Source-linked AI summary
Controllability of Brain Networks
Shi Gu, Fabio Pasqualetti, Matthew Cieslak, Scott T. Grafton, Danielle S. Bassett
TL;DR
The paper addresses how structural brain-network organization constrains transitions between cognitive states, a question for which fundamental principles remain elusive. It applies network control theory to white-matter network structure and finds distinct control roles associated with dense, weak, and boundary connectivity. These results link cognitive-system organization with different modes of controlling brain-network trajectories.
Problem
Fundamental principles constraining how large-scale neural systems move between cognitive states remain elusive.
Method
The study uses network control theory and structural brain networks estimated from diffusion spectrum imaging and tractography to predict cognitive-dynamics control properties.
Results
Densely connected default-mode areas facilitate many easily reachable states, weakly connected cognitive-control areas facilitate difficult-to-reach states, and boundary areas support integration or segregation.
Takeaways & Limitations
Structural differences between cognitive systems correspond to distinct roles in controlling dynamic trajectories of brain-network function.
Takeaways & Limitations
The predictions depend on the accuracy of the estimated structural connectivity matrix and the equation defining dynamics, including assumptions about streamline counts.
Abstract
from arXiv · showhide
Cognitive function is driven by dynamic interactions between large-scale neural circuits or networks, enabling behavior. Fundamental principles constraining these dynamic network processes have remained elusive. Here we use network control theory to offer a mechanistic explanation for how the brain moves between cognitive states drawn from the network organization of white matter microstructure. Our results suggest that densely connected areas, particularly in the default mode system, facilitate the movement of the brain to many easily-reachable states. Weakly connected areas, particularly in cognitive control systems, facilitate the movement of the brain to difficult-to-reach states. Areas located on the boundary between network communities, particularly in attentional control systems, facilitate the integration or segregation of diverse cognitive systems. Our results suggest that structural network differences between the cognitive circuits dictate their distinct roles in controlling dynamic trajectories of brain network function.
Introduction
The paper asks how structural brain-network organization constrains transitions between cognitive states and applies network control theory to model those dynamics. It predicts distinct control roles for densely connected, weakly connected, and boundary regions.
- Fundamental principles governing how neural systems move along cognitive, disease, or recovery trajectories remain elusive.
- Network control theory models how local brain-region interactions can drive the global brain network along desired trajectories through diverse system states.
- The study constructs structural brain networks from triplicate diffusion spectrum imaging scans of 8 healthy adults and 234 cortical and subcortical regions.
- Controllability is assessed through the Gramian, whose invertibility indicates whether control inputs can drive the network to a target state.
- Average controllability identifies hubs that can steer the brain into many states, whereas modal controllability identifies weakly connected areas that can reach difficult states.
- Boundary controllability identifies intermediate-degree regions that can steer different cognitive systems toward decoupled or integrated states.
Regional Controllability of Cognitive Systems
Controllability profiles are differentially distributed across cognitive systems: default mode hubs support many easy-to-reach states, cognitive-control regions support difficult transitions, and attentional regions regulate integration or segregation.
- Boundary controllability: Boundary controllability identifies regions that steer the brain toward states in which cognitive systems are integrated or segregated, with hubs concentrated in ventral and dorsal attention systems.Boundary-control regions are neither the highest-degree hubs nor the lowest-degree non-hubs, but lie near network-community boundaries.
- Cognitive-system distribution: 30% of average control hubs lie in the default mode system, 32% of modal control hubs in fronto-parietal and cingulo-opercular systems, and 36% of boundary control hubs in ventral and dorsal attention systems.Hubs were defined as the 30 regions with the largest controllability values and normalized by cognitive-system size.
- Differential recruitment: Default mode regions form strong average-controllability hubs, cognitive-control regions form strong modal-controllability hubs, and attentional-control regions form strong boundary-controllability hubs.A repeated-measures analysis reported significant main effects of cognitive system and controllability diagnostic, with a significant interaction reported in the cited passage.
- Interpretation: The framework predicts that structural differences among default mode, cognitive-control, and attentional-control systems support distinct roles in controlling dynamic brain trajectories.The paper describes these roles as movement toward easy states, movement toward difficult states, and integration or segregation of cognitive systems.
