kareem.zaghloul@nih.gov). This study did not generate new unique reagents.

Experimental model and subject details

Participants

Ten participants (3 male; 39 ± 3 years old; mean ± SEM) with drug resistant epilepsy underwent a surgical procedure in which platinum recording contacts were implanted subdurally on the cortical surface. In all cases, the clinical team determined the placement of the contacts to best localize epileptogenic regions. In all of the participants investigated here, the clinical region of investigation was the temporal lobes. Data were collected at the Clinical Center at the National Institutes of Health (NIH; Bethesda, MD). The research protocol was approved by the Institutional Review Board, and informed consent was obtained from the participants.

Method details

Intracranial Recordings and Pre-Processing

Intracranial EEG (iEEG) signals were sampled at 1000 Hz (Nihon Kohden) and the raw recordings were initially referenced to a common contact placed subcutaneously. Subdural contacts were arranged in both grid and strip configurations with an inter-contact spacing of 10 mm (PMT Corporation, Chanhassen, MN). In order to compare functional networks with similar spatial configurations and anatomical coverage across participants, we restricted our analysis only to electrode contacts belonging to a subdural grid placed over the lateral temporal cortex. This ensures that any investigations of connectivity are performed at a similar spatial scale with a similar spatial distribution of electrode contacts across participants (approximately a two-dimensional rectangular lattice, see Figure 2A). In each participant, this corresponded to a 32-contact grid, although in some individuals, one to four contacts were manually cut out from the grid during surgical placement. This provided 293 subdural grid contacts for potential analysis (29 ± 1 per participant). Contact localization was accomplished by co-registering the post-op CTs with the pre-op MRIs and mapped to both MNI and Talairach space. The resulting contact locations were subsequently projected to the cortical surface of a reconstruction of each individual participant’s brain [58].

We captured 30-second long continuous iEEG recordings from all electrodes in each participant and defined each 30-second recording as a block. We collected 10 blocks on two separate days for analysis, for a total of 20 blocks per participant. Data blocks collected from the same day were captured during a single two-hour recording, which we defined as a session, and were separated from one another by 10 minutes on average. We confirmed that each recording session was captured at least 12 hours before or after a clinically documented seizure. An epileptologist reviewed video captured during each recording session to ensure that participants were awake for the entirety of that session. We discarded blocks that contained obvious artifacts or epileptiform discharges.

In each session, we rejected electrodes exhibiting abnormal signal amplitude and/or large line noise. First, we divided each session into one-second epochs and for each electrode calculated the amplitude of the continuous time series over each epoch. Any electrode with a voltage trace whose average amplitude was more 1.5 than standard deviations away from the mean across all electrodes was flagged for visual inspection; any electrode which displayed clear artifacts throughout a session was subsequently rejected. We used the Chronux 2.11 toolbox to apply a local detrending procedure to remove slow fluctuations (≲ 2Hz) from each electrode’s time series and used a regression-based approach to remove line noise at 60 Hz and 120 Hz [47]. We subsequently inspected time-frequency spectrograms for each electrode, and manually rejected any electrode still exhibiting significant line noise power. A low-pass type I FIR filter (order = 108, f_cutoff = 105 Hz) was used to remove higher order line harmonics as well as high frequency noise. We discarded an electrode from all blocks if we rejected that electrode in any one block. Finally, to ensure that our results were not confounded by abnormal activity related to seizures, we also excluded all electrodes identified by the clinical team as lying within the seizure onset zone from our analyses. We retained 25 ± 1 electrodes for analysis in each participant. To approximate a reference free montage we subtracted the common average signal, calculated using the retained electrodes in each participant, from each electrode’s voltage trace, and finally, to further mitigate any confounds due to artifacts and epileptiform discharges we removed and interpolated any time points where the voltage exceeded 4 scaled median absolute deviations [59, 60].

