Envelope analysis of the human alpha rhythm reveals EEG Gaussianity

The origin of the human alpha rhythm has been a matter of debate since Lord Adrian attributed it to synchronous neural populations in the occipital cortex. Although some authors have pointed out the Gaussian characteristics of the alpha rhythm, their results have been repeatedly disregarded in favor of Adrian’s interpretation; even though the first EEG Gaussianity reports can be traced back to the origins of the field. Here we revisit this problem using the envelope analysis — a method that relies on the fact that the coefficient of variation of the envelope (CVE) for continuous-time zero-mean Gaussian noise (as well as for any filtered sub-band) is equal to , thus making the CVE a fingerprint for Gaussianity. As a consequence, any significant deviation from is linked to synchronous neural dynamics. We analyzed occipital EEG and iEEG data from massive public databases. Our results showed the human alpha rhythm can be characterized either as a synchronous or as a Gaussian signal based on the value of its CVE. Furthermore, Fourier analysis showed the canonical spectral peak at ≈ 10[Hz] is present in both the synchronous and Gaussian cases, thus demonstrating this same peak can be produced by different underlying neural dynamics. This study confirms the original interpretation of Adrian regarding the origin of the alpha rhythm but also opens the door for the study of Gaussianity in brain dynamics. These results suggest a broader interpretation for event-related synchronization/desynchronization (ERS/ERD) may be needed. Envelope analysis constitutes a novel complement to Fourier-based methods for neural signal analysis relating amplitude modulation patterns (CVE) to signal energy.


Introduction
The neuronal dynamics that generate the human alpha rhythm, first described by Berger (1929), have been a matter for discussion since Lord Adrian attributed it to synchronous neural populations in the occipital cortex. Adrian undertook a thorough examination of the alpha rhythm and attributed its origin to neural synchronization (Adrian and Matthews, 1934); an idea that rapidly became mainstream and accepted as the standard interpretation for EEG phenomena. Indeed, neural synchronization is a fundamental mechanism in EEG signal generation (Buzsáki et al., 2012) and a widespread phenomenon at many levels of biological organization (Singer, 1999;Varela et al., 2001). Adrian also reported that the rhythm can be entrained by a flicker to frequencies as high as 25 [Hz]. The famous cybernetician Norbert Wiener hypothesized that the alpha rhythm is produced by a synchronous population of coupled nonlinear oscillators with similar natural frequencies (Wiener, 1948;Strogatz, 1994) -a claim supported by Kuramoto's early model of coupled oscillators systems (Kuramoto, 1975).
Nevertheless, some authors have reported that the alpha rhythm exhibits Gaussian characteristics, but this possibility has been consistently disregarded for any EEG phenomena under the assumption that the amplitude of a signal emerging from an asynchronous population is either zero or, at best, too low to be measured (Cohen, 2017;Lachaux et al., 1999;Singer, 1999). On the contrary, the amplitude of a signal emerging from a population of asynchronous oscillators is proportional to √ n (Elul, 1972), as was recognized early in the EEG literature by Motokawa (1943). In fact, Motokawa contended that the alpha rhythm originated from a population of uncorrelated oscillators (Motokawa and Mita, 1942). In an elegant approach, Saunders (1963) compared the distribution of raw signal amplitude values of "well developed" alpha rhythm epochs against the Gaussian distribution and found a close fit. From this perspective, some researchers have suggested that the underlying neural generator may be better described as an alpha filter (Lopes Da Silva et al., 1973): a system that receives random input and produces the characteristic narrow band signal centered ≈ 10[Hz]. This interpretation of the alpha rhythm generator as a bandpass filter can be traced back to Prast (1949). Dick and Vaughn (1970) went one step further and estimated the envelope function by the complex demodulation method. They found the envelope values of alpha rhythm closely fit the Rayleigh distribution, which is important because the envelope function of a Gaussian signal follows that distribution (Schwartz et al., 1995). Furthermore, they proposed to study the changes of the variance of the envelope to explore the time-variant behaviour of the alpha rhythm with an approach similar to ours, but which did not exploit the properties of the envelope of Gaussian noise.
Another remarkable characteristic of the alpha rhythm is what Adrian and Matthews described as "waxing and waning" or "pulsating" activity, i.e. a distinctive amplitude modulation pattern. While sometimes ignored, there is some interindividual variability among alpha amplitude modulation patterns (Niedermeyer, 2005). Davis and Davis (1936) reported as many as four distinct alpha rhythm types based on their amplitude modulation, including cases where little or no waning is present. These distinct patterns are also reflected in the frequency of the alpha rhythm envelope (Schroeder and Barr, 2000). Consequently, these different alpha modulation regimes result in characteristic raw signal and envelope waveforms. Interestingly, in recent years, some authors have pointed out the importance of the signal waveform to detect features in neural signals that may not be evident using Fourier-based methods (Cole and Voytek, 2017). Indeed, as we will show, the signal waveform does offer information about neural dynamics; information that can be extracted analyzing the envelope signal.
In a previous report, Díaz et al. (2018) used rat EEG to describe a novel framework to characterize EEG signals: the envelope analysis. This analysis relies on the fact that the coefficient of variation of the envelope (CVE) of a continuous-time zero-mean Gaussian noise signal (and any of its sub-filtered bands) is invariant and equal to (4 − π)/π ≈ 0.523 (Schwartz et al., 1995). Thus, the Gaussian CVE is a universal constant and acts as a mark of Gaussianity. To capture this time-variant behaviour, the signal under study was segmented and the CVE for each epoch estimated, as well as the mean of the envelope (ME) to measure energy. As Gaussianity is a hallmark of asynchrony, any deviations from the CVE of Gaussianity reveals synchronous processes. Díaz et al. (2007) showed that a signal emerging from a population of phase-locked nonlinear oscillators exhibits low-CVE values, while high-CVE signals emerged only from unsteady states.
Hence, CVE (morphology) and ME (energy) can be combined to produce a novel phase-space in which signals can be categorized: the envelope characterization space. This space unifies generator dynamics, energy and signal morphology in a single framework. Using this framework, we propose here to classify EEG epochs not by frequency alone but by the two parameters defining the envelope characterization space: CVE and the mean energy of its envelope.

