Fig 1.
A) Turbulence provides a good description of the seemingly chaotic dynamics of fluids as first described by Leonardo da Vinci [9] (left panel drawing of turbulent whirls). The physical principles giving rise to turbulence are given by high-dimensional spacetime non-linear coupled systems. In turbulence, a fundamental property is its mixing capability which yields the energy cascade through turning large whirls into smaller whirls and eventually energy dissipation (middle panel). Furthermore, the turbulent energy cascade has been shown to be highly efficient across scales, as evidenced by a power law (right panel). B) Empirical brain dynamics was recently shown to exhibit turbulence [8]. The fMRI resting state analysis over 1000 healthy participants (left panel) shows the presence of highly variable, local synchronisation vortices across time and space (middle panel). Equally, the turbulent brain regime also gives rise to an efficient information cascade obeying a power law (right panel). C) Furthermore, Hopf whole-brain models [16] (left panel) were able to fit both turbulence and the empirical data at the same working point (right panel). D) We model brain activity as a system of non-linear Stuart Landau oscillators, coupled by known anatomical connectivity. E) The Stuart Landau equation (top panel) is suited for describing the transitions between noise and oscillation. By varying the local bifurcation parameter, a, the equation will produce three radically different regimes: Noise (a<<0), fluctuating subcritical regime (a<0 & a~0) and oscillatory supercritical regime (a>0) (bottom panel). F) We evaluated the fitting capacity of the three model regimes in terms of functional connectivity and turbulence (with the dashed line showing the empirical level of turbulence). G) However, it is well-known that physical systems can be more deeply probed by perturbing them. Therefore, we used strength-dependent perturbations to disentangle the generative roles of the fluctuation (subcritical) and oscillations (supercritical) models. We observed the evolution of two key perturbative measures, susceptibility and information capacity, as a function of the applied global sustained perturbation. H) Finally, in order to generate experimentally testable hypotheses, we used local strength-dependent, non-sustained perturbations and measured the elicited dynamics in terms of the empirical perturbative complexity index [25]. Specifically, we simulated 600 volumes with the perturbation active, and we then evaluated the evolution of the signals in the following 200 volumes without perturbation and computed the difference between the PCI after and before perturbation.
Fig 2.
Model Schaefer1000 fitting of noise, fluctuations, and oscillatory for 1) Turbulence and 2) FC.
A-C) We explored the bi-dimensional parameter space defined by β and G for noise, fluctuating and oscillatory regime (bifurcation parameter a = -1.3, a = -0.02 and a = 1.3, respectively, indicated in upper row). We computed the level of amplitude turbulence error as the absolute difference between the empirical and simulated turbulence. Yellow stars indicate the (β, G) combination that reaches the lowest turbulence error in each regime. D) The upper subpanel shows the model fitting scheme in fine Schaefer1000 parcellation (the render on a flatmap of the hemisphere stands for a scheme of brain regions considered in this parcellation). The bottom subpanel displays the barplot that indicates the statistical distribution of the level of amplitude turbulence obtained by simulating 20 trials with 100 subjects for each model regime with the parameters set at the corresponding working point. We also display the results of two model-based surrogates created by increasing the shear parameter of each model regime. The red dashed line indicates the empirical level of amplitude turbulence averaged across participants. The subcritical, supercritical and empirical level of turbulence are not statistically different (Wilcoxon test, P = 0.33), the rest of the comparison are statistically significant (Wilcoxon test, P<0.001). E-G) We explored the bi-dimensional parameter space defined by β and G for noise, fluctuating and oscillatory regime computed the FC fitting as Euclidean distance between the empirical and simulated FC. Yellow stars indicate the (β, G) combination that reaches the lower turbulence error in each regime (the optimal working point obtained in panels A-C). H) The barplot indicates the statistical distribution of the FC fitting obtained by simulating 20 trials with 100 subjects for each model regime at the corresponding working point defined as the minimum turbulence error. We also display the results for the model-based surrogates. All comparisons are statistically significant (Wilcoxon, P<0.001). I) Visualization of the change of the local Kuramoto order parameter, R, in space and time reflecting amplitude turbulence in a single simulation at the optimal working point of each regime (noise, fluctuating and oscillatory cases) and one participant (empirical). This can be appreciated from continuous snapshots for segments separated in time rendered on a flatmap of the hemisphere.
Fig 3.
Model Desikan-Killiany fitting of fluctuations and oscillatory for 1) Metastability and 2) FC fitting in.
