2.1 Analysis of Jansen-Rit single-node dynamics

We characterized the effect of plasticity by studying the model at the single-node level, considering only the α population (r^α = 1). Without plasticity, this corresponds to the original Jansen-Rit model in its most common implementation and parameter set (with a slight modification of neural gains and time constants, as presented in Table 2). Figure 2A shows a bifurcation diagram of this model against the p parameter (external input), showing the stability of the fixed point x₀ (excitatory inputs to pyramidal neurons), the frequency of the detected oscillations, and two examples of output voltage. As it has been previously described [29,34], the model presents robust oscillations of ~10 Hz within a suitable range of p.

Download:

• PNG
  larger image
• TIFF
  original image

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.

https://doi.org/10.1371/journal.pcbi.1013463.g002

The addition of ISP into the model dramatically reshapes the bifurcation diagram of the model (Fig 2B, 2C). Now, oscillations are sustained robustly in a much wider range of p (note the difference of scale in the X axes of Fig 2B vs 2A). The frequency of oscillation is not only dependent on the input p, but also on the target value ρ for the firing rate of the pyramidal population. Here, ρ denotes a homeostatic target value for the time-averaged pyramidal firing rate (regulated by ISP) and should not be interpreted as a target oscillation frequency; instead, changing ρ shifts the E/I operating point via adaptive inhibition, thereby moving the system across dynamical regimes with different emergent oscillation frequencies. When ρ is set to 3.5 Hz, oscillations are set very robustly around the α band (10 Hz) (Fig 2C). Also, with p = 220 Hz, the modulation of amplitude resembles the behavior of the original model at the same input value [28,29].

Across the bifurcation diagrams in Fig 2, oscillatory stability is shaped by bifurcation of the limit cycle rather than by a saddle-node on invariant circle mechanism. In Fig 2A, the stability change of the limit cycle around p ≈ 140 corresponds to a fold of cycles (saddle-node of limit cycles). In Fig 2B, between the saddle-node bifurcation at p = 6.62 and the torus bifurcation at p = 111, deterministic simulations do not converge to a stable attractor and instead display persistent oscillatory switching between two regions of state space (around x₀ ≈ 0.005 and x₀ ≈ 0.15). Because this regime is not central to the aims of the present work, we only report its consequences in S1 Fig. In Fig 2C, the system always exhibits a stable attractor (one or two stable fixed points or a stable limit cycle) for the explored input range, and no divergent behavior is observed. We also analyzed bifurcations with respect to the target firing rate ρ (S2 Fig). As in [10], varying ρ shifts the E/I operating point and can drive transitions between non-oscillatory (fixed-point) and oscillatory regimes, typically via a Hopf bifurcation (along with additional bifurcations in some parameter ranges). Consistent with this, ρ also modulates the oscillation frequency, with higher frequencies generally obtained at larger ρ values.

Then, we analyzed the effect of combining the two cortical subpopulations, α and γ, on nodal oscillatory frequency. We ran simulations without the ISP to characterize the EEG power spectrum shape (Fig 3A). Using single-node simulations, we modified the r^α parameter to control the contribution of α versus γ neurons within the cortical columns. For r^α < 0.25, the single node oscillates within the γ EEG frequency band (> 30 Hz). On the other hand, r^α > 0.9 produces mainly α oscillations (with a frequency peak around 10 Hz). For intermediate values of r^α, the model exhibits a richer power spectrum, with distributed power around all the possible canonical frequency bands. Using the multi-frequency model, we ran single-node simulations with r^α = 0.5 to check the model's capability to clamp firing rates to target values. Across different values of external stimulation, p, and target firing rates, ρ, the model adjusts the feedback inhibition (the inhibitory-to-excitatory local connectivity constant C₄) to reduce/increase excitability and finally match the desired firing rates ρ (Fig 3B, 3C).

Download:

• PNG
  larger image
• TIFF
  original image

Fig 3. Single-node dynamics.

