Table 2.
Model’s parameters.
Fig 1.
Whole-brain neural mass model.
A) The model is constrained with structural connectivity (SC, obtained from DTI), fMRI/EEG functional connectivity (FC), and the EEG power spectrum. Brain areas were parcellated using the AAL90 brain parcellation. B) Each brain area in the whole-brain model consisted of a modified version of the Jansen-Rit neural mass model. Each region is modeled using two subpopulations of neural masses with a frequency peak of oscillation within the α and γ bands of the EEG spectrum. Both subpopulations include the interactions between pyramidal and interneuron populations. The model also included inhibitory synaptic plasticity (ISP), as an additional equation to model the time courses of the feedback inhibition onto pyramidal populations, to reach a desired average firing rate. The brain plots in (A) and (B) were generated using BrainNet Viewer for MATLAB [32]. The DTI-based network in (A) was generated using FSL-FMRIB [33].
Fig 2.
Bifurcation analysis of the model with and without plasticity.
A) Bifurcation diagram, main oscillatory frequency, and two sample activity traces at the indicated p input values, for the original Jansen-Rit model without plasticity and consisting only of α population (10 Hz). Below the traces, a power spectral density (PSD) plot is shown. In the left panel, red thick lines and black lines represent stable and unstable fixed points, respectively. Colored lines represent maxima and minima of stable (green) and unstable (blue) periodic orbits. The colors are maintained in the center panel to show the frequency (in Hz) of the associated oscillations. Dots denote the bifurcations encountered: purple = saddle-node bifurcations and saddle-node of limit cycles; black = Hopf bifurcation. Simulations include a noise factor added to the p parameter. B) Same as in (A), with the addition of the homeostatic plasticity mechanism and a target for pyramidal target firing rate ρ = 2.5 Hz. Blue dots denote a Torus or Neimar-Sacker bifurcation. The shaded rectangle in the left panel denotes the range of p that is plotted in (A). C) Same as in (B), with the target ρ = 3.5 Hz. Red dot denotes a branching or pitchfork bifurcation.
Fig 3.
A) Modulation of oscillatory frequency in single-node simulations. Normalized PSD as a function of α versus γ subpopulation proportion, rα, in the single-region model. The values shown here are the average of 50 random seeds. Note that PSD estimates are computed on a discrete frequency grid (Welch), so narrow-band ridges may appear step-like with rα. B) Noise-free simulations with fixed rα = 0.5, showing the target firing rate ρ versus the time-averaged pyramidal firing rate produced by the model. The ISP is “clamping” the ⟨rate⟩ ≈ ρ, which is an indicator of a successful feedback inhibition regulation. C) Same simulations as in (B), showing the target firing rate ρ versus the time-averaged feedback inhibition, given by the inhibitory-to-excitatory local connectivity parameter C4 (the steady-state value reached by ISP to achieve the clamping in B). D) Peak frequency versus external input for the “fast” γ subpopulation (rα = 0) and the “slow” α subpopulation (rα = 1). The dashed vertical lines correspond to the traces in E) for rα = 1, and F) for rα = 0.
Table 1.
List of brain regions of AAL90 parcellation. The atlas comprises 90 cortical and subcortical brain areas (45 per hemisphere). Regions marked by * were included for fMRI simulations, but not for EEG.
Fig 4.
Modulation of oscillatory frequency in whole-brain simulations.
Normalized power spectral density (PSD) as a function of: A) global coupling, K, in the whole-brain model, with a fixed target of ρ = 2.5 Hz. B) target (desired) firing rate in the whole-brain model, with a fixed K = 0.5. C) inhibitory synaptic plasticity (ISP) time constant, τ. The values shown here are the average of 50 random seeds. D) Example of EEG traces for a fixed K = 0.5, rα = 0.5, τ = 2 sec, and different values of ρ. E) Relationship between the nodal strength, from the structural connectivity matrix, and the time-averaged feedback inhibition (mean C4), for different values of global coupling K. F) Slopes of the linear regressions in (E), as a function of global coupling. 10 random seeds were used for these simulations. G) Convergence time vs ISP timescale. The convergence time (in seconds) was defined as the first time the ROI-averaged C4 reached 63.2% of its total change from the initial to the final steady state in a noise-free simulation.
Fig 5.
Goodness of fit of the model to empirical EEG FC.
A) Model without inhibitory synaptic plasticity (ISP). The first row shows the structural similarity index (SSIM; captures the goodness of fit), and the second row represents the mean EEG FC of simulations. The axes correspond to the global coupling parameter, K, and the proportion of α versus γ subpopulations, rα. The value of SSIM = 1 indicates a good fit of the model. B) Goodness of fit using the model with ISP, and a fixed target firing rate, ρ, of 2.5 Hz. The values shown here are the average of 50 random seeds.
Fig 6.
Comparison of the single- and multi-frequency models in fitting empirical EEG FC.
A) The structural similarity index (SSIM) was used to measure the goodness of fit (SSIM = 1, perfect fit). B) Complementary to SSIM, we reported Pearson’s correlation between simulated and empirical matrices (r = 1, perfect fit). The columns represent the different EEG frequency bands. C) Simulated EEG FCs from the single-frequency model. D) Simulated FCs from the multi-frequency model. E) Empirical EEG FC matrices. ***: |Cohen’s D| > 1.2. Each point corresponds to a different model realization (50 random seeds).
Fig 7.
Band-limited model time series and relative band power from the ISP-fit (K = 0.675, rα = 0.575, ρ = 2.5 Hz).
Representative 10-s segment of simulated EEGs, band-pass filtered into canonical frequency bands: δ (0.5-4 Hz), θ (4-8 Hz), α (8-13 Hz), β (13-30 Hz), and γ (30-40 Hz). The bottom panel shows the EEG α band envelope (α band power fluctuations).
Fig 8.
Fitting the EEG and fMRI sleep dataset.
A) Fitting to awake condition. The first column corresponds to fMRI FC fitting, assessed using the structural similarity index (SSIM). The second column shows the goodness of fit to the EEG power spectrum. The last one corresponds to fMRI FC fitting using a threshold of EEG power fitting of 0.85. B) Fitting to EEG and fMRI NREM stage 3 (N3) data. The blue and red dots represent the best parameters (highest SSIMs when applying the EEG-fitting mask) for wakefulness (K = 1.44, ρ = 2.56 Hz) and N3 (K = 0.52, ρ = 2.1 Hz), respectively. The values shown here are the average of 50 random seeds.
Fig 9.
Example simulations for the EEG and fMRI sleep dataset.
A) Empirical fMRI FC matrices for wakefulness and N3 sleep stages. B) Best simulated matrices. C) Empirical EEG traces (for one representative participant). D) Averaged empirical EEG power spectrum. E) Examples of EEG time courses from simulations. F) Simulated EEG power spectrum. The model’s parameters were K = 1.44 and ρ = 2.56 Hz, for wakefulness, and K = 0.52, ρ = 2.1 Hz, for N3 sleep. The values shown here are the average of 50 random seeds.