Metastability, fractal scaling, and synergistic information processing: What phase relationships reveal about intrinsic brain activity

Functional neuroimaging aims to provide brain-based markers for diagnosis, stratification, tracking disease progression, and predicting treatment outcomes. Static functional connectivity (FC) obscures the dynamic nature of brain activity, motivating the study of dynamic FC (dFC). Yet the interpretation and reliability of dFC measures are debated due to issues such as sampling variability, lack of ground truth, and heterogeneous methodologies. Adopting a complexity-science perspective, this study investigates whether diverse phase-based dFC metrics are representative and reliable across multiple acquisitions, how they relate to each other, and the implications of these relationships. Particular focus is placed on metastability (capturing the balance of segregation and integration in dynamical systems) and integrated information (reflecting synergistic and transfer information processing), to test if they provide converging evidence about intrinsic brain activity. Using four resting-state fMRI acquisitions from 99 healthy unrelated Human Connectome Project participants and considering parcellations with and without cerebellar regions, the study evaluates reliability and interrelationships among metrics from dynamical systems, stochastic processes, and information dynamics.

Prior work has questioned the reliability and interpretability of dFC metrics, noting challenges in clinical translation. Test-retest studies report mixed results, and validation often relies on null models or replication across datasets. Complexity science provides a framework integrating tools from statistical physics, dynamical systems, and information theory to characterize brain dynamics. Metastability has been posited as a hallmark of healthy brain function and a key indicator in computational models, balancing segregation and integration. Integrated information has been proposed to quantify similar integration/segregation balances and relate to consciousness, with recent refinements (Φ^R) capturing synergistic and transfer information processing. Despite individual validation of methods, few studies jointly assess diverse metrics within the same subjects and across multiple acquisitions to examine reliability and their interrelations.

Data and participants: Resting-state fMRI from the Human Connectome Project (HCP) 100 unrelated subjects (ages 20–35, 54 females). Each subject underwent four 14 min 24 s scans over two consecutive days (TR=0.72 s, 1200 frames, 2 mm isotropic multiband acquisition). Phase-encoding orders were counterbalanced; data were minimally preprocessed (gradient nonlinearity correction, motion realignment, fieldmap correction, structural alignment, nonlinear MNI registration), denoised with ICA-FIX, then parcellated. One subject was excluded due to corrupted data. Parcellations and filtering: AAL anatomical parcellation was used including cortex, subcortex, and cerebellum (AAL116) and a version excluding cerebellum (AAL90). Regional time series were bandpass filtered to 0.01–0.08 Hz using FFT-based filtering. The analytic signal was obtained via Hilbert transform, and Carson’s empirical rule was checked to confirm non-violation of Bedrosian theorem. Phase-based measures and functional connectivity: Instantaneous phase was obtained from the analytic signal. Phase synchrony (Kuramoto order parameter magnitude) quantified within-community synchronization. Phase-locking (PL) between regions n and p at time t was defined as the cosine of their phase difference, ranging from −1 to +1. Instantaneous phase-locking iPL(t) yielded an N×N×T tensor for each subject. Dimensionality reduction (LEIDA): Leading Eigenvector Dynamic Analysis (LEIDA) was applied to iPL(t). The leading eigenvector V1(t) (N×1) captures the dominant phase-locking pattern at each time point, partitioning regions into two communities by sign. K-means clustering (K=2–10; 300 replications; up to 400 iterations) of V1(t) across time yielded K centroids (modes). Based on silhouette values and prior work, K=5 modes were retained. Time points were assigned to the nearest centroid via cosine distance. Modes were visualized in cortical and voxel space (downsampled to 10 mm voxels; eigenvectors averaged over time points assigned to each cluster). Metrics from phase-locking dynamics: (1) Fractional occurrence of each mode ψk (proportion of time points). (2) Duration of each mode (mean length of consecutive visits, in seconds). (3) Reconfiguration speed defined as 1 − correlation between iPL(t) and iPL(t+1), characterizing smoothness of transitions. (4) Detrended fluctuation analysis (DFA) α to assess fractal scaling of reconfiguration speed, after testing for linear power-law scaling via Bayesian model comparison (BIC); only subjects with extended linear scaling contributed to DFAα summaries. Metrics from phase synchrony (data-driven communities): The 5 recurrent modes ψ1–ψ5 were used as dynamic communities (regions can belong to different communities over time). For each community: Synchronization (time-average Kuramoto order parameter), Metastability (standard deviation over time of the Kuramoto order parameter). Global metastability was the mean across communities. Chimerality (cluster synchronization) was the variance across communities of the Kuramoto order parameter at each time t. Instantaneous phase coherence across communities was computed as the average phase across communities exceeding a synchronization threshold λ>0.8 at time t; the phase coherence coefficient was the fraction of time this condition occurred. Coalition entropy quantified the diversity of cluster synchronization configurations over time (in bits). Integrated information: Integrated information Φ^R was computed from five binarized time series (one per mode). A value of 1 was assigned when a community’s synchronization exceeded λ>0.8. Φ^R indexed synergistic and transfer information processing over integration timescales τ; the maximal Φ^R across τ (1–500 TRs) was retained per subject and run. Statistical analyses: Reliability was assessed via intraclass correlation coefficients (ICC): ICC(1,1) for agreement of mode extraction across runs; ICC(3,1) initially planned for global metric consistency but replaced with non-parametric permutation-based paired t-tests due to non-normality (Shapiro-Wilk tests; Greenhouse-Geisser corrections in repeated measures ANOVA). Repeated measures ANOVA evaluated mode-specific metrics across runs. Permutation-based paired t-tests (1000 permutations) identified significant run differences. Linear mixed-effects models (lmerTest in R) predicted integrated information from standardized predictors, with RUN as a random intercept; quadratic terms were evaluated where appropriate (SYNC^2 retained). Model fit and diagnostics used the performance package. Visualizations included t-SNE embeddings of dPL streams to illustrate reconfiguration walks in phase space. Code and mode centroids were made available on GitHub.