- Average controllability: Average controllability identifies regions that steer the brain toward many easily reachable states and is greatest in densely connected hub regions, particularly in the default mode system.Average controllability is associated with a baseline organization that supports movement to many reachable states.
The Role of Community Structure in Brain Control
The paper uses three controllability diagnostics to connect brain network organization with distinct control goals, including reaching easy or difficult states and integrating or segregating modules. These diagnostics are normalized across subjects and examined across multiple atlas resolutions.
- Average controllability: Average controllability identifies regions influential across many target states and is approximated using Trace(W_K) because the controllability Gramian is nearly singular.The adopted measure reflects control over network dynamics across different target states.
- Modal controllability: Modal controllability identifies nodes able to control the network’s dynamic modes and drive activity toward hard-to-reach configurations.It is computed from the adjacency matrix’s eigenvector structure.
- Boundary controllability: Boundary controllability measures whether control nodes can decouple dynamic trajectories of disjoint brain regions.It is evaluated after partitioning the brain network and identifying boundary nodes.
- Normalization and comparison: The three diagnostics produce one scalar value per brain region, which is ranked within subject and averaged across subjects for comparison.This procedure supports direct comparisons among diagnostics and participants.
- Network construction and scale: The study constructs structural brain networks from diffusion tractography of 234 cortical and subcortical regions and examines reproducibility across Lausanne atlas resolutions.The main parcellation uses N = 234 regions, while the supplement evaluates other spatial scales.
Global Controllability Across Spatial Scales
Across Lausanne atlas resolutions, global controllability decreases with spatial scale, while the anatomical distributions of average, modal, and boundary controllability remain visually reproducible. The diagnostics also show greater within-subject than between-subject correlations across scans.
- Global controllability: Mean global controllability decreases as the Lausanne atlas spatial scale increases.The reported mean averages across subjects, scans, and brain regions.
- Spatial-scale reproducibility: Average, modal, and boundary controllability retain visually similar anatomical distributions across all five assessed spatial scales.The main manuscript uses Scale 125 with N = 234 regions, while the supplement examines the wider Lausanne atlas family.
- Relations with network degree: Degree is strongly positively correlated with average controllability, strongly negatively correlated with modal controllability, and not strongly correlated with boundary controllability at Scale 125.The supplement reports these correlations across resolutions from Scale 33 to Scale 500.
- Test-retest reliability: All three controllability diagnostics show significantly greater within-subject than between-subject correlation across scanning sessions.This pattern indicates statistical reproducibility across sessions and differences across individuals.
Reproducibility of Differential Recruitment of Cognitive Systems to Network Control
Control roles are differentially recruited across cognitive systems, and this pattern is reproduced when control hubs are defined using alternative thresholds. Repeated-measures ANOVA supports system-by-diagnostic differences in control roles.
- Statistical validation: For the top 25 nodes, the system-by-diagnostic interaction is F(18) = 42.1475 (p = 0), while for the top 35 nodes it is F(18) = 36.9762 (p = 0).Both analyses use repeated-measures two-way ANOVA with system and diagnostic as categorical factors and scan replicate as a repeated measure.
- Interpretation: The findings suggest that structural differences among default mode, cognitive control, and attentional control systems may support distinct roles in controlling dynamic brain trajectories.The paper interprets the system-by-diagnostic interaction as evidence that controllability types may be differentially utilized or enabled by cognitive systems.
Robustness of Results to Alternative Weighting Schemes
The results remain robust when anatomical-network edge weights are adjusted for region size: controllability diagnostics retain their differential recruitment across cognitive systems.
- Alternative weighting scheme: The alternative weighting scheme divides each adjacency-matrix element by the summed sizes of its two connected regions.This correction addresses potential bias from larger regions producing more estimated streamlines.
- Robustness of results: Average, modal, and boundary controllability hubs are differentially located in default-mode, cognitive-control, and attentional-control systems.These values are averaged over three replicates for each individual, with error bars representing the standard deviation of the mean over subjects.
- Robustness of results: The hub-recruitment pattern is consistent when control hubs are defined as the top 25, 30, or 35 nodes by control value.The figure caption reports this consistency across the three hub-count definitions.
- Robustness of results: The three controllability diagnostics remain differentially recruited to known cognitive systems under the region-size correction.The same pattern is observed when compared with the original weighting scheme.
