Fig 1.
Alpha-band oscillations in human IRF and in a predictive coding model.
(A) Cross-correlating the white-noise sequence of a stimulus with simultaneously recorded EEG produces an IRF, which reverberates at 10 Hz for several successive cycles (1 representative subject shown here, electrode POz). (B) A simple predictive coding model in which the hierarchically higher level makes predictions y(t) about the input received by the lower level, and the residual x(t) (prediction error) is used to update the next prediction. Residual and prediction signals are transmitted to the next or previous level (respectively) with a communication delay ΔT. Such a model, with physiologically plausible parameters, generates an oscillatory IRF at 10 Hz. (C) The oscillatory IRF produced by the model, with communication delay ΔT = 12 ms, and neural membrane time constant τ = 17 ms. (D) Systematic exploration of these two parameters suggests that alpha reverberation is a robust phenomenon (red colors) within a biologically plausible range of values. EEG, electroencephalography; IRF, impulse response function.
Fig 2.
Alpha-band traveling waves in EEG IRFs.
(A-B) IRF as traveling waves can be observed over the 7 midline electrodes of the 10–20 system (Oz to Fz; 1 representative subject). From the 2D map obtained by stacking signals from the 7 electrodes, we compute a 2D FFT and derive a log ratio between spectral quadrants that quantifies the direction of the waves. Positive values are associated with FW waves and negative values with BW ones. (C) BW (red) and FW (blue) values (max value in the corresponding quadrant of the 2D FFT) computed over 20 subjects. Participants were watching a white-noise luminance sequence (see Fig 1A). (D) Log ratios computed from the values in (C). The BF of a 1-sample t test against zero confirms the presence of FW waves. See S1 Data. BF, Bayes factor; BW, backward; EEG, electroencephalography; FFT, fast-Fourier transform; FW, forward; IRF, impulse response function.
Fig 3.
Alpha-band traveling waves in a hierarchical predictive coding model.
(A) Multilevel version of the model: the same parameters (ΔT = 12 ms and τ = 20 ms) are used throughout. The model is fed either a time-varying input (left) or a time-varying prior signal (right), reflecting top-down expectations computed in other parts of the brain. (B) Two-dimensional maps with only input (left column) or prior signals (right column): traveling waves are visible both in the IRF (computed by cross-correlating prediction signals with input or prior signals, top row) and in the raw prediction signals (considered as a proxy for the EEG, lower row). All the values have been z-scored level-wise for visualization purposes only. The numbers in white show the log ratio for each simulation: positive or negative numbers reveal respectively FW or BW waves. (C) Systematic exploration of parameters ΔT and τ (with τD kept constant at 200 ms). The upper panels show the temporal frequency of the peak spectral component in the 2D Fourier spectrum, with the same color scheme as in Fig 1D. In the lower panels, the direction of traveling waves was similarly explored (color indicates log ratio, with absolute values below 0.2 set to 0 for increased readability). When the model was fed the input or the prior, we observed mostly positive or negative values, respectively. (D) Explorations of model behavior as parameter τD is varied systematically (x-axis, from 20 ms to 200 ms, stride of 5). From top to bottom, the panels represent log ratios, max spectral power values, and temporal frequency of the max spectral value from the 2D FFT. Each value is an average over 20 simulations (with ΔT and τ kept constant at 12 and 20 ms, respectively). The code of the model and the simulation is available at https://github.com/artipago/echoPred. BW, backward; EEG, electroencephalography; FFT, fast-Fourier transform; FW, forward; IRF, impulse response function.
Fig 4.
Quantification of traveling waves in human EEG data and in model simulations.
(A) The first row shows the real (solid line) and shuffled (dashed line) distributions of log ratios computed over the IRF obtained during 2 sets of simulations, cross-correlating prediction signals (a proxy for the EEG in our model) either with the sensory inputs (first column), or with the top-down prior signals (second column). The second row shows the difference between the two, focusing on positive values, i.e., the traveling-wave events in the data that occur more often than predicted by the null distribution. The proportion of significant BW and FW waves are shown in red and blue, respectively. (B) Same as (A), but for the IRF computed in the INPUT human EEG dataset (during human EEG experiments, we only have direct access to the visual input signals, but the top-down priors, if any, remain unknown and cannot be used for cross-correlation). (C,D) Same as (A,B), but based on log ratio values computed directly in 1s EEG epochs, rather than in the IRF (for the model, we use the prediction signals yi as a proxy for the EEG signals). Experimental human EEG results (B,D) follow the same qualitative pattern as the simulations (A,C). All data are available at: https://osf.io/nc4rg/?view_only=bb06fe996d1f49b285dc25464eec7ac7. BW, backward; EEG, electroencephalography; FW, forward; IRF, impulse response function.
Fig 5.
The tables summarize the results of the data (left) and simulation (right) for each dataset/condition. In each situation, the average log ratio is presented numerically as well as color-coded. Red and blue squares indicate respectively a higher incidence of BW and FW waves. (As explained previously, it is not possible to compute an IRF in the CLOSED EYES condition.) All data are available at https://osf.io/nc4rg/?view_only=bb06fe996d1f49b285dc25464eec7ac7. BW, backward; EEG, electroencephalography; FW, forward; IRF, impulse response function.