EEG data
We applied the envelope analysis to the EEG data of the "Leipzig Study for Mind-Body-Emotion Interactions" (LEMON) dataset (Babayan et al., 2019). This is a cross-sectional, publicly available dataset that contains resting-state EEG data from 216 younger and older healthy adults and is part of the MPI Leipzig Mind-Brain-Body database. The study protocol was in accordance with the declaration of Helsinki. All participants provided written consent prior to assessments and were screened for drug use, neurological, cardiovascular, and psychiatric conditions.
The recordings were obtained using 62 active electrodes, 61 EEG channels in accordance with the 10-10 system (Oostenveld and Praamstra, 2001)  processing. Outlier channels were rejected, and time intervals with extreme deflection or frequency bursts were removed after visual inspection. Additionally, PCA and ICA was applied for further artifact rejection.
The reader can consult (Babayan et al., 2019) for details.

Synthetic EEG data from a population of weakly coupled oscillators
Synthetic EEG signals were created using the order parameter of a population of weakly coupled nonlinear oscillators following the Matthews-Mirollo-Strogatz model (Matthews et al., 1991). Simulations were implemented using the DifferentialEquations.jl package (Rackauckas and Nie, 2017). A system of 800 oscillators was instantiated, whose natural frequencies followed the uniform distribution with a bandwidth ∆ = 0.8. Solutions for the order parameter were obtained by a fourth order Runge-Kutta method(timestep=1/250) using three distinct coupling constant K: 1.1(phase-locking), 0.1(incoherence), and 0.9(chaos). A 24[s] epoch was isolated from each case, and white noise and pink noise were added to reproduce the normal EEG spectral background.

iEEG data
The "atlas of the normal intracranial electroencephalogram" (AN-iEEG) dataset (Frauscher et al., 2018) was also analyzed to further study the human alpha rhythm generator dynamics. The AN-iEEG dataset is a massive resting-state iEEG dataset collected from candidates for epilepsy surgery at three different hospitals in Canada and France. The recordings were obtained while the subjects were awake with their eyes closed and were selected to exclude epileptic activity. The signals were recorded using either sEEG, subdural strips, or subdural grids at the different hospitals, and the electrode locations were projected into a common brain map. Segments 60[s] long were selected visually from each recording. The data was filtered at 0.5-80 [Hz] with a FIR filter and decimated to 200 [Hz]. An additional adaptive filter was used to reduce power-line noise.

