Fig 1.
Resting EEG as a sum of damped independent alpha oscillatory processes.
(a) Power spectral density of the model electrocortical impulse response cos(2πfα t) exp(−γt)Θ(t) for γ = {0.1, 1, 10, 50}s−1, where Θ(t) is the Heaviside step function. Resting EEG is modeled as a continuous sum of such processes over some suitable interval of dampings [γl, γh]. In general we aim to find the appropriate weights {wi} that account for the shape of the resting EEG power spectrum. (b) A 1/f spectrum (dashed blue) can be approximated as a sum of 1/f2 relaxation processes (solid blue) that are uniformly distributed in damping over some interval which for the displayed case is [γl = 10−2, γh = 102] s−1. For clarity we have assumed fα = 0 Hz.
Fig 2.
Validation of the inverse method by the recovery of prior damping distributions.
(a) Unimodal prior damping distribution. (b) Bimodal prior damping distribution where both modes of same peak magnitude (c) Bimodal prior damping distribution where the relative magnitude of the peak of the most weakly damped mode is 0.1 that of the peak of the heavier damped mode. (d) Trimodal prior damping distribution where all modes are of the same peak magnitude. (e) As for (d) but relative peak mode heights are 0.01, 0.1 and 1 respectively. (f) As for (d) but relative peak mode heights are 1, 0.1 and 0.01 respectively. Blue lines are the prior distributions, whereas red dotted lines represent the recovered distributions using the inverse Tikhonov regularisation method and λopt are the optimally chosen regularisation parameters. For further details refer to Methods.
Fig 3.
Example damping distributions for strong alpha blockers.
(a-f) Damping distributions obtained from EC (blue) and EO (red) power spectra for six individuals (three per data set) who presented with strong alpha blocking. Plotted alongside the experimental power spectra are the respective model fits (EC—blue, EO—red) generated using the EC and EO damping distributions in the forward problem Eq 14. The inferred damping distributions are clearly bimodal in shape, a feature common across both data sets. The first mode of the EC damping distribution is generally peaked at a smaller damping value and has a larger respective area when compared to the EO distributions.
Fig 4.
Resting EEG power spectral densities are well described by a sum of damped alpha oscillatory processes.
(a) Histograms of the minimum residual sum of squares (RSS) error between experimental and model spectra for EC and EO states in pooled data set analysis. The median RSS error of EC = 0.0079 and EO = 0.0090 (Δmedian = −0.0011, p = 10−6) are associated with objectively good fits. (b) Examples of model fit quality for three distinct cases. Plots are ordered top to bottom with decreasing fit error (EC spectra presented).
Fig 5.
Experimental and model spectral scaling in EC and EO states.
(a) Probability distribution of experimental spectral scaling exponent (βE) for EC (blue) and EO (red) states with data set 1 and 2 pooled for analysis. (b) Same as (a) but for optimal model fit spectral scaling exponent (βM). (c) Scatter plot of the experimentally determined spectral scaling parameter (βE) versus the model calculated spectral scaling parameter (βM) for EC (dots—blue) and EO resting state (dots—red). The experimental and model fit scaling parameters are well correlated across both recording conditions (EC: r = 0.64, p = 10−6 / EO: r = 0.81, p = 10−6). Statistical significance calculated via a nonparametric permutation test.
Fig 6.
Spectral and topographic properties of alpha blocking in healthy participants.
(a-f) Examples of spectral variation between EC and EO resting states (O1 electrode) with three subjects shown for each data set. Plots are ordered top to bottom with decreasing alpha blocking (JSD). The exemplar cases demonstrate the contrast between strong and weak alpha blockers. (g-j) Topographic maps showing the occipital dominance of alpha blocking for a typical individual (g-h) and at the group level (i-j) across data set 1 and 2. Group level average is computed across subjects on an electrode-wise basis. Note, in order to highlight the consistent pattern of occipital dominance topographic maps are all displayed using different scales.
Fig 7.
Alpha blocking is driven by changes in the distribution of alpha oscillatory damping.
Topographic maps of mean difference between the weakly damped measure in EC and EO states (a) Data set 1 (b) Data set 2. Red asterisks indicate significant (p < 0.05) differences in the weakly damped measure between EC and EO states calculated using the Max/Min-t permutation test (see Methods) and corrected for multiple comparisons. (c) Scatter plot of electrode-wise Jensen-Shannon divergence and the difference in the weakly damped measure between EC and EO, averaged across all participants in each data set (red = data set 1, blue = data set 2). In general the two measures are well correlated with each other: data set 1, r = 0.838, p = 10−6; data set 2, r = 0.901, p = 10−6. Note that the group-level quantities differ between the data sets, in particular the <JSD>. The reason for this is not clear. It is possible that it is due to differing experimental configurations of the two studies, or more likely, it is due to a larger prevalence of strong alpha blockers in data set 2 (Fig 6j). Statistical significance was validated using a nonparametric permutation testing.