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Developmental increases in white matter network controllability support a growing diversity of brain dynamics
Evelyn Tang, Chad Giusti, Graham Baum, Shi Gu, Eli Pollock, Ari E. Kahn, David Roalf, Tyler M. Moore, Kosha Ruparel, Ruben C. Gur, Raquel E. Gur, Theodore D. Satterthwaite, Danielle S. Bassett
TL;DR
The paper asks how developing white matter architecture supports increasingly diverse brain dynamics and emerging cognition. It analyzes diffusion-imaging networks from 882 youth ages 8–22 using controllability and synchronizability measures, alongside network-evolution modeling. The results show increasing support for diverse dynamics, regional specialization, and a negative cognitive-performance association for stable subcortical controllers, beyond simple modularity changes.
Problem
The study addresses how white matter architecture supports the emergence of cognitive abilities and changing brain dynamics during development.
Method
The authors analyze diffusion-imaging structural networks from 882 youth ages 8–22 using controllability, synchronizability, null models, and network-evolution modeling.
Results
White matter networks show increasing average and modal controllability, decreasing synchronizability, and developmental patterns that persist after controlling for modularity.
Takeaways & Limitations
Development appears to favor structural network organization supporting a wider range of brain dynamics, with distinct regional control specialization and weaker stable subcortical control associated with higher cognitive performance.
Abstract
from arXiv · showhide
As the human brain develops, it increasingly supports coordinated control of neural activity. The mechanism by which white matter evolves to support this coordination is not well understood. We use a network representation of diffusion imaging data from 882 youth ages 8 to 22 to show that white matter connectivity becomes increasingly optimized for a diverse range of predicted dynamics in development. Notably, stable controllers in subcortical areas are negatively related to cognitive performance. Investigating structural mechanisms supporting these changes, we simulate network evolution with a set of growth rules. We find that all brain networks are structured in a manner highly optimized for network control, with distinct control mechanisms predicted in child versus older youth. We demonstrate that our results cannot be simply explained by changes in network modularity. This work reveals a possible mechanism of human brain development that preferentially optimizes dynamic network control over static network architecture.
RESULTS
Across development, white matter networks increasingly support diverse state transitions while becoming less synchronizable, and these patterns exceed simple modularity-based explanations.
- Controllability in brain networks: Average controllability and modal controllability are negatively related across regions but positively related across individuals.Regional correlation: ρ = −0.76; whole-brain individual correlation: r = 0.87.
- Synchronizability and changes across development: Networks with higher synchronizability showed lower average controllability and lower modal controllability.Correlations were r = −0.85 and r = −0.82, respectively.
- Synchronizability and changes across development: Average controllability increased with age (r = 0.28), modal controllability increased with age (r = 0.22), and synchronizability decreased with age (r = −0.37).The developmental associations were reported while controlling for brain volume, head motion, sex, and handedness.
- Synchronizability and changes across development: These developmental changes indicate support for a broader range of dynamics and persist after regressing out network modularity.The supported range spans nearby to distant states, alongside reduced support for globally synchronized states.
Super-controllers and cognition
Development concentrates increasing controllability in already strong regional controllers, while stable subcortical controllers show the clearest negative association with cognitive performance.
- Super-controllers and cognition: Regions with higher controllability showed the greatest developmental increases, producing ‘super-controllers’ for average and modal controllability.Age correlations with regional controllability were ρ = 0.48 for average controllability and ρ = 0.33 for modal controllability.
- Super-controllers and cognition: Age-related controllability changes were distributed across many tracts: 95 of 24,027 non-zero edges connected 43% of brain regions.The analysis controlled for sex, handedness, brain volume, and head motion.
- Super-controllers and cognition: Prefrontal and anterior cingulate controllers increased in modal controllability and decreased in average controllability with age.This pattern may indicate narrowing or specialization of preferred control roles, according to the authors.
- Super-controllers and cognition: Higher cognitive performance was associated with weaker stable-controller average controllability, particularly in largely subcortical regions (ρ = −0.16).This association controlled for age and passed false discovery rate correction, unlike the super-controller analyses.
A network growth model for modeling development
The authors use evolutionary network optimization to test whether developmental brain-network trajectories favor high controllability and low synchronizability. Simulated trajectories closely track human brain data, indicating near-optimal controllability while synchronizability remains finite.
- Model and optimization: An evolutionary algorithm models network development by optimizing mean average controllability, mean modal controllability, and synchronizability in forward and backward directions.The Pareto-optimization procedure runs for 1500 edge steps in each direction, preserving the network’s edge distribution while modifying topology.