EEG data analysis
We followed the pipeline reported by Díaz et al. (2018). All computational procedures were implemented using the Julia programming language (Bezanson et al., 2017)  CVE values were color encoded using the "buda" color scheme (Crameri, 2018) as low-CVE/purple, mid-CVE/salmon, and high-CVE/gold.

Surrogate Data for Gaussianity
To obtain a probability model for Gaussianity, 10 6 instances of a 6000-point vector were simulated, sampling each point from a normal standard distribution N (0, 1). As in the experimental data, a sampling frequency of 250[Hz] was assumed. These in silico generated epochs were treated as previously described in subsection 2.2 for the experimental data. The same filter was applied to each epoch to obtain the alpha band. The envelope was calculated, 2[s] segments were cut from both ends, and the CVE and ME values were computed. Then, the probability density function(PDF) and the cumulative density function (CDF) were calculated for the CVE values of these alpha-filtered white Gaussian noise epochs. The 0.005 and 0.995 quantiles were estimated to obtain the lower and upper limits of a 99% confidence interval to assess the null hypothesis H 0 : an EEG epoch is indistinguishable from white Gaussian noise.

Envelope characterisation space. Scatterplots, density plots and vector field
The envelope characterization space is a phase space that includes CVE and EEG amplitude measured for each epoch. EEG amplitude is represented by the mean of the envelope(ME), which is log-transformed and normalized using each subject and each subject's channel statistics. This phase space is visualized by scatterplots and density plots. The density plots were built estimating 2D-histograms with a 500x500 matrix. Rows and columns were filtered with a 51-coefficient binomial kernel, and the matrix was plotted using an alternating colour/white palette to produce a contour-like plot. In addition, as the (CVE,log(ME)[zscored]) data for each subject and each channel's recording formed an independent data cloud containing

Time-frequency analysis
Time-frequency analysis was applied to a complete recording that did not suffer any time interval removal(960[s]).
The eyes-closed and eyes-open blocks were re-concatenated to recover the initial interleaved protocol. Then, the data was divided in 20[s] epochs with an overlap of 50% and treated as in Prerau et al. (2017) setting the time-bandwidth=10[s] and number of tapers=20, resulting in a frequency resolution of 0.5 [Hz]. Only frequencies 20≥f≥1 were kept and the spectral power was log-transformed and colour encoded (dark blue < cyan <yellow < dark red).

Comparison of Fourier analysis and envelope analysis
One raw representative epoch for each CVE class was selected and re-analyzed using both Fourier analysis and envelope analysis. The spectrum for each epoch was calculated before filtering, while after filtering the CVE and energy were obtained and visualized in the envelope characterization space. The same steps were applied to the synthetic EEG data from a population of weakly coupled nonlinear oscillators.

iEEG data analysis
The iEEG data was treated as the EEG data, with some exceptions. The CVE surrogate distribution was re-calculated setting the sampling frequency to 200[Hz] to match the AN-iEEG sampling frequency. Filtering, segmentation, and envelope-parameter computation remained untouched. In total, 416 epochs were analyzed. In the envelope characterization space, energy values were log-transformed but not normalized, as the AN-iEEG dataset contains eyes-closed data only. Three representative epochs recorded from the calcarine cortex were used to compare Fourier analysis with envelope analysis.