A) Modulation of oscillatory frequency in single-node simulations. Normalized PSD as a function of α versus γ subpopulation proportion, r^α, 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 r^α. B) Noise-free simulations with fixed r^α = 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 (r^α = 0) and the “slow” α subpopulation (r^α = 1). The dashed vertical lines correspond to the traces in E) for r^α = 1, and F) for r^α = 0.

https://doi.org/10.1371/journal.pcbi.1013463.g003

Finally, we examined the “slow” limit-cycle frequencies for each subpopulation, in addition to the fast α/γ regimes. For both the α (r^α = 1) and γ (r^α = 0) subpopulations, the model exhibits two oscillatory regimes depending on external input p (Fig 3D–3F). At lower input (p = 120; slow regime), the α subpopulation shows the expected low-frequency limit cycle with a peak around ~3 Hz (δ/θ range), consistent with the classical Jansen-Rit slow cycle. The γ subpopulation also presents a slow limit cycle, but with a higher peak frequency around ~20 Hz (β range). At higher input (p = 300; fast regime), the α and γ subpopulations express their respective fast limit cycles (α ~ 10 Hz; γ > 30 Hz), indicating that the γ tuning shifts both fast and slow oscillatory regimes while preserving the presence of a distinct slow cycle.

2.2 Whole-brain multi-frequency model with ISP

Here, we investigated the behavior of the multi-frequency model with ISP when many regions were coupled. In this setting, the ISP mechanisms were used to prevent hyperexcitability. Based on the results of the previous section, we first fixed the target firing rate ρ to 2.5 Hz and r^α = 0.5 [7,35] (Fig 4), as they generated robust oscillatory activity in a wide spectrum of frequencies. To connect different regions, we employed structural connectivity (SC) data of 45 healthy controls from the ReDLat consortium [36]. Data were parcellated into 90 regions using the AAL90 brain atlas [37] (Table 1), excluding all subcortical areas except the amygdala and hippocampus. We swept the global coupling parameter, K, from no coupling (K = 0) to strong coupling (K = 1) (Fig 4A). For K = 0, the local dynamics are constrained to solely γ oscillations. Increasing the global coupling to K = 1 generates α and θ rhythms. In the middle, with K = 0.5, a region of coexistence of faster (β, γ) and slower (θ, α) oscillations emerged at the whole-brain level. The spread of the frequency of oscillation, for intermediate values of r^α, might be a consequence of the small stochastic fluctuations in the external input, to mimic background variability in afferent drive. This does not alter the underlying oscillatory regime, but it broadens PSD peaks and can produce a small apparent spread in the dominant frequency estimated from finite-length time series, especially near transitions.

Download:

• PNG
  larger image
• TIFF
  original image

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.

https://doi.org/10.1371/journal.pcbi.1013463.t001

Download:

• PNG
  larger image
• TIFF
  original image

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, r^α = 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.

https://doi.org/10.1371/journal.pcbi.1013463.g004

When fixing global coupling to K = 0.5, r^α = 0.5, and then manipulating target firing rates, ρ, we can shift brain dynamics from slower (for ρ < 2.5 Hz) to faster (for ρ > 2.5 Hz) regimes of activity (Fig 4B). Remarkably, we found no effect of the ISP time constant, τ, on the oscillatory frequency of the model (Fig 4C). Example EEG traces are depicted in Fig 4D, for K = 0.5, and r^α = 0.5. There, low ρ values produce δ-θ oscillations, intermediate values α rhythms, and the highest values produce faster β-γ activity.

Across whole-brain simulations, we examined how the ISP shapes the spatial distribution of feedback inhibition and how the ISP timescale affects the adaptation dynamics (Fig 4E–4G). For each value of global coupling K, the time-averaged feedback inhibition (mean C₄) was computed and related to the nodal strength of the structural connectivity matrix (Fig 4E). Mean C₄ increased with nodal strength, and this relationship was well approximated by a linear regression for each K value. Then the slope of this regression was expressed as a function of K (Fig 4F), showing that feedback inhibition changes systematically with global coupling. Importantly, while this mapping could in principle be used to pre-compute a set of node-specific C₄ values for a given K, the regression slope (and thus the inferred C₄ distribution) also depends on other model parameters (e.g., the α/γ mixture r^α), limiting the generalizability of any fixed pre-computed inhibition across the full parameter space. To assess the role of the ISP time constant, we quantified the convergence time of the plasticity variable in noise-free simulations (Fig 4G). Convergence timescale (seconds) was defined as the first time at which the ROI-averaged C₄ reached 63.2% of its total change from the initial value to its final steady-state value. Across the tested time constant values τ, this parameter primarily modulated the convergence timescale, while the steady-state oscillatory regime (and the normalized PSD, Fig 4C) remained largely unchanged once the system had converged.