- Invariant phase-locking modes: Five LEIDA-derived spatiotemporal phase-locking modes (K=5) were highly reproducible across the four runs. With AAL90, agreement between runs was almost perfect (0.99>ICC>0.97). With AAL116 (including cerebellum), agreement remained almost perfect (1>ICC>0.94), though occurrence probabilities varied, affecting mode ordering. Modes mapped meaningfully onto canonical resting-state networks (e.g., ψ1 global in-phase; ψ2 involving DMN/Limbic/hippocampi/cerebellum; ψ3 involving frontoparietal/limbic/striatum/cerebellum; ψ4 sensorimotor/ventral attention; ψ5 visual). - Global metric stability: Global metastability showed no statistically significant differences across the four runs when cerebellum was included, making it the most stable global metric. Excluding cerebellum reduced its reliability. Other global metrics (synchronization, phase coherence coefficient, coalition entropy, integrated information, reconfiguration speed, chimerality) exhibited significant run-to-run differences in at least some comparisons. - Mode-specific metrics unreliable: Fractional occurrence, duration, and other mode-specific metrics were not consistently reliable across all modes and runs (significant differences persisted after Bonferroni correction), limiting their generalizability for cross-sectional comparisons. - Reconfiguration dynamics: Reconfiguration speeds of phase-locking FC were generally slow, indicating smooth transitions with intermittent sharp changes (“knots and leaps”). DFA revealed fractal scaling (DFA α>0.5) in subjects exhibiting linear power-law behavior, indicating persistent fluctuations, long-range correlations, and deviation from Gaussianity. However, linear power-law scaling was absent in 40–50% of subjects in a given run, highlighting heterogeneity. - Inter-metric relationships: Most metrics were significantly correlated. Integrated information showed strong positive correlations with phase coherence coefficient, synchronization, and coalition entropy. Chimerality and reconfiguration speed tended to be negatively related to other metrics and were not correlated with each other, suggesting sensitivity to complementary dynamical features. - Predictive model for integrated information: A linear mixed-effects model (RUN as random intercept; no detectable random effect) predicted integrated information (Φ^R) from SYNC^2 (β=0.05, p=0.003), coalition entropy (β=0.87, p<0.001), and chimerality (β=−0.11, p<0.001), with marginal R^2=0.85. - Cerebellum’s role: Including cerebellar regions improved the reliability of global metastability, suggesting a cerebellar contribution to global synchronization dynamics.