CVE for alpha-filtered Gaussian noise
The main motivation behind the envelope analysis rests upon the statistical characteristic of Gaussian signals and their envelopes. Therefore, we first review these mathematical foundations. For any zero-mean continuous-time Gaussian signal (and any of its filtered sub-bands) ( Fig. 1.a), the respective envelope signal follows a Rayleigh distribution ( Fig. 1.b). Since the coefficient of variation for the entire family of Rayleigh distributions is invariant and equal to (4 − π)/π ≈ 0.523, the coefficient of variation of the envelope (CVE) of a infinite zero-mean Gaussian signal always display this value that acts as a fingerprint for Gaussianity ( Fig. 1.c). Thus, CVE is a dimensionless, scale-independent parameter to assess Gaussianity. Nonetheless, CVE = (4 − π)/π only holds for infinite signals, and the CVE distribution for discrete-time Gaussian signals must be estimated by computer simulations. This distribution varies both with epoch length and bandwidth (as well as sampling frequency).
We first estimated the CVE probability density functions for Gaussian signals of different lengths (5, low-CVE, mid-CVE, and high-CVE ( Fig. 1.e). Additionally, each CVE class displays distinctive signal morphologies produced by different amplitude modulation patterns, and thus reflect distinct underlying generative processes ( Fig. 1.f).

Alpha-band CVE confidence interval and CVE for resting state EEG
To characterize the human alpha rhythm using the CVE, we first obtained confidence intervals of Gaussianity to test the experimental resting-state EEG epochs from the LEMON dataset. For our simulations, we used 24[s] epochs with a passband in ≈ 8-13 following the definition of Kane et al. (2017). As expected, the CVE probability density function for alpha-filtered Gaussian noise epochs obtained by simulations is bell-shaped and unimodal with mean= 0.520, which is near the fingerprint of Gaussianity (4 − π)/π ≈ 0.523 (Fig. 2

Envelope characterization space and vector field
The envelope characterization space is a two-dimensional phase space based on two envelope signal parameters: (i) coefficient of variation of the envelope (CVE) and (ii) mean of envelope (ME). Both values have to be combined in a single framework as the CVE is a scale-independent statistic. Scatterplots in Fig. 3 show the envelope characterization space for eyes-closed and eyes-open conditions (all channels together).
In Contour plots permit further analysis of the envelope characterization space (Fig. 4). Eyes-closed data (blue) strongly cluster in the Gaussian region around 1 standard deviation above the mean. While eyes-open data (orange) also clusters in the Gaussian region, most epochs group around 1 standard deviation below the mean.
This difference of about 2 standard deviations in the overall amplitude change between both conditions is also found when data from each subject and each channel are analyzed independently. Vectors from the eyes-open centroid to the eyes-closed centroid were traced and plotted in the envelope characterization space (Fig. 5). Each of these vectors was classified as low-CVE, mid-CVE or high-CVE based on the CVE coordinate of the eyes-closed centroid. While there is some variability in the magnitude and phase of these vectors(especially high-CVE vectors), the overall trend is an amplitude change of about 2 standard deviations in all three classes of vectors. The presence of low-CVE vectors shows that some subjects generate preferentially rhythmic alpha rhythms, although many subjects generate data that is either Gaussian or pulsating overall.

Spectrogram and CVE during open/closed conditions
The spectrogram of a complete recording (960[s]) shows that alpha power increases during the eyes-closed condition regardless of the CVE values involved (Fig. 6,left panel), as CVE is straddles on all three CVE intervals. During the eyes-open condition, CVE tends to settle inside the Gaussian region. Spurious high-CVE epochs are often found in the transition between conditions as those epochs contain both eyes-open (low energy) and eyes-closed (high energy) data, thus appearing pulsating. Three representative eyesclosed epochs associated with high alpha power display the morphology/amplitude modulation characteristic of their respective CVE values (Fig. 6,right panel).