Spike field coherence

For two participants, we additionally collected single unit and local field potential (LFP) recordings from a micro-array implanted on the temporal lobe. For both participants we analyzed two hours of micro-array activity which were captured simultaneously with the iEEG recordings. We digitally recorded microelectrode signals at 30 kHz using the Cereplex I and a Cerebus acquisition system (Blackrock Microsystems), with 16-bit precision and a range of ±8 mV. To obtain LFPs we performed the same line noise removal and channel selection procedure as for the iEEG channels. Each microelectrode’s raw voltage data was re-referenced by subtracting the average signal across all microelectrode channels and then downsampled to 1000 Hz. To extract neuronal spiking activity, we bandpass filtered the signal between 600 to 30000 Hz and re-referenced each electrode’s signal offline by again subtracting the global average. Following a whitening a procedure, each channel was separately spike-sorted offline using a fully-automated spike sorter, Mountainsort [61]. The analysis was limited to units with a signal-to-noise ratio ≥ 1, noise-overlap score ≤ 0.1, and an isolation score ≥ 0.95.

The spike field coherence between single unit spiking activity and the LFP is defined as the ratio of the power in the spike triggered average (STA) to the average power in the LFPs. We used two-second epochs to compute the STA, and then estimated the power spectral densities using Welch’s overlapped averaged periodogram method [47, 48]. Briefly, we divided the time series into overlapping one-second epochs (80% overlap), taper each epoch with a Hamming window, calculate the spectral density for each epoch at each frequency from 3 to 100 Hz in steps of 1 Hz and finally take the average density over all epochs to produce a single spectrum. We found the preferred phase for any frequency by finding the instantaneous phase of each spike’s LFP at t = 0 and taking the circular average of the phase values for all LFPs. We removed spike waveforms by interpolating the STA between t = −2ms and t = 8ms with a cubic spline.

To examine the relation between spiking activity and power within the alpha band, we first extracted the instantaneous alpha power for each unit’s corresponding LFP by convolving the time series with complex Morlet wavelets (wavelet number = 6) and taking the modulus squared of the analytic signal. We computed the power time series at both frequencies identified within the alpha band at which spiking activity appeared preferentially locked (see orange bars in Figure 1C) and averaged the results to produce a single time series. We extracted the instantaneous firing rate by convolving every individual spike train with a 500 ms long Gaussian kernel (FWHM = 175 ms). Next, for each unit we divided the resultant time series into non-overlapping 30 second windows, calculated the correlation between alpha power and spike rate for each window, and averaged the correlation over all windows to produce a single value for each identified unit.

Spectral coherence

We computed the magnitude squared spectral coherence between every electrode pair using the time series data from each electrode during each block (Matlab function ‘mscohere’) [47, 48]. The coherence between two time series, x(t) and y(t), is a function of frequency:

$$C(f)=\frac{{\left|P_{XY}(f)\right|}^2}{P_{XX}(f)P_{YY}(f)}\#$$ (1)

where P_XX and P_YY are the power spectral densities and P_XY is the cross-spectral density. The spectral densities were estimated using Welch’s method as described in the previous section. In this manner, we generated a single coherence spectrum for each electrode pair and each block. We defined the frequency at which the spectral coherence peaks within the alpha band as f_c.

We were interested in quantifying the extent to which the peak in the coherence spectrum exceeds the overall shifts in coherence across all frequencies that are due to non-specific and non-rhythmic (broadband) correlated activity between electrode pairs. To this end, we computed a type of peak prominence, C_excess, that reflects how much the coherence at the peak frequency exceeds the background 1 ⁄ f ^α activity. We refer to this measure as excess coherence. To compute this measure, we identified and removed the dominant peak in the alpha band for each average coherence spectrum in addition to the coherence values of the two adjacent frequencies flanking the peak. We then linearly interpolated across the removed points and subtracted the interpolated value from the original peak value to derive the peak prominence, C_excess (see Figure 2B). The excess coherence should be strictly positive, but for a small subset of electrode pairs with very low overall coherence the estimate yielded negative values due to the 1 ⁄ f ^α fall-off for low frequencies. In these cases we manually set C_excess to zero.