The study addressed whether diverse phase-based dFC metrics are representative and reliable across acquisitions, how they interrelate, and the implications for interpretation. Five LEIDA-derived phase-locking modes provided a robust, invariant basis across runs, suitable as templates for future studies. Among global metrics, only global metastability was consistently stable across runs when cerebellum was included, positioning it as a promising neuromarker candidate for intervention studies and states of consciousness. Most other global and mode-specific metrics varied across runs, indicating nonergodicity and cautioning against cross-sectional neuromarker discovery based on sample means. The relationships between dynamical and informational metrics showed that integrated information is strongly linked to synchronization and coalition diversity, and inversely to chimerality, offering converging evidence that dynamical and informational complexity are intertwined. Notably, integrated information was not directly predicted by metastability, suggesting that while both reflect integration/segregation balance, they capture distinct aspects of brain dynamics. Characterizing dFC as a stochastic process revealed slow, continuous, non-random reconfiguration of phase-locking with long-range temporal correlations in many subjects, though power-law linearity was not universal. Including cerebellar regions enhanced the reliability of metastability, aligning with literature on cerebellar involvement in widespread cortical and limbic interactions. Collectively, the findings support a complex systems account of resting-state brain dynamics and underscore the importance of individual trajectories over cross-sectional averages for neuromarker development.

This work integrates tools from dynamical systems, stochastic processes, and information dynamics to yield a coherent picture of resting-state fMRI dynamics. Five robust phase-locking modes provide an invariant basis for studying brain dynamics across acquisitions. Global metastability emerges as the most representative and stable global metric (especially with cerebellum included), supporting its utility as a candidate neuromarker. Strong interrelationships between dynamical and informational metrics and a predictive model for integrated information demonstrate convergence of evidence and interpretability within neuroscience. The observed nonergodicity of many metrics cautions against cross-sectional neuromarker discovery and motivates longitudinal, individualized approaches. Future research should examine gender differences, frequency-specific behavior, the causes of non-linear power-law scaling, computational modeling of metric relationships, and applications to longitudinal life-trajectories and clinical populations.

- ICC considerations: ICC is a relative metric; large between-region differences can inflate ICC even if within-region differences exist. Similar conclusions were supported by Pearson correlations. - Community definition: Communities were derived from LEIDA phase-locking modes rather than established intrinsic connectivity networks. Communities are not mutually exclusive (ψ1 includes all regions), potentially violating assumptions of some metrics but arguably reflecting realistic dynamic coalitions. - Integrated information computation: Used a discrete, thresholded model with binarization at λ=0.8 and selected maximal Φ^R across integration timescales (1–500 TRs) without a maximum statistic correction, limiting inferences about τ. - Differences from prior work: Used instantaneous phase-locking (not sliding-window correlations), a larger parcellation (AAL116 vs 68 regions), and did not pool data across subjects, potentially affecting speed and scaling estimates. A substantial fraction (40–50%) of subjects lacked linear power-law scaling per run. - Null models: No new null models were developed here; methods employed have prior validation with null or surrogate data. Direct null-model testing for this dataset was not performed. - Heterogeneity and non-stationarities: Loss of linearity in power-law scaling may reflect trends, non-stationarities, or non-linear transformations; understanding these causes was beyond scope. - Generalizability: Many metrics were not stable across runs at the individual level, emphasizing nonergodicity and limiting cross-sectional generalization. - Additional factors: Potential influences include arousal, physiological states, ongoing cognition, time-of-day effects, and acquisition sequences; global metastability appeared relatively robust to these factors.