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Optimally controlling the human connectome: the role of network topology

Richard F. Betzel, Shi Gu, John D. Medaglia, Fabio Pasqualetti, Danielle S. Bassett

arXiv:1603.05261v1q-bio.NC

TL;DR

The paper asks how anatomical network topology relates to the effort required for brain-state transitions. It uses network control theory to model transitions among cognitive systems and finds that weighted degree, communicability, and rich-club organization relate to transition energy.

  • Problem

    The authors ask which anatomical and topological features help explain why some transitions between cognitive-system states are more effortful than others.

    Source: §I · pp. 1–2

  • Method

    Network control theory is used to calculate minimum-energy inputs for finite-time transitions among selected brain-system states and to examine alternative initial and target states.

    Source: §I · pp. 1–2, Control tasks · p. 12

  • Results

    Control energy is associated with weighted degree, communicability predicts compensation after input suppression, and disrupting rich-club connections increases energy for optimal state transitions.

    Source: §I · p. 2, DISCUSSION · pp. 7–8

  • Takeaways & Limitations

    The findings link brain-network topology, especially hub organization, to the energy associated with transitions among brain systems.

    Source: Rich club promotes low-energy… · p. 7, DISCUSSION · p. 8

    The standard communicability measure on binary networks exponentially downweights longer paths, and alternative weighting schemes may reveal additional insight.

    Source: Methodological considerations · p. 11

Abstract

from arXiv · show

To meet ongoing cognitive demands, the human brain must seamlessly transition from one brain state to another, in the process drawing on different cognitive systems. How does the brain's network of anatomical connections help facilitate such transitions? Which features of this network contribute to making one transition easy and another transition difficult? Here, we address these questions using network control theory. We calculate the optimal input signals to drive the brain to and from states dominated by different cognitive systems. The input signals allow us to assess the contributions made by different brain regions. We show that such contributions, which we measure as energy, are correlated with regions' weighted degrees. We also show that the network communicability, a measure of direct and indirect connectedness between brain regions, predicts the extent to which brain regions compensate when input to another region is suppressed. Finally, we identify optimal states in which the brain should start (and finish) in order to minimize transition energy. We show that the optimal target states display high activity in hub regions, implicating the brain's rich club. Furthermore, when rich club organization is destroyed, the energy cost associated with state transitions increases significantly, demonstrating that it is the richness of brain regions that makes them ideal targets.

I. INTRODUCTION

The introduction frames brain-state transitions as a network-control problem: anatomical topology may shape how much input different transitions require. The model represents regional activity over time and evaluates finite-time transitions among cognitive-system states using minimum-energy inputs.

I. INTRODUCTION

  • The authors use network control theory to identify minimum-energy inputs for driving the brain to and from selected cognitive-system states.Regional input energy represents a control point’s contribution to a task.Source: §I · pp. 1–2
  • The study addresses a gap in node- and edge-level understanding by examining finite-time transitions among accessible brain-system states rather than only binary controllability.Earlier approaches emphasized global network statistics or all transitions over an infinite time horizon.Source: §I · pp. 1–2
  • Weighted degree is associated with control energy, while communicability predicts how strongly regions compensate when input to another region is suppressed.Communicability measures the strength of direct and indirect pathways between nodes.Source: §I · p. 2

II. MATHEMATICAL MODEL

  • Regional activity follows linear time-invariant dynamics constrained by a weighted adjacency matrix of white-matter tracts, with time-varying inputs applied at selected control points.The state vector records each brain region’s activity level over time.Source: §II · p. 2
  • The model evaluates 56 directed transitions among eight cognitive-system states over a finite horizon, using minimum-energy inputs between initial and target states.The state repertoire is limited to accessible configurations corresponding to previously identified brain systems.Source: §II · p. 2, Figure 1 · p. 3

Predicting energy from topology

Control energy depends on both a region’s task class and its network strength. When input to regions is suppressed, remaining regions compensate, and their energy changes track communicability.

Predicting energy from topology

  • Within classes, log control energy correlated positively with strength for initial and bulk regions (r = 0.85(15), r = 0.79(15)) and negatively for target regions (r = −0.84(13)).Across all controlled regions, energy ranked Etarget > Einitial > Ebulk; free evolution left bulk regions nearest their target state, followed by initial and target regions.Source: Predicting energy from topology · p. 3, Figure 2 · p. 4

Simulated suppression and compensatory effects

  • Simulated suppression excludes selected regions from direct control, leaving them to be manipulated indirectly through their neighbors.The framework models a situation analogous to externally induced suppression such as TMS, while still allowing suppressed regions to receive surrounding inputs.Source: Simulated suppression and compensatory… · p. 4

Suppression of individual brain regions

  • Suppressing individual regions generally increased total energy, with the greatest percent changes occurring among bulk regions.The change in one region’s energy after another region was suppressed corresponded with weighted communicability across tasks (r = 0.95(0.01)); the relationship persisted within initial, target, and bulk classes.Source: Suppression of individual brain regions · p. 4, Figure 3 · p. 5

Suppressing initial or target classes

Suppressing an entire initial or target class increased the burden on remaining controllers, especially bulk regions. When both classes were suppressed, bulk-region energy tracked communicability to the suppressed regions.