Directed functional connectivity

For each block and electrode pair, we computed the absolute value of the correlation between the corresponding time series data in overlapping one second epochs (80% overlap) without any temporal offset between them. In this case, the time delay is defined as τ = 0 and the average absolute correlation for that time delay over all such epochs in a block is $\overline{|R(0)|}$. We use the absolute value, rather than the raw correlation, to capture the magnitude of the correlation between the two time series rather than their relative polarity. We also performed the same calculation while imposing time delays between the voltage traces from each pair of electrodes from −250 ms to 250 ms in 1 ms steps. The result is a metric, $\overline{|R(\tau )|}$, which quantifies the average absolute correlation between two signals over many epochs, as a function of time delay. This metric is similar to windowed-scaled estimates of the cross-correlation function, which have been shown to be superior for analyzing correlations between non-stationary time series [62].

In order to compare correlations between different electrode pairs we computed a normalized measure of absolute correlation as function of time delay:

$$W(\tau )=\frac{\overline{|R(\tau )|}-{\mu }_{\overline{\left|R\right|}}}{{\sigma }_{\overline{\left|R\right|}}}\#$$ (2)

where ${\mu }_{\overline{|R|}}$ and ${\sigma }_{\overline{|R|}}$ represent the mean and standard deviation of $\overline{|R|}$ over all τ. We refer to this normalized measure as the cross-coupling function (See Figure 3A and Figure S3 for examples). The maximum W(τ) over all τ’s is a measure of how consistently correlations between the two signals are time-locked at the preferred delay, τ_w, relative to all other delays, and we define this value as the coupling, W_max, between that electrode pair. In a previous study, we used a non-linear metric based on windowed-scaled estimates of time-lagged mutual information and showed that qualitatively similar results can be obtained using absolute correlation [46]. The reduced computational demands, as well as the more straightforward interpretation of the correlation based metric motivated its use here. For each block we analyze only pairs whose maximal coupling occurred at a non-zero latency and phase lag. Any connections that may arise spuriously due to volume conduction or from common average referencing will exhibit a maximum coupling at zero latency [3, 63], and so we exclude all such pairs from our analyses.

To determine the relation between alpha power and coupling over time, we first computed a time dependent metric of coupling at the preferred delay, τ_w, for every coupled electrode pair in every block using overlapping 500 ms epochs (step size 100 ms), yielding a time series of coupling values, W_max(t). We also extracted the instantaneous power at f_c in each electrode of each pair by convolving both time series with a complex Morlet wavelet (wavelet number = 6) centered at f_c. We averaged the two power time series, resulting in a single average power time series for that electrode pair for each block. Because the time dependent coupling W_max(t), is calculated using overlapped 500 ms windows, we subsampled the power time series using a moving average (500 ms windows, 90% overlap) in order to compute the temporal correlations between coupling and alpha power.

To examine if coupling for a given electrode pair is affected by the phase of the underlying alpha oscillations at each site, we first aligned the voltage traces in each pair according to the preferred delay τ_w. We then used a complex Morlet wavelet (wavelet number = 6) centered at the identified peak frequency of coherence, f_c, (see Spectral Coherence) to extract the instantaneous phase of each signal. Based on these phases, we designated each point of the time series data in each electrode as belonging to one of four phase bins (rise −3π⁄4 to −π⁄4, peak −π⁄4 to π⁄4, fall π⁄4 to 3π⁄4, or trough 3π⁄4 to −3π⁄4). We identified periods where the instantaneous phase of both signals fell within the same phase bin and concatenated the time series data from each of these periods. In this manner, for each electrode pair, we derived separate time series data for each phase bin. We used these separate time series to then compute the absolute value of the correlation between each electrode pair at each pair’s preferred delay as a function of phase.

To ensure that correlations are not being biased toward any phase due to the concatenation of the time series, we constructed a surrogate distribution of this metric for each participant using the following approach. For every block and phase bin we counted the total number of time points that had a matching phase for a single electrode pair. For each surrogate distribution, we then randomly selected the same number of samples from each time series, while enforcing the constraint that the samples are taken consecutively for a duration of at least T⁄4, where T is the period of the alpha cycle. This ensures that, on average, each surrogate signal retains the autocorrelation structure of the real signals. We repeated this process, creating ten surrogate correlations for every phase bin in every block.