- Model and optimization: The simulated trajectory optimizing controllability and minimizing synchronizability tracked the human brain data, with mean and variance of distance equal to 0.0049 ± 0.0376.This constrained path supports reconfiguration of white matter connectivity as a possible developmental mechanism for increasing flexible movement between diverse brain states.
- Model and optimization: 0.52 was the forward-to-backward distance ratio when optimizing average and modal controllability, making controllability decreases nearly twice as easy as increases.The ratio was computed in the average-controllability versus modal-controllability plane with synchronizability excluded.
- Model and optimization: 0.46 was the forward-to-backward distance ratio in three dimensions, indicating that increasing synchronizability was markedly easier than decreasing it.The three-dimensional optimization included mean average controllability, mean modal controllability, and synchronizability.
- Model and optimization: Brain-network topologies were well optimized for high controllability and low synchronizability within networks sharing the same edge distribution.Forward evolution increased both controllability measures and decreased synchronizability, whereas backward evolution optimized in the opposite direction.
- Model and optimization: Final evolved controllability values were close to brain-network maxima, whereas synchronizability remained less fully limited.Final evolved values were 31.7 for average controllability and 0.985 for modal controllability, compared with brain-network maxima of 32.6 and 0.983, respectively.
Pareto optimization with other metrics
The authors compare controllability-based optimization with weighted-degree, efficiency, and modularity alternatives. Controllability produces structured developmental-like trajectories and more constrained network evolution than these related metrics.
- Comparisons with alternative metrics: Optimization using global efficiency and network modularity displayed far less structure than optimization using controllability and synchronizability.The authors therefore describe brain networks as best characterized by optimization for control of diverse neural dynamics among the tested models.
- Comparisons with alternative metrics: Figure 5 compares original and final evolved networks across controllability and synchronizability, highlighting overlapping controllability values but separated synchronizability values.The comparison supports near-optimal control alongside a retained finite level of synchronizability.
- Comparisons with alternative metrics: High weighted degree appears necessary but not sufficient for average controllability, while low weighted degree appears necessary but not sufficient for modal controllability.Average controllability correlates positively with ranked weighted degree, whereas modal controllability correlates negatively.
- Comparisons with alternative metrics: Weighted-degree plots showed little structure or discernible relationship across individuals, unlike the clean developmental arcs for controllability and synchronizability.Degree-based forward and backward trajectories moved noisily across the plotted planes rather than following constrained curves.
- Comparisons with alternative metrics: Separate weighted-degree optimization simulations produced non-overlapping trajectories, unlike simulations optimizing controllability metrics.A highlighted degree-based trajectory zigzagged across the plane, while controllability trajectories overlapped across simulations.
- Comparisons with alternative metrics: Controllability-based optimization constrains evolutionary trajectories more strongly than increasing maximum or decreasing minimum weighted degree.The comparison preserves the network’s mean weighted degree while altering node-degree extremes through edge swaps.
Steeper trajectories in children versus older youth
The study tests whether age-related changes in controllability and synchronizability can be explained by modularity, using alternative Pareto-optimization trajectories. Modularity does not compellingly account for the observed developmental relationships.
- Controlling for modularity: Pareto optimization of modularity and global efficiency produces trajectories that do not resemble the functional form of empirical brain-network data.The modularity–efficiency trajectories are fairly linear, unlike the similar exponential forms observed for controllability and synchronizability.
- Controlling for modularity: Modularity and global efficiency are not parsimonious candidates for the network-level mechanisms of structural rewiring during development.This conclusion follows from analyses of modularity-controlled age relationships and alternative optimization trajectories.
- Controlling for modularity: Together, the results suggest that modularity does not compellingly explain the age-related relationship between controllability and synchronizability.The authors explicitly characterize modularity as an inadequate explanation for the observed developmental pattern.
DISCUSSION
The discussion connects controllability profiles to cognitive control and developmental specialization while emphasizing the limits of the linear dynamical model and topology-only simulations. It frames increased controllability as a possible developmental optimization whose behavioral and biophysical implications require further study.
- DISCUSSION: The linear brain-dynamics model constrains predictive power to short time scales and states near the operating point.The model is presented as useful for first-order dynamics but limited as a biophysical account of broader brain dynamics.
- DISCUSSION: High modal controllers in executive areas are predicted to drive brain dynamics toward distant states, whereas high average controllers along the medial wall target nearby states.These structural predictions align modal control with distributed cognitive-control dynamics and average control with intrinsic dynamics.