Suppressing initial or target classes

  • Bulk regions showed the greatest energy changes when either the initial or target class was suppressed, and change in energy tracked total communicability to the suppressed class.For target-class suppression, correlations were r = 0.76(0.24) for remaining initial regions and r = 0.38(0.36) for bulk; for initial-class suppression, they were r = 72(0.30) for remaining target regions and r = 0.32(0.41) for bulk.Source: Suppressing initial or target classes · p. 5, Figure 4 · p. 6

Suppressing initial and target classes

  • With both initial and target regions suppressed, only bulk nodes controlled the transition, and bulk energy correlated with communicability to the suppressed classes (r = 0.57(0.25)).Bulk nodes then had to turn target regions on and initial regions off; the authors interpret them as compensators despite not changing their own state.Source: Suppressing initial and target classes · p. 5
  • Bulk regions’ energy was negatively correlated with the weighted distance of their neighbors from their target states (r = −0.33(0.18)).The weighted distance combines neighbors’ distances from their target states with the strength of each connection to the bulk region.Source: Suppressing initial and target classes · p. 6

Suppressing initial or target classes

Optimal class assignments

The paper identifies low-energy initial and target assignments, then tests whether network structure—especially rich-club organization—supports those transitions. Optimal targets are highly connected, and disrupting rich-club connections raises transition energy.

Optimal class assignments

  • Optimal assignments associated target probability positively with weighted degree (r = 0.88(0.03)) and bulk probability negatively (r = −0.73(0.07)); initial assignments were not closely associated with degree.Assignments were found by minimizing E = vT v using simulated annealing; randomized degree-preserving networks had greater energies than observed networks across class compositions (max p < 1 × 10−15).Source: Optimal class assignments · p. 7

Rich club promotes low-energy transitions

  • Rewiring rich-club connections while preserving degree increased energy whenever at least eight regions were assigned as initially active (max p < 1 × 10−15).The result held for rich clubs defined from k = 80 to k = 88; the study focused on the binary rich club at k = 84.Source: Rich club promotes low-energy… · p. 7

DISCUSSION

  • Across optimal assignments, initial states were diverse, while target probability correlated strongly with weighted degree; rewiring rich-club connections consistently increased transition energy.The discussion interprets the findings as indicating that connections among hub regions support low-energy transitions.Source: DISCUSSION · pp. 7–8

From descriptive to predictive network models.

The paper extends network research from static descriptions toward models of how topology shapes control among plausible brain states. It finds that control energy depends on a region’s class and strength, compensation tracks communicability, and rich-club organization is associated with lower-energy transitions.

From descriptive to predictive network models.

  • The study extends work on static network properties and passive dynamics by examining control among a limited set of seemingly neurophysiologically plausible states.The authors contrast this approach with assessing controllability across the entire state space.Source: From descriptive to predictive network… · p. 8

control

  • Control energy correlated positively with strength for initial and bulk regions, but negatively for target regions.The relationship between strength and energy therefore depended on a region’s assigned class.Source: control · p. 9
  • Communicability between regions was highly correlated with compensation when input to another region was suppressed, including compensation by direct and indirect partners.The authors propose that indirect, potentially long-distance connections may help explain effects of focal lesions observed away from the lesion site.Source: control · p. 9
  • Bulk regions showed the greatest percent increase in energy under suppression, suggesting a disproportionate compensatory role outside the regions directly involved in a control task.The authors propose that competition among these compensatory roles could contribute to later processing burdens after injury or neurodegeneration.Source: control · p. 9

The rich club as control backbone

  • Dissolving the rich club increased energy for optimal class assignments, while rich-club regions were more likely to serve as targets for transitions from diverse initial states.The authors suggest that hubs’ many connections help make them suitable targets.Source: Figure 6 · p. 9, The rich club as control backbone · p. 10

Methodological considerations

The study uses diffusion spectrum imaging and tractography to construct structural connectivity networks, while noting limitations in tract reconstruction and model assumptions. Its conclusions point to future investigation of how structural differences relate to control and behavior.

Methodological considerations

  • Diffusion spectrum imaging and tractography can produce spurious tracts or miss existing ones, although the authors report no other non-invasive method for reconstructing human structural connectivity networks.They anticipate that improved tractography and imaging may address these shortcomings.Source: Methodological considerations · p. 10
  • The authors used linear dynamics despite acknowledging that brain activity is generally considered nonlinear, because their emphasis was topology and the approach permits tractable analysis.They note that linear approximations near equilibrium points are common for otherwise intractable systems.Source: Methodological considerations · pp. 10–11
  • For binary networks, the communicability measure weights longer paths exponentially less than shorter paths; alternative weighting schemes may provide additional insight.Source: Methodological considerations · p. 11

Conclusion

  • The conclusion identifies individual differences in brain structure, control implementation, behavior, and response to neuromodulation as future topics, with personalized medicine as a potential application.Source: Conclusion · p. 11

Data acquisition and processing

  