Simulations of coupling and coherence

In order to illustrate the relationship between time-locked coupling and coherence, we constructed simulated signals and examined the temporal and spectral properties of these signals as well as the measures of coupling and coherence between the (Figure S3). In the simplest case, two signals composed of autocorrelated noise plus sinusoids of the same frequency with a constant phase shift Δϕ (corresponding to a time delay τ_ϕ), have a single peak in the coherence spectrum, at f = f_c (Figure S3 column A). If these two time series share no other signal components (since the noisy signals have non-zero autocorrelation but are not cross-correlated), then there will be no coherence at any other frequency. In the time domain, the coupling function is oscillatory with no clear peak and no preferred time delay due to the periodicity of the signals. Phase locked oscillations at a fixed frequency by themselves can convey little information since the bandwidth for communication is small, and any frequency or amplitude modulations that may increase information capacity would result in correlations between other frequency components besides f_c.

Conversely, two signals may be correlated over a broad frequency band with a fixed time delay but share no oscillatory components (Figure S3 column B). In the time domain, the coupling function will have a sharp peak at the time delay, τ_w. In this case, the coherence spectrum will exhibit an overall increase in coherence across all frequencies rather than a peak at any single frequency. Coherence is proportional to the Fourier transform of the cross-correlation, which underlies the coupling function, and therefore peaks in coupling correspond to an overall shift in the coherence spectrum for all frequencies where the signal is above the noise. Adding in background 1 ⁄ f ^α noise to better mimic the spectral content of biological systems limits the frequencies that exhibit shifts in coherence since the coherence function decays as a function of the signal to noise ratio (Figure S3 column C). These simulated signals demonstrate that it is possible to have high correlations in the absence of a clear peak in coherence, as in the case of broadband correlated activity, or high coherence in the absence of broadband correlated activity, as in the simple case of two noisy sinusoids. Hence, while these time and frequency domain measures are not independent, analyzing them separately allows us to more easily distinguish frequency specific relationships from broadband correlations. This is analogous to how the auto-correlation function and the power spectral density are related via a Fourier transform, yet they are commonly analyzed together in order to highlight different aspects of a signal.

If large scale communication is being mediated via an oscillatory mechanism such as communication through coherence, then we would expect that communicating brain regions should exhibit both coherent oscillations as well as correlated broadband activity, and the phase delay between the oscillations should match the time delay between the correlated components. To illustrate this case, we constructed a final simulation comprised of a linear combination of noisy, correlated, and oscillatory signals, with the constraint that the phase delay between the oscillations, τ_ϕ, is equal to the time delay of maximum coupling, τ_w (Figure S3 column D). As expected, the spectral coherence in this case demonstrates a clear peak at f_c, and an overall shift for all frequencies where the signal power exceeds the noise. In the time domain, this spectral domain profile corresponds to a clear and prominent peak in coupling at the time delay, τ_w, superimposed on an oscillatory background. If large scale communication between brain regions is modulated by an oscillatory mechanism, we would expect to see these spectral and time domain profiles in the neural data.

Simulation parameters

A. Phase-shifted sinusoids with auto-correlated noise:

$$X_{\text{osc}}=\cos \left(2\pi f_{\text{c}}t\right)+Z_1(t)$$

$$Y_{\text{osc}}=\cos \left(2\pi f_{\text{c}}t-\varphi \right)+Z_2(t),\varphi =2\pi f_{\text{c}}{\tau }_{\text{w}}$$

where Z_1,2(t) are produced by convolving white noise, $\mathcal{N}(0,1)$, with a 25 sample Gaussian (FWHM = 9) to introduce autocorrelations.

B. Correlated broadband signals

The correlated signals were created by generating samples from a 2-dimensional Gaussian distribution of the form:

$$\left(\begin{array}{c}{X}_{\mathit{corr}}\\ Y_{\mathit{corr}}\end{array}\right)=\mathcal{N}(\mathit{\mu },\mathbf{\Sigma });\mathit{\mu }=\left(\begin{array}{c}0\\ 0\end{array}\right),\mathbf{\Sigma }=\left(\begin{array}{cc}1& 0.75\\ 0.75& 1\end{array}\right)$$

We then smooth these signals with a 25 sample Gaussian to achieve a non-zero autocorrelation at non-zero lags and shift them in order to introduce the time lag τ_w.