- DISCUSSION: Brains predicted to switch easily to nearby mental states are also predicted to switch easily to distant mental states, despite distinct regions supporting average and modal controllability.The positive relationship between the two control strategies was unexpected because they are associated with different brain regions.
- DISCUSSION: Development from ages 8 to 22 is associated with regional super-controllers specialized for different control strategies and coordination length-scales.Average super-controllers are broadly frontal-parietal, while modal super-controllers are located in prefrontal areas.
- DISCUSSION: Whether structural changes enable learning or learning alters white-matter architecture remains unresolved and requires longitudinal empirical studies.The discussion identifies this as a future research direction linking skill acquisition to control architectures.
- DISCUSSION: The topology model uses edge swaps and should be expanded with spatial constraints and edge addition or deletion to better represent growth and pruning.The authors state that more detailed biophysical investigation is needed for a complete characterization.
METHODS
The methods use publicly available Philadelphia Neurodevelopmental Cohort data from 882 participants aged 8–22, with diffusion MRI acquired on a standardized scanner and cognitive scores available for 880 participants.
- METHODS: The sample comprises 882 Philadelphia Neurodevelopmental Cohort subjects between 8 and 22 years old.The data are publicly available through the Database of Genotypes and Phenotypes.
- METHODS: Using only high-quality data reduces diffusion-noise effects but does not eliminate limitations of deterministic or probabilistic tractography.The authors note that noise reduction improves tract estimation and reduces false positives.
- METHODS: Diffusion tensor imaging and other MRI data were acquired on the same 3 T Siemens Tim Trio scanner with a 32-channel head coil.The DTI protocol used a twice-refocused spin-echo single-shot EPI sequence.
- METHODS: Cognitive performance was measured with the Penn Computerized Neurocognitive Battery and summarized using a bifactor-derived efficiency score.Cognitive scores from 880 participants passed quality-control measures.
Connectome construction
The study constructs weighted structural connectomes from diffusion MRI and models regional brain activity with a stabilized linear dynamical system. Control nodes and controllability are defined within this network representation.
- Connectome construction: 64-direction DTI data were processed with deterministic whole-brain fiber tracking, initiating exactly 1,000,000 streamlines per subject after removing streamlines shorter than 10mm.A 234-region parcellation was constructed, and alternative edge definitions and probabilistic tracking methods were also evaluated.
- Dynamical model: The network state is a vector of regional activity evolving over time, with the adjacency matrix encoding structural connections among regions.The state can represent measures such as electrical charge, oxygen level, firing rate, or BOLD magnitude.
- Connectome construction: The connectome is represented by a symmetric weighted adjacency matrix whose elements count white matter streamlines between brain regions.The approach assumes streamline counts are proportional to structural connectivity strength.
- Dynamical model: The authors use a noise-free linear discrete-time, time-invariant model and rescale the adjacency matrix to assure Schur stability.The input matrix identifies selected control points, and the input vector specifies the control strategy.
- Dynamical model: Controllability is defined as the ability to drive the network toward a target state using external input, assessed through the invertibility of the controllability Gramian.Control nodes are selected one at a time, so the input matrix becomes one-dimensional for individual-node analyses.
Controllability metrics
The paper distinguishes average and modal controllability as complementary measures of how brain regions can move network activity across an energy landscape. These measures capture transitions toward nearby states versus difficult-to-reach dynamic configurations.
- Controllability metrics: Average controllability describes the ease of transitioning to many nearby states on an energy landscape.It is interpreted using the network’s control energy and computed from the controllability Gramian.
- Controllability metrics: Average controllability is adopted as Trace(WK), which is inversely related to Trace(WK^-1) and corresponds to the network impulse-response energy or H2 norm.The trace of the inverse Gramian is difficult to compute accurately because the Gramian is typically ill-conditioned.
- Controllability metrics: Modal controllability describes a node’s ability to control each evolutionary mode and drive network dynamics toward hard-to-reach configurations.It is computed from the eigenvectors and eigenvalues of the adjacency matrix.
- Controllability metrics: Whole-brain mean average and modal controllability are obtained by averaging the corresponding regional values across the network.The same framework supports regional and whole-brain summaries.
Discrete transitions and centralized versus
The paper frames brain states as patterns of activity across cortical and subcortical regions and considers transitions among discrete or attractor-like states. It also distinguishes single-region control from distributed control and identifies distributed control as a future direction.
- Discrete transitions: A brain state is represented as an activity pattern across 234 cortical and subcortical areas, while state transitions may occur between nearby or distant states.Nearby and distant states are distinguished using distances between state vectors rather than requiring all transitions to be large.
