Fig 1.
Stochastic processes and covariance functions.
A) Example of a continuous-time oscillatory process (blue line) sampled at discrete equally-spaced time points though noise corrupted measurements (red dots). B–E) Samples (colored) and expected values (black) of the stochastic processes. The processes are a damped harmonic oscillator, second order integrator, first order integrator and residuals respectively. The samples start from an excited state and decay back to their respective stationary distribution. F) Illustration of the decomposition of a complex signal’s covariance function into simpler additive components. This corresponds to an additive decomposition of the measured time series. The second order integrator process has been excluded from this panel for visualization purposes.
Fig 2.
Simulation results on the decomposition of spatiotemporally mixed signals.
Performance of SGPD, SSA, EMD, ICA and PCA in recovering an oscillatory signal that was mixed with complex spatiotemporal noise. The performance is quantified as correlation with the ground truth. The red line is the median correlation across trials, the boxes contain correlations between the first and the third quartile. There are four conditions: A) low spectral and low spatial overlap; B) high spectral and low spatial overlap; C) low spectral and high spatial overlap; D) high spectral and high spatial overlap.
Fig 3.
Simulation results on recovery of spatial profiles of non-contiguous dynamic components.
Performance of SGPD, ICA and PCA in recovering an oscillatory signal with a non-contiguous spatial profile. A) Ground truth spatial profile of the dynamic component. B) Performance of SGPD, PCA and PCA in recovering the spatial profile of a dynamic component with bi-modal spatial profile on a two-dimensional cortical sheet. The performance is quantified as spatial correlation with the ground truth.
Fig 4.
Simulation results on the estimation of modulations in oscillatory amplitude.
A) Effect size of temporal GP and DPSS multitaper spectral analysis as function of mean percentage amplitude difference between simulated conditions. The parameters of the temporal GP-based decomposition (blue line) were estimated from the raw simulated time series. The spectral smoothing of the multitaper method (green line) was chosen for each to maximize the effect size. The red line is the effect size for a multitaper method with constant spectral smoothing of 0.6 Hz. B) Effect size ratio between temporal GP and (optimized) multitaper method as function of the mean amplitude difference between conditions.
Fig 5.
Simulation results on localizing the source of an oscillatory amplitude modulation.
A) Spatial maps of the simulated brain sources. The left map shows the spatial extent of the amplitude-modulated source while the two right maps show the interfering sources. The dipole orientation was set to be orthogonal to the mesh surface. B) Visualization of sensor activity as a mixing of the three sources. The dots represent MEG sensors. The color of the dots show the sign (red for positive and blue for negative) together with the magnitudes. The time series was taken from an occipital sensor. C) Scatter plot of the accuracy of SGPD and Harmony. The index was computed by dividing the total reconstructed effect within the amplitude-modulated cortical patch by the sum of total effects in the non-modulated patches. D) Scatter plot of the sharpness of SGPD and Harmony. The sharpness index was obtained by dividing the total reconstructed effect within the amplitude-modulated cortical patch by the total effect elsewhere. For the purpose of visualization, in both scatterplots, we excluded some outliers (> 5 × median). These outliers arise when the denominator of one of the indices becomes too small. The outliers have been removed from the figure but they were involved in the calculation of the medians for the two methods.
Fig 6.
Estimation of the model covariance functions.
Parametric fit of the MEG auto-covariance functions of Participant 1 and Participant 2. The red lines refer to the estimated parametric model and the blue lines reflect the empirical auto-covariance of the measured time series. A single auto-covariance was obtained from the multi-sensor data by performing a principal component analysis and averaging the empirical auto-covariance of the first 50 components, weighted by their variance. The parameters of the model were estimated using a least-squares simulated annealing optimization method. The graphs have been scaled between 0 and 1 by dividing them by the maximum of the individual empirical auto-covariance.
Fig 7.
Reconstructed source-level neural activity of Participant 1. A) Reconstructed time series of the four dynamic processes localized in a right parietal cortical vertex. B) Reconstructed spatiotemporal dynamics of alpha oscillations along the x axis. This choice of axis is arbitrary and has been chosen solely for visualization purposes. The source-reconstructed activity has been normalized by dividing it by the maximum of the absolute of the spatiotemporal signal.
Fig 8.
Caudal-to-rostral progression of alpha amplitude attentional modulation.
A) Group average of alpha amplitude attentional modulation as function of time. B,C) Alpha amplitude attentional modulation for participants 1 and 2, respectively. D) Spatial map obtained by computing the slope of the average alpha difference between cued and non-cued conditions as a function of time for each cortical vertex.
Table 1.
Comparison of signal decomposition methods.
Fig 9.
Spherical harmonics and covariance functions.
Visualization of the spherical harmonics morphed onto the cortex and the resulting spatial correlation structure. A) Example of spherical harmonics on the brain cortex for frequency numbers from 0 to 2. For each frequency number l there are 2l+1 harmonics with “phase” number m ranging from −l to l. As clear from the picture, the spatial frequency increases as a function of the frequency number. In all our analyses we truncated the harmonic expansion after the 11th frequency number. B,C) Prior correlation structure induced by Eq (24). Panel B shows the prior correlations on the cortical surface from a cortical point identified by a red dot. Panel C shows the same function on the spherical hull. The spatial correlations are determined by the frequency discount function f(l); here we used the same smoothing parameters as all analyses in the paper: k = 2 and υ = 3.