C. Background noise

The 1 ⁄ f ^α noise, (X_pink, Y _pink), was generated by passing white noise through a FIR filter whose frequency response was approximately 1 ⁄ f ^1.5 for high frequencies. The filter is a type I FIR filter (order=34) with an ideal amplitude response of the form 1 ⁄ (1 + 2f)^1.5 estimated using a least squares algorithm. The factor of 1 in the denominator introduces a knee for frequencies below ≈ 30Hz.

D. Linear combination

Letting $\tilde{X}$ represent the variable X normalized to a range of [−0.5,0.5], the combined signals are constructed as:

$$X_{\mathrm{mix}}=0.5{\tilde{X}}_{\text{osc}}+{\tilde{X}}_{\text{corr}}+{\tilde{X}}_{\text{pink}}$$

$$Y_{\mathrm{mix}}=0.5{\tilde{Y}}_{\text{osc}}+{\tilde{Y}}_{\text{corr}}+{\tilde{Y}}_{\text{pink}}$$

Broadband vs narrowband coupling

Coherence is proportional to the Fourier transform of the cross-correlation, which underlies the coupling metric. Therefore, as described in the previous sections, these time and frequency domain measures are not independent. In order to separate frequency specific interactions from broadband correlations we computed the excess coherence, C_excess, which captures how much the coherence exceeds the overall shift in the coherence spectrum that arises from broadband correlated activity (see Figure S3).

To further ensure that the relationships between coupling and excess coherence that we find do not arise merely from the presence of strong alpha activity at two electrodes, we computed a modified version of the coupling metric, W_notch. This metric is calculated in the same way as W_max except that we first apply a notch filter at the frequency f_c determined for each participant to remove the narrowband alpha oscillations (type I FIR minimum order filter, f₁^pass = f_c - 4, f₂^pass = f_c + 4; Matlab functions ‘firpmord’ and ‘firpm’). In this way, W_notch quantifies coupling in the absence of narrowband alpha activity. We should note that there is substantial broadband power near f_c, and filtering out these components may negatively impact true broadband correlations between the signals. Nevertheless, even when directly removing the alpha rhythm from the time series we found a significant correlation between alpha coherence and coupling across electrode pairs in each participant, as well as a good match between the phase delay of the alpha oscillations and the time delay of the non-alpha components (Figure S6).

Quantification and statistical analysis

Statistics

Results are reported as mean ± SEM unless otherwise noted. Descriptions of statistical tests and summary statistics are provided in results under the corresponding subsections and in Table 1. The significance threshold was set at p = 0.05.

Model based thresholding of coupling and excess coherence

In order to determine which electrode pairs have significant coupling or coherence, we used a Gaussian mixture clustering approach. This approach is motivated by the fact that for individual participants the distributions of coupling and coherence are very heavy tailed, and in many cases appear bimodal (see Figure 3C and Figure S3). This suggests that there are two populations, and that a two Gaussian model may be a good fit to the data. We implemented this by separately fitting the distributions of coupling and excess coherence with two component Gaussian mixtures, and using the posterior probability to assign values to each Gaussian component. We define values of coupling or excess coherence that are assigned to the Gaussian component with the larger mean as being significant. A threshold near the local minimum in these distributions represents a natural cut point, above which electrode pairs can be considered to have significant coupling or excess coherence. It is important to note that the goal of this procedure is not to find the model which best fits the data. Rather, the goal is to find the simplest model that fits the data well enough to allow for a robust identification of a threshold. As long as the threshold approximates the dip in the probability distributions, our results will not be appreciably affected.

Data and code availability

Except where otherwise noted, computational analyses were performed using custom written MatLab (MathWorks) scripts. This work utilized the computational resources of the NIH HPC Biowulf cluster (http://hpc.nih.gov). Processed data used in this study and custom MATLAB code used for analysis are available upon request.

Highlights.

• Analysis of human electrophysiology at multiple spatial scales

• Pairs of regions with correlated broadband activity are also alpha coherent

• The phase delays of the oscillations match the time delays of broadband activity

• Large-scale correlations and spiking activity are modulated by the phase of alpha