In summary, slower oscillatory regimes emerge from a high proportion of α over γ neurons (r^α), high global coupling K, and low ρ, whereas faster oscillations appear in the opposite direction. In addition, ISP shapes a node-specific inhibition profile that scales with nodal strength in a K-dependent manner, while the ISP timescale primarily controls the convergence time of C₄ to its steady state.

2.3 Fitting to EEG source functional connectivity data

To further validate our model, we assessed its capacity to fit FC across the whole EEG power spectrum (Fig 5). We used the same SC data [36] and brain parcellation [37] described in the previous section, and the source-reconstructed EEG time series from the same participants [36]. We filtered the simulated and empirical EEG signals in the canonical frequency bands (δ, θ, α, and β). We calculated the signals’ envelopes, which were high-pass filtered (cut-off frequency of 0.5 Hz), and FCs were computed using the amplitude envelope correlation [38]. We compared the model without (Fig 5A) and with (Fig 5B) ISP. The r^α and K parameters were modulated to find the best combinations to reproduce the EEG FC. Simulated and empirical matrices were contrasted using the structural similarity index (SSIM, with SSIM = 1 being a perfect fit, 0 being the opposite) [39]. We aimed to simultaneously maximize SSIM for all EEG frequency bands (maximizing the average SSIM across bands).

Download:

• PNG
  larger image
• TIFF
  original image

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, r^α. 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.

https://doi.org/10.1371/journal.pcbi.1013463.g005

The model without ISP is characterized by a parameter space with a small region of correlated activity (Fig 5A). The model is unable to reach correlated activity before becoming hyperexcited, as no compensatory mechanisms were introduced to avoid that. Further, the model cannot generate correlated activity when mixing α and γ subpopulations in cortical columns; that is, near 0 mean FC for r^α values around 0.5 suggests model frustration (inability to generate correlated activity). This limitation reflects the local and global network dynamics of the model without ISP: for large portions of the (K, r^α) space, increased effective excitation drives the system toward over-saturation and loss of sustained oscillations, which prevents the emergence of stable band-limited envelope fluctuations and, consequently, correlated activity. Therefore, the restricted parameter region in Fig 5A appears primarily due to the model’s inability to sustain oscillatory dynamics without an adaptive inhibitory mechanism, rather than a bias induced by additional fitting targets or anatomical priors. Including ISP in the model, with ρ = 2.5 Hz, enriches the parameter landscape with a wider range of parameters where correlations can emerge, from hypo to hyperconnectivity patterns, including intermediate connectivity regimes of activity (Fig 5B). Interestingly, correlations for intermediate values of r^α are now viable. Increasing global coupling in this scenario produced multi-frequency EEG FC connectivity, with maximal SSIM values in the proximity of r^α = 0.5 and K = 0.5.

We used the optimal parameters to compare the single-frequency model (without ISP, K = 0.425, r^α = 1, SSIM = 0.14 averaged across frequency bands) and the multi-frequency model (with ISP, K = 0.675, r^α = 0.675, ρ = 2.5 Hz, SSIM = 0.56 averaged across frequency bands) to reproduce EEG FC (Fig 6). The SSIM and Pearson’s correlation values between the two models are presented in Fig 6A. Regardless of the chosen metric to compare the goodness of fit, the multi-frequency model outperformed the classical single-frequency model across all frequency bands (|D| > 1.2, denoting a huge effect size) (Fig 6A). Example FCs are presented for the single and multi-frequency models, including the empirical FC matrices (Fig 6B).

