Data-driven inference of brain dynamical states from the r-spectrum of correlation matrices

TL;DR

The paper addresses the loss of dynamical information when using fixed correlation thresholds by introducing the r-spectrum, which treats the threshold r as a control parameter and identifies a characteristic percolation point r_c where multiple independent signatures converge. Through analysis of resting-state fMRI from N=996 healthy individuals and simulations with a whole-brain GH model, the framework shows r_c covaries with temporal autocorrelation AC(1) and tracks proximity to criticality; the model results further demonstrate r_c moving with the control parameter T near the critical point. The study also demonstrates robustness across modeling choices by testing alternative r_c definitions and shows percolation-based observables (S_2, σS_1, L(G)) converge at r_c, linking spatial organization to dynamical state. Overall, the r-spectrum provides a threshold-free, physically grounded method to compare brain dynamical states and relate them to aging, integration–segregation spectra, and critical dynamics.

Abstract

We present a data-driven framework to characterize large-scale brain dynamical states directly from correlation matrices at the single-subject level. By treating correlation thresholding as a percolation-like probe of connectivity, the approach tracks multiple cluster- and network-level observables and identifies a characteristic percolation threshold, rc, at which these signatures converge. We use $r_c$ as an operational and physically interpretable descriptor of large-scale brain dynamical state. Applied to resting-state fMRI data from a large cohort of healthy individuals (N = 996), the method yields stable, subject-specific estimates that covary systematically with established dynamical indicators such as temporal autocorrelations. Numerical simulations of a whole-brain model with a known critical regime further show that $r_c$ tracks changes in collective dynamics under controlled variations of excitability. By replacing arbitrary threshold selection with a criterion intrinsic to correlation structure, the r-spectra provides a physically grounded approach for comparing brain dynamical states across individuals.

Figures (14)

• Figure 1: Typical workflow schematic to compute the $r$-spectra for a single subject brain data. From the BOLD time series in Panel A the correlation matrix is obtained (Panel B). The cartoon of Panel C shows the two types of binary graphs computed for a range of correlation thresholds $r$. One is calculated from the unperturbed thresholds $r$ (top graphs), and the others (bottom graphs) are stochastic realizations computed for small random variations around the thresholds ($r \mp \Delta r$). Panel D shows, for a typical subject, the "$r$-spectra": it plots, as a function of $r$, the order parameter $S_1$ (top), and the three observables used to identify the graph percolation threshold, which correspond to the values of $r$ (arrow) for the maximum values (circles) of $S_2$, $\sigma S_1$ and $L(G)$. Results from the first session of subject code #100206 of the WU-Minn rs-fMRI dataset WuMinn.
• Figure 2: Across subjects $r_c$ covaries with the temporal correlation of the average brain BOLD activity $AC(1)$. Panels A-D show, for a subset of ten typical subjects, the order parameter $S_1$ (top), $S_2, \sigma S_1$ and $L(G)$ as a function of $r$. Panel E shows for 64 subjects the value of $AC(1)$ versus the values of $r_c$ derived from the maximum values of the observables $S_2, \sigma S_1$ and $L(G)$.
• Figure 3: Panel A: The brain dynamical states evaluated by the percolation thresholds $r_c$ strongly correlate in each of 996 individuals with the $AC(1)$ values computed from the global brain BOLD activity. Panel B: On average, the percolation threshold $r_c$ decreases with age. Age groups composition: 22-25 years (218 subjects), 26-30 years (429 subjects) and 31+ years (349 subjects, 340 subjects of age 31-35 and 9 subjects of age 36+). Bars represent within-group standard errors.
• Figure 4: Identification of the model' critical point $T_c$ (vertical dashed line) by the maximum in the autocorrelation of the order parameter fluctuations as well as the largest eigenvalue ($\lambda _1$) of the correlation matrix. Panels A-B show the average activation fraction $F_a$ serving as the order parameter and $AC(1)$ as a function of the control parameter $T$. The box in A denotes the range of $T$ explored in the results of Fig. \ref{['fig:5']}.
• Figure 5: The model dynamical state can be successfully estimated from the percolation thresholds $r_c$ (panel A), which in turn strongly correlates with the $AC(1)$ (panel B) and pair wise correlation (panel C) values computed from the time series of the mean field activity $F_a$. The red dashed vertical line in panel A indicates the value of the model critical point $T_c$. Continuous lines show mean $\pm$ standard deviation for all points on a window of of size $\Delta T=0.0025$ (Panel A), $\Delta AC(1)=0.002$ (Panel B) or $\Delta C=0.01$ (Panel C).