- Discrete transitions: Evidence from bistable and multistable behavior supports describing neural activity as periods spent in attractor states followed by transitions between them.In bistable perception, fMRI activity is fit by a model with basin states and estimable transition rates.
- Centralized versus distributed control: The study analyzes control capabilities predicted from single brain regions, but this does not exclude coordinated control by groups of regions.The same linear time-invariant framework has been extended to distributed or multi-point control in other work.
- Centralized versus distributed control: Distributed control is beyond the scope of the current study and is identified as a useful direction for future efforts.The stated scope concerns fundamental understanding of control predicted from individual regions.
- Centralized versus distributed control: Internal control is described as a feature of the brain itself, including homeostasis and cognitive control that can drive transitions between dynamical states.The related literature discusses both localized control systems and views that control is not confined to small regions or modules.
Network synchronizability
Synchronizability measures a network’s tendency to persist in a single synchronous state and is quantified from the spread of Laplacian eigenvalues. The analysis normalizes for overall coupling strength and uses permutation null models and fitted developmental trajectories.
- Network synchronizability: Synchronizability measures the ability of a network to persist in a single synchronous state, analyzed through the positive eigenvalues of its Laplacian matrix.The master stability function provides the stability framework without requiring detailed specification of individual dynamical units.
- Network synchronizability: The normalized spread of Laplacian eigenvalues is used to quantify how synchronizable a network generally will be.The metric incorporates the eigenspectrum rather than relying only on connection density.
- Network synchronizability: The synchronizability calculation normalizes for average coupling strength per node to account for overall network strength.The analysis separately plots numerator and denominator components with age to distinguish eigenspectrum variation from connection-density differences.
- Statistical analysis: Permutation null models preserve either network degree or strength while randomly reallocating edge weights within those constraints.These models are used to assess the statistical significance of results.
- Statistical analysis: Correlations were generally Pearson correlations, with Spearman correlations used for markedly skewed distributions such as regional modal controllability and cognitive performance.Regional age correlations were corrected for multiple comparisons using false discovery rate q = 0.05.
- Curve fitting: Generalized additive models with penalized regression splines were used for nonlinear developmental fits, while exponential fits were used for data and optimization trajectories.The authors note that Pareto optimization trajectories were not truly exponential but retained exponential fits for plotting.
Pareto-optimization parameters
The analysis uses Pareto-optimization trajectories and graph-model ensembles to examine how network topology and weighting relate to controllability and synchronizability. Models are compared under fixed network size, density, and, in some analyses, controlled edge-weight distributions.
- Pareto-optimization parameters: 100 parallel computations used independent random edge swaps, with forward controllability trajectories following the same path.Curve-fitting was performed only on forward trajectories.
- Pareto-optimization parameters: Backward trajectories were variable and truncated when the controllability gradient became negative; the longest trajectory was retained.Some trajectories turned around or showed erratic jumps after decreasing controllability.
- Graph-model ensembles: Eight weighted graph models included random, lattice, small-world, modular, geometric, and preferential-attachment topologies.The models were selected from general graph theory and models proposed for human brain-network topology.
- Graph-model ensembles: Networks were fixed at 128 nodes with edge counts chosen to match densities observed in large-scale human brain graphs.The fixed node count also accommodated models defined only for powers of two.
- Graph-model ensembles: The weighting pipeline reweighted graph models to disentangle topology from edge-weight distributions while preserving relative weight magnitudes.Analyses used streamline-count and Gaussian-weight ensembles, with 100 instantiations per model.
Data availability
The study provides public access to controllability-metric scripts and uses diffusion tensor imaging data from the publicly available Philadelphia Neurodevelopmental Cohort. The passage set also records author contributions, funding acknowledgments, and the absence of competing financial interests.
- Data availability: Scripts for calculating controllability metrics are publicly available through the authors’ research-projects webpage.The scripts include the controllability metrics used in the study.
- Data availability: Diffusion tensor imaging data came from the Philadelphia Neurodevelopmental Cohort, publicly available through the Database of Genotypes and Phenotypes.The cohort is identified as the source of the diffusion imaging data.
- Acknowledgments: The authors acknowledge support from multiple foundations, government agencies, and research organizations.Acknowledged support includes the MacArthur Foundation, Sloan Foundation, Army Research Laboratory, NIH, ONR, and NSF.
- Author contributions: Author contributions covered research design, data collection and preprocessing, tractography, coding, analysis, and manuscript preparation.Different contributors handled tractography, controllability metrics, Pareto optimizations, cognitive data, and DTI preprocessing.
- Conflicts of interest: The authors declared no competing financial interests.This is the paper’s stated conflict-of-interest disclosure.