Fourier/envelope analysis comparison
As some authors have argued in favour of time domain methods over frequency domain methods to study brain rhythms (Jones, 2016), we re-analyzed three representative epochs (same data in Fig. 6) using Fourier analysis and envelope analysis (Fig. 7). A closer inspection of their pre-filtering spectra reveals that all three epochs contain the canonical spectral peak at ≈ 10[Hz]. Furthermore, their spectral profiles are almost indistinguishable from each other ( Fig. 7.b). The CVE, however, can easily discriminate among these epochs, as shown in the envelope characterization space (Fig. 7.c). A similar trend was obtained through analysis of synthetic signals obtained from the order parameter of a synchronization MMS model of coupled oscillators ((Matthews et al., 1991)). While the three alpha-filtered epochs display different morphologies ( Fig. 7.d), their pre-filtering spectra cannot be separated from each other (Fig. 7.e). Noticeably, the spectral peak at ≈ 10[Hz] is much narrower than for experimental data. Again, these three synthetic epochs can be distinguished by their CVE in the envelope characterization space (Fig. 7.f).

Envelope analysis of intracranial EEG data
Epochs obtained from occipital intra-cortical EEG data (AN-iEEG dataset) are highly clustered in the Gaussian region, while only a small percentage of epochs (5%) stretch into the low-CVE region (Fig. 8). We also analyzed AN-iEEG data specifically recorded from the calcarine cortex (Fig. 8, right panel) We applied Fourier analysis and envelope analysis to calcarine cortex data and found the same result as when we analyzed experimental EEG and synthetic data (Fig. 9). The spectral profile for calcarine epochs is almost the same for all, and the spectral peak at ≈ 10[Hz] is much broader than the scalp-level EEG and synthetic counterparts ( Fig. 9.b). Nonetheless, these three iEEG epochs display different amplitude modulation regimes; a feature that is revealed in their CVE (Fig. 9.c).

Discussion
In this study we used envelope analysis to study the human alpha rhythm. The core of our method is the observation that, for Gaussian noise (or any its filtered sub-bands), the coefficient of variation of the envelope adopts the value (4 − π)/π ≈ 0.523 independently of the mean power of the EEG epoch under study. We demonstrated in Figures 1 and 2 the basic statistical ideas needed to work with the CVE of a given signal and how to build confidence intervals by computer simulations for the hypothesis that a given EEG epoch is indistinguishable from Gaussian noise. Furthermore, we introduced the concept of the envelope characterization space as a method to classify EEG epochs according to their mean energy and their CVE.
Interestingly, in this phase space, the data clouds derived from eyes-open and eyes-closed condition are very different. Our results underscored the classic result that EEG epochs for eyes-closed are more energetic.
We found (Figure 3 and 4) that most epochs for the eye-closed condition are similar to a Gaussian signal (i.e signal that looks like noise) or have a phasic temporal profile. Also in the envelope characterization space, we quantified the mean transitions between eyes-open and eyes-closed conditions ( Figure 5) and found that most transitions begin as a random noise and end up as either random noise or pulsating activity.
An important observation is that signals with similar frequency contents could have different CVE (Figure 6 and 7). The envelope analysis is not restricted to scalp EEG signals. In effect, iEEG signals can be classified using their CVE ( Figure 8)  in the supposed cradle of α rhythm (Figure 9). Interestingly, most EEG epochs analyzed in this work carried the hallmark of Gaussianity, thus supporting Motokawa's interpretation of asynchrony. However, an important number of epochs were found to be highly rhythmic, as shown by their low-CVE values, in accordance with Lord Adrian's interpretation of neural synchronization. It seems that envelope analysis invites us to expand our viewpoints beyond the ideas of event-related synchronization (ERS) and desynchronization (ERD) and consider that the internal traffic of neural signals may be more akin to the coupling among randomly interacting oscillators.

Concluding remarks
Here, we used envelope analysis to characterize the human alpha rhythm. Envelope analysis offers an elegant time-domain method to assess Gaussianity based on the study of the envelope signal (Hidalgo et al., 2022). Previous studies regarding EEG Gaussianity can now be reassessed by envelope analysis (e.g. Elul, 1969). Furthermore, envelope analysis puts neural synchronization and Gaussianity in the framework of coupled oscillators systems (K,MMS), and it allows the continuous monitoring of neural processes purely by CVE. Our results confirm the interpretations of both Adrian (synchronization by coupling) and Motokawa (random addition of elements leading to Gaussian noise), opening a new door to explore dynamics in EEG and other neural signals (e.g. LFP, iEEG, MEG) (Rao and Edwards, 2008).