Download:

• PNG
  larger image
• TIFF
  original image

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).

https://doi.org/10.1371/journal.pcbi.1013463.g006

To illustrate the qualitative temporal structure of the ISP-fit model beyond FC metrics, we additionally report representative EEG time series and amplitude envelopes for the best-fitting parameter set (K = 0.675, r^α = 0.575, ρ = 2.5 Hz) (Fig 7). We show a 10-second segment of simulated EEG, band-pass filtered into canonical frequency bands (δ-γ). We also plot the α-band envelope from the same simulation, which captures the slow α-band power fluctuations (amplitude dynamics).

Download:

• PNG
  larger image
• TIFF
  original image

Fig 7. Band-limited model time series and relative band power from the ISP-fit (K = 0.675, r^α = 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).

https://doi.org/10.1371/journal.pcbi.1013463.g007

In this way, we demonstrated that ISP preserved E/I balance and enabled correlated activity without hyperexcitability. Also, the ISP mechanism combined with the multi-frequency model can generate multiband EEG FC across a wide range of parameters.

2.4 Dual fitting to EEG and fMRI

As an additional demonstration and validation of our model, we investigated its ability to jointly reproduce fMRI and EEG dynamics during NREM sleep. We used simultaneous EEG-fMRI recordings during the transition from wakefulness to different stages of NREM sleep. fMRI recordings were collected from 71 individuals, with sleep stages labeled for each fMRI volume based on simultaneously recorded polysomnographic data [40]. Here, we fitted our model to the awake (W) and N3 sleep (deep sleep). We fixed r^α = 0.5, and systematically explored combinations of K and ρ drawn from a predefined parameter grid (41 × 41 parameter grid, with K ranging from 0 to 3 and ρ ranging from 2 to 3). We employed a connectome of young healthy participants from another study [41,42]. We simulated EEG-like signals and then used a hemodynamic model [43] to transform the firing rates into fMRI BOLD-like signals. We aimed to fit two different observables, one from each recording modality. First, the empirical averaged EEG power spectrum at the sensor level (32 EEG channels), and second, the fMRI BOLD FC (considering the AAL90 brain parcellation). From the empirical and simulated power spectrum, we computed the θ, α, and β relative power, concatenated the values into a vector, and then compared the vectors using Clarkson’s distance [44]. So, we attempted to reproduce, using whole-brain simulations, the relative contribution of θ, α, and β oscillations in EEG. For fMRI BOLD FC, we compared simulated and empirical matrices using the SSIM.

The results presented in Fig 8 suggest that using fMRI FC generates regions of the parameter space with model degeneracy in terms of goodness of fit (SSIM). That is, we observed dissimilar combinations of parameters with the same goodness of fit, regardless of whether the fitting was focused on W or N3 sleep. However, EEG fitting can be used as a mask to delimit the possible combinations of parameters that might reproduce empirical FC. In Fig 8, we used a measure of similarity (1 – Clarkson’s distance) as a representation of the goodness of fit for simulated EEG, with values closer to 1 corresponding to a perfect fit to the empirical EEG relative power across frequency bands. Using a threshold of 0.85 for EEG fitting, we obtained a restricted parameter space where just a few parameters can simultaneously generate EEG and fMRI dynamics that closely match empirical data. For wakefulness, the best fit was found for higher coupling and medium firing rates (K = 1.44, ρ = 2.56 Hz, SSIM = 0.34, Pearson’s r = 0.45, Fig 8A). In contrast, N3 sleep was better characterized by low coupling and low firing rates (K = 0.52, ρ = 2.1 Hz, SSIM = 0.29, Pearson’s r = 0.38, Fig 8B).

Download:

• PNG
  larger image
• TIFF
  original image

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.

https://doi.org/10.1371/journal.pcbi.1013463.g008

Simulation outcomes are displayed in Fig 9. The empirical (Fig 9A) and simulated (Fig 9B) FC matrices showed decreased connectivity strength in N3 sleep. The empirical EEG traces (example of one participant) and the group-averaged PSDs are shown in Fig 9C. Simulated EEG-like signals showed high amplitude slow oscillations for N3 compared to wakefulness (Fig 9D). Finally, the simulated power spectra presented the typical characteristics of the awake condition (peak in α) and deep sleep (power concentrated in the lowest frequency bands).