A-C) We explored the bi-dimensional parameter space defined by β and G for fluctuating, supercritical fluctuations and oscillatory regime (bifurcation parameter a = -0.02, a = 0.02 and a = 1.3, respectively, indicated in the upper row) and computed the level of metastability error as the absolute difference between the empirical and simulated metastability. Yellow stars indicate the (β,G) combination that reaches the lowest metastability error in each regime. D-F) We explored the bi-dimensional parameter space defined by β and G for fluctuating, supercritical fluctuations and oscillatory regime computed the FC fitting as Euclidean distance between the empirical and simulated FC. Yellow stars indicate the (β,G) combination that reaches the lowest metastability error in each regime (the optimal working point obtained in panels A-C). G) The upper subpanel shows the model fitting scheme procedure in coarser Desikan-Killiany parcellation (the render on flatmap of the hemisphere stands for a scheme of brain regions considered in this parcellation). The bottom subpanel displays the barplot that indicates the statistical distribution of the metastability error obtained by simulating 20 trials with 100 subjects for each model regime with the parameters set at the corresponding working point. We also display the results of two model-based surrogates created by increasing the shear parameter of each model regime. The comparison between fluctuations, supercritical fluctuations and oscillations model’s regimes at fitting the metastability shows that the two regimes are equally good (Wilcoxon, fluctuations vs oscillations P = 0.21; fluctuations vs supercritical fluctuations P = 0.26; and supercritical fluctuations vs oscillations P = 0.46), while the rest of the comparisons are statistically significant (Wilcoxon, P<0.001). H) The barplot indicates the statistical distribution of the FC fitting obtained by simulating 20 trials with 100 subjects for each model regime at the corresponding working point defined as the minimum metastability error. We also display the results of two model-based surrogates created by increasing the shear parameter of each model. All comparisons are statistically significant (Wilcoxon, P<0.001).
Fig 4.
Global and sustained strength-dependent perturbation.
A) We applied global strength-dependent, sustained perturbation in Schaefer1000 parcellation, and B) the same perturbation in Desikan-Killiany parcellation. C-D) The evolution of local and global Susceptibility (fine parcellation, panel C and coarse parcellation, panel D, respectively) as a function of perturbation strength. In dark purple is shown the response of the subcritical fluctuating regime, while in light purple, the behaviour of the supercritical oscillating regime. The subcritical regime is clearly more susceptible than the supercritical regime that is almost unaltered by the perturbation. E-F) The evolution of global absolute Information Capacity (fine parcellation, panel E and coarse parcellation, panel F, respectively) as a function of perturbation strength. In dark orange is shown the response of the subcritical fluctuating regime, while in light orange, the behaviour of the supercritical oscillating regime. The subcritical regime clearly changes the Information Capacity with the perturbation strength comparing with the supercritical regime that is almost unaltered by the perturbation.
Fig 5.
Local and Sustained/non-sustained strength-dependent perturbation.
A) The evolution of Susceptibility (first row) and the absolute Information Capability (second row) as a function of the perturbation strength and the perturbed pairs of homotopic nodes. The middle left panel displays the results for the subcritical regime (fluctuations, a = -0.02, G = 2.2 and β = 0), and the middle right panel shows the response of the supercritical regime (oscillations, a = 1.3, G = 0.4 and β = 2.2). The right panels present the perturbative node hierarchy rendered onto the brain cortex for both measures (first and second row) for the case of a perturbation strength of 0.01 indicated with a box in middle left panel. B) Non-sustained PCI: The PCI is obtained by perturbing by pairs of homotopic nodes and different forcing amplitude. In the left column, the PCI results are obtained by perturbing the subcritical model in its corresponding working point with an external periodic force applied by pairs of homotopic nodes as a function of the amplitude of this forcing. In the right column, the same measurement is displayed but, in this case, for the supercritical model in its corresponding working point. The right panel shows the node-perturbative hierarchy in terms of PCI of each region for the maximum value of the forcing amplitude (indicated with black box in the middle-left panel) rendered onto a brain cortex.
Fig 6.
The correlation between the node-level PCI and different sources of regional-level heterogeneity.
A-B) The correlation between node-level PCI and the node-level Susceptibility and Information Capacity are computed with significant negative correlation. C) The correlation between the node-level PCI and the first principal component of genes expression node information was computed. No correlation was found between variables. D) The same occurs in the correlation computed between the node-level PCI and the ratio between the T1/T2 MRI. E) The correlation between the node-level PCI and the node anatomical strength is computed obtaining a significant level of negative correlation. F) The correlation between the node-level PCI and the node functional connectivity strength (GBC) is computed obtaining a significant level of negative correlation.
Fig 7.
Revealing the causal mechanistic principles of empirical results modulating resting state networks using stimulation.
A) For the reference, the seven Yeo resting state networks are rendered in the medial and lateral surface of the right hemisphere of the brain. B) Local and Sustained stimulation differentially enhances the resting state networks. The difference in the level of FC between the perturbed and unperturbed case is shown for the seven Yeo resting state networks as a function of the perturbed node. The subcritical regime enhances the FC for all networks and nodes, while the supercritical regime is much smaller and almost constant across nodes and networks. C) The seven subpanels show boxplots of the mean FC on each one of the seven Yeo resting state networks of the two model regimes in the unperturbed and perturbed case. The subcritical regime shows higher levels for the 7 networks and while the supercritical case remains almost unaltered with the perturbation. The significance of the results was assessed using the Wilcoxon rank-sum test, where *** represents p<0.001. D) Left column shows a representative region (in red) in different resting state networks being perturbed in the fluctuating regime, which gives rise to a stabilisation of the respective network. The middle column is showing a rendering of the normalised difference between the perturbed and unperturbed activity in terms of RSN FC, thresholded to top 15%. The difference between the perturbed and unperturbed RSN FC can be seen in the spiderplots (right column), where the elicited activity is maximal for the stimulated network.