Declaration of competing interest
None of the authors have potential conflicts of interest to be disclosed. Notice these distributions also vary with the sampling frequency(not shown). (e) The CVE distribution for discrete-time Gaussian noise allows constructing confidence intervals (both dashed lines) to assess the null hypothesis of Gaussianity. Hence, with respect to CVE, three different classes of behavior can be established: low-CVE (purple), mid-CVE (salmon), and high-CVE (gold). (f) These CVE classes correspond to distinctive amplitude modulation patterns that are linked to different generative processes. Low-CVE signals are synchronous, rhythmic signals coming from phase-locking processes, while mid-CVE signals correspond to Gaussian noise, and high-CVE signals are distinctively non-sinusoidal and pulsating.  For each corresponding epoch, the mean of the envelope (ME) and CVE were calculated. Then, the ME was log-transformed and Z-scored using the mean and standard deviation from the whole recording (eyes-closed and eyes-open), resulting in normalized envelope amplitude values for each subject and each channel data. Left panel: Eyes-closed data (22893 epochs) show a CVE excursion from 0.3 to 1.2. The epochs corresponding to low-CVE values also have an energy one standard deviation above the mean. The big majority of epochs have CVE values corresponding to Gaussian or pulsating signals. Some epochs display an energy level below the mean energy, which corresponds to subjects whose alpha rhythm is either weak or non-existent (as widely reported in the literature). Additionally, high energy epochs are present in all three CVE classes. Right panel: Eyesopen data (22634 epochs) mostly show amplitude below the mean energy, as expected from the experimental condition, and with a clear tendency towards mid-CVE and high-CVE values. Low-CVE values are almost absent. Dotted line at 0.520, the mean of the surrogate alpha-filtered Gaussian noise CVE distribution.

Figures
Dashed lines at 0.460 and 0.586 depict the 99 % confidence interval for Gaussianity. Transparency was set to 50% to highlight cluster density.    : Envelope analysis reveals both Gaussian and synchronous dynamics in event-related activity. Three eyes-closed EEG epochs were re-analysed using Fourier analysis(raw data) and envelope analysis(alpha-filtered data). (a) Three experimental alpha-filtered epochs and their respective envelopes are shown(same data in Fig. 6), each epoch belongs to one CVE class (rhythmic, Gaussian, phasic). (b) The three epochs contain an energy peak ≈ 10[Hz] and their spectra are almost indistinguishable from each other. (c) On the other hand, the envelope analysis for the alpha-filtered data does discriminate between the three epochs by CVE. (d), (e), (f) Data obtained from the synchronization model (MMS, (Matthews et al., 1991) ). Epochs with different CVE, but similar spectra and showing a marked peak at 10Hz but the epochs can be separated by their CVE in the envelope characterization space. Dotted line at 0.520, the mean of the surrogate alpha-filtered Gaussian noise CVE distribution. Dashed lines at 0.460 and 0.586 depict the 99 % confidence interval for Gaussianity.  : Envelope analysis reveals both Gaussian and synchronous dynamics in intracranial event-related activity in the alpha band. Three eyes-closed AN-iEEG dataset epochs recorded from the calcarine cortex were re-analysed using Fourier analysis (raw data) and envelope analysis (alpha-filtered data). (a) Three experimental alpha-filtered epochs and their respective envelopes are shown, each epoch corresponding to one CVE class (rhythmic, Gaussian, phasic). (b) The three epochs contain energy at ≈ 10[Hz] and their spectra are very similar. The spectral peak is much broader than the observed in the EEG or synthetic data (Fig. 7). (c) Although their spectra are similar, the envelope analysis for the alpha-filtered data discriminates between the three epochs by CVE in the envelope characterization space. Dotted line at 0.520, the mean of the surrogate alpha-filtered Gaussian noise CVE distribution. Dashed lines at 0.460 and 0.586 depict the 99 % confidence interval for Gaussianity.