Download:

• PNG
  larger image
• TIFF
  original image

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.

https://doi.org/10.1371/journal.pcbi.1013463.g009

Ethics Statement

ReDLat: Ethical approval was obtained from the institutional ethics committee at each ReDLat-participating center, and all participants provided written informed consent in accordance with the Declaration of Helsinki.

Young connectomes: All participants gave written informed consent before taking part, and the study protocol received approval from the SWPS University Ethical Committee in accordance with the Declaration of Helsinki.

Sleep data: All participants gave written informed consent. The study was approved by the local ethics committee of Goethe University Frankfurt am Main.

4.1 Participants and datasets

4.2 Functional connectivity estimation

For EEG, we filtered signals in the common EEG frequency bands: δ (0.5-4 Hz), θ (4–8 Hz), α (8–13 Hz), and β (13–30 Hz) using a 3rd-order passband Bessel filter. We used the Hilbert transform to get the signals’ envelopes, and then we filtered the envelopes with a 3rd-order Butterworth high-pass filter (cut-off frequency at 0.5 Hz), and finally, we built the FC by computing the Pearson’s correlation between all pairs of areas [38].

For fMRI, we first filtered the signals between 0.01-0.08 Hz using a 3rd-order Bessel filter, and then we built the FC matrix using Pearson’s correlation between all pairs of areas.

4.3 Power spectrum analyses

For both empirical and simulated data, the EEG power spectrum was calculated using the Welch method with 2-sec length time windows with 50% overlap [76]. The power was extracted for each EEG band and divided by the total power of the spectrum between 0.5 and 30 Hz.

4.4 Whole-brain model

Whole-brain activity at the source level was modeled using a modified version of the Jansen-Rit neural mass model [7,12,30]. In this model, each brain region consists of two subpopulations of neural masses, each tuned to oscillate in either the α (around 10 Hz) or γ (around 45 Hz) frequency bands of the EEG spectrum. The contribution of these subpopulations to the generation of the postsynaptic potential (PSP) of pyramidal neurons is weighted by the parameter r^α, which reflects the proportion of α versus γ subpopulations within the brain areas, as indicated in [30]. Specifically, r^α = 1 indicates a full contribution from the α subpopulation, while r^α = 0 indicates none. Each subpopulation is composed of pyramidal neurons, along with excitatory and inhibitory interneurons.

The PSPs, v, are integrated and transformed into firing rates using a sigmoid function:

where ζ_max is the maximal firing rate output, r is the slope, and v_th is the voltage threshold. The inverse operation is conducted by a PSP block, which convolves the firing rates of the neurons with an impulse response function defined as:

for the excitatory PSPs, and

for inhibitory PSPs. Here, A (B) and a (b) correspond to the maximal amplitude and inverse characteristic time constant of the excitatory (inhibitory) PSPs, respectively. On a macroscopic level, brain areas i and j are connected using an SC matrix M derived from DTI data. The coupling strength was scaled by a global parameter K and, considering that long-range projections are primarily excitatory [77,78], connections between brain regions involved only pyramidal neurons [7]. Each region received background input p(t), with values randomly sampled from a normal distribution with a mean ⟨p(t)⟩ = 220 Hz and a standard deviation σ_p = 31 Hz. The complete system of equations for the α subpopulation was:

The first pair of equations model the excitatory feedback loop, the second pair models the outputs of pyramidal neurons, and the third represents the inhibitory feedback loop. Neuron populations were connected via constants C₁, C₂, C₃ and C₄, which are scaled by a local connectivity constant, C = 135, with C₁ = C, C₂ = 0.8C, C₃ = 0.25C, and C₄ = 0.25C. Identical equations apply to the γ subpopulations, differing only by the γ superscript. The final output of the model produced EEG-like signals in source space, featuring a richer power spectrum. The model’s parameters are summarized in Table 2.