Interpretations Dependent on Model Assumptions
The paper’s interpretations depend on a structural connectivity matrix and an equation of state defining dynamics on that structure. The approach assumes streamline counts approximate structural connectivity strength, with important exceptions.
- Model assumptions: Network controllability predictions depend on the accuracy of the structural connectivity matrix and the equation of state defining network dynamics.These are the two features distinguishing the controllability model from static graph-theoretical approaches.
- Model assumptions: The analysis assumes that the number of white-matter streamlines is proportional to structural connectivity strength.The paper states that this assumption has important exceptions but is most reasonable for cortico-cortical control.
- Model assumptions: The equation of state is based on prior work demonstrating its utility for predicting resting-state functional connectivity.
Correlation Between Degree and Average Controllability
The paper explains the observed positive correlation between node degree and average controllability mathematically: average controllability is linked to a diagonal element of an inverse matrix, whose first-order approximation includes node degree.
- Mathematical explanation: Because A is stable, a first-order approximation can be used to analyze the relevant diagonal element.
- Mathematical explanation: The diagonal element is approximated by a term containing the sum of squared connection weights associated with node j.
- Implication: A positive correlation between node degree and average controllability is mathematically expected in the studied networks.The paper defines node degree as d_j = Σ_i A_ij.
Lower Bound on the Largest Eigenvalue of the Controllability Gramian
The paper establishes a lower bound for the largest eigenvalue of the controllability Gramian while contrasting it with the much smaller smallest eigenvalue.
- Eigenvalue bound: The largest eigenvalue of the controllability Gramian is lower bounded by 1.The paper introduces this bound while noting that the smallest eigenvalue is much smaller than the largest.
Additional Algorithmic Details for Boundary Control Method
The boundary-control algorithm combines community detection, recursive Fiedler bipartitions, and a local threshold ratio to select control nodes. Its parameters and partitions are evaluated for stability before controllability calculations.
- Community detection: The modularity quality function assigns each brain region to one community and favors partitions with high within-community edge weight relative to a null model.The resolution parameter γ controls the structural scale of community detection.
- Community detection: 100 Louvain-like optimizations are used for each scan, followed by a consensus partition to address near-degenerate modularity solutions.Repeated optimization produces a representative partition based on statistical comparison with a null model.
- Parameter stability: Mean partition similarity was high and variance low for γ values between 1.5 and 2 across all five atlases.Partition similarity was evaluated across γ values from 0.5 to 2 in increments of 0.1 and across five spatial scales.
- Boundary criteria: The boundary-point threshold uses a ratio ρ because nearly all weighted brain-network nodes connect to both communities.Values that are too small have little effect, whereas values that are too large identify only a few boundary points.
- Boundary criteria: Boundary controllability values were highly similar across ρ values from 0.15 to 0.25, with a minimum Pearson correlation of approximately 0.68.The main analysis used ρ = 0.2, and the robustness analysis varied ρ in increments of 0.01.
- Algorithm: The method begins with community detection, then recursively applies Fiedler bipartitions within communities to add boundary nodes.Boundary nodes connect to nodes in other communities, and the recursion targets improved local controllability.
Association of Brain Regions to Cognitive Systems
The paper maps 234 brain areas from 42 cortical structures onto nine cognitive systems to compare controllability diagnostics across systems. Because many regions participate in multiple functions, these assignments are treated as pragmatic approximations rather than exclusive functional labels.
- Assignment procedure: Assignments are based primarily on the cognitive-system decomposition of Power et al. (2012), supplemented by wider literature for ambiguous regions.Lateral orbitofrontal cortex and pars orbitalis are examples where broader literature informed a single assignment.
- Scope of mapping: The region-to-system association is a gross approximation and should not be interpreted as indicating that areas have single functions.The mapping is used pragmatically to assess whether controllability diagnostics differ across cognitive systems.
- System mapping: The analysis associates 234 brain areas from 42 cortical structures with nine cognitive systems.The systems include fronto-parietal, cingulo-opercular, dorsal and ventral attention, default mode, motor and somatosensory, auditory, visual, and subcortical systems.
- Representative assignments: Several regions are assigned to systems consistent with their described network roles, including default mode hubs and fronto-parietal or cingulo-opercular control areas.Examples include the precuneus in the default mode system, frontal pole in fronto-parietal, and anterior cingulate regions in cingulo-opercular.