The rationale of combining these two subpopulations is grounded in the organization of cortical layers, where different neuronal subpopulations exhibit distinct input-output dynamics and generate frequency-specific rhythms [79]. Superficial layers tend to produce faster oscillations, such as β and γ, while deeper and granular layers are associated with slower α and θ activity [80]. Here, we use two subpopulations as a parsimonious modeling choice to broaden the nodal spectral repertoire while keeping the whole-brain model tractable for fitting. This does not mean that this approach should be treated as an account of the unique origin of each rhythm in every region. Similar multi-timescale approaches have been explored previously in whole-brain modeling to increase spectral richness at each node [59]. In our model, the EEG reflects a weighted summation of these laminar contributions, and the resulting broadband spectrum emerges from the combined activity of spatially and synaptically distinct rhythm-generating circuits. The state variables x₀(t), x₁(t), and x₂(t) are defined as a weighted contribution of α and γ subpopulations:

The model EEG-like signal can be computed as [7,30]:

Additionally, following [9], we integrated inhibitory synaptic plasticity into the model. This mechanism controls the firing rate of pyramidal neurons, preventing hyperexcitability, which can occur when global coupling increases. Synaptic plasticity was introduced as an additional differential equation:

This equation dynamically updates feedback inhibition to regulate the firing rate of pyramidal neurons, thus preventing saturation of the sigmoid function (hyperexcitability). Here, represents the inverse learning rate, ζ_inh(t) and ζ_pyr(t) are the firing rates of inhibitory interneurons and pyramidal neurons at time t, respectively, ρ is the target firing rate, and β is a bounding exponent controlling convergence to C_4,min. We used β = 1 (soft bound), though other values are possible [81]. The key parameters for plasticity are the learning rate, τ, and the target firing rate, ρ, which were set by default to τ = 2 sec and ρ = 2.5 Hz [9,35]. The firing rates are defined as

We solved the system of differential equations using the Euler-Maruyama method with a 1 msec integration step. The filtering and FC estimation procedures were the same as those used for the empirical signals.

4.5 Hemodynamic model

We simulated fMRI-like signals using the pyramidal firing rates ζ_pyr,i(t) and a generalized hemodynamic model as presented by Stephan et al. [43]. In this model, an increase in the firing rate ζ_pyr,i(t) initiates a vasodilatory response s_i leading to blood inflow f_i, and subsequent changes in blood volume v_i and deoxyhemoglobin content q_i. The system of differential equations governing this process is

where the time constants τ_s = 0.65, τ_f = 0.41, τ_v = 0.98, and τ_q = 0.98 correspond to signal decay, blood inflow, blood volume, and deoxyhemoglobin content, respectively. The stiffness constant κ (resistance of the veins to blood flow) and the resting-state oxygen extraction rate E₀ were set to κ = 0.32 and E₀ = 0.4. The BOLD-like signal for the node i, denoted as B_i(t), is a nonlinear function of q_i(t) and v_i(t):

where V₀ = 0.04 represents the fraction of deoxygenated venous blood at rest, and k₁ = 2.77, k₂ = 0.2, k₃ = 0.5 are kinetic constants.

We solved the system of differential equations using the Euler method with a 10-msec integration step. The resulting fMRI-like signals were then down-sampled to match the empirical signal's sampling rate (TR = 2.08 sec). The filtering and FC estimation followed the same procedures applied to the empirical signals.

4.6 Simulations

We first simulated the activity of an isolated node of the model (K = 0) without plasticity. There, we simulated 180 sec and discarded the first 60 sec (for a final length of 120 sec), downsampled the signals to a 200 Hz sampling rate (to reduce memory consumption), and calculated the Normalized Power Spectrum (NPS, obtained using the Welch method and dividing by the maximum value of spectral density). Simulations were run against variations of r^α in 41 equidistant values between 0 and 1 (Fig 3).

We then ran a whole-brain simulation, where nodes represent brain areas that are connected using the ReDLat SC matrix. We simulated 660 sec, discarded the first 60 sec, and obtained the NPS (averaged across brain areas) against variations in different parameters (average of 50 seeds in Fig 4): we varied K over 41 equidistant values between 0 and 1 (upper right, fixing r^α = 0.5, ρ = 2.5 Hz, τ = 2 sec); ρ over 41 equidistant values between 1 and 4 Hz (lower left, fixing r^α = 0.5, K = 0.5, τ = 2 sec); and τ over 41 equidistant and log spaced values between 0.01 and 50 sec (lower right, fixing r^α = 0.5, ρ = 2.5 Hz, K = 0.5).

Using the same whole-brain simulations described above (ISP enabled), we quantified how the spatial distribution of feedback inhibition relates to the structural connectivity topology. For each simulation, we computed the nodal strength of each brain area as the sum of its structural connectivity weights (row-sum of the SC matrix). We then computed the time-averaged feedback inhibition (mean C₄) for each node by averaging C₄ over time after 20 sec simulations, discarding the initial 5 sec transient, using 10 random seeds. For each value of the global coupling K, we fitted a linear regression between nodal strength and mean C₄ across nodes and stored the regression slope (Fig 4). Unless otherwise stated, these analyses were performed with r^α = 0.5, ρ = 2.5 Hz, and τ = 2 sec.

To quantify how the ISP time constant τ affects adaptation dynamics, we ran noise-free whole-brain simulations (σ = 0) with ISP enabled and fixed parameters K = 0.5, r^α = 0.5, and ρ = 2.5 Hz, using only one random seed (noise-free simulation). Simulations had a total duration of 120 sec. We swept τ over the values τ ∈ [10^-3, 10^-2, 10^-1, 1,10, 100] sec. For each τ, we computed the ROI-averaged C₄(t) and defined a convergence timescale as the first time at which C₄(t) reached 63.2% (1 − 1/e) of its total change from an initial baseline to its final steady value , where was estimated from the first 2 sec and from the last 10 s of the simulation (Fig 4).

We fitted the model to the average FC matrices of healthy individuals in the ReDLat dataset [60], simulating 660 sec and discarding the first 60 sec of adaptation. We fixed ρ = 2.5 Hz and τ = 2 sec and simultaneously varied the model’s global coupling K and r^α between 0 and 1 (41 values each), first without ISP and then with ISP (Fig 5B). This was carried out by fitting the source-EEG envelope FC, after filtering the signal in the different EEG frequency bands, and downsampling to a 100 Hz sampling rate.

For the sleep EEG-fMRI dataset (Fig 8) [82], we swept the coupling parameter, K, from 0 to 3 (41 values in total), and the target value for ISP, ρ, from 2 to 3 (41 values as well), fixing r^α = 0.5 and τ = 2 sec. We simulated 660 seconds and discarded the first 60 sec for stabilization. We used the pyramidal neurons’ firing rates to simulate fMRI-like signals through the Balloon-Windkessel model [43], averaging FCs across the 50 simulation seeds. We used firing rates for both α and γ subpopulations scaled by r^α. The SC matrix came from the connectome of the young participants dataset [11].

4.7 Model fitting

For EEG and fMRI FC matrices, we maximized the SSIM [83], used to compare empirical and simulated FC matrices in whole-brain simulations [73].

For the simultaneous EEG and fMRI dataset, after running the model, we aimed to fit the EEG power spectrum and fMRI FC. First, we calculated the relative power of each EEG band using the Welch method, generating a spectral vector of three entries (relative θ, α, and β power) for empirical and simulated data. Then, we compared the distance between the empirical and simulated relative power using the Clarkson Distance (say, x is the simulated power vector and y the empirical power vector):

where is the Euclidean norm of x. We took 1 - as the goodness of fit between spectral vectors .

Finally, we calculated the distance between empirical and simulated spectral vectors for each parameter combination and further analyzed only parameter combinations with a goodness of fit ≥ 0.85. From this subset of parameters, we maximized the SSIM between empirical and simulated FCs.

4.8 Bifurcation analysis

Bifurcation analysis was performed using the XPP-Auto software, and the figures were made using custom code in Python.

4.9 Statistical analysis

Given that statistical p-values could be artificially inflated by sample size in computer simulations (varying the number of seeds), instead of statistical tests, we report results using Cohen’s D for effect size. Usually, Cohen’s D is interpreted as very small (|D| < 0.2), small (0.2 < |D| < 0.5), moderate (0.5 < |D| < 0.8), large (0.8 < |D| < 1.2), very large (1.2 < |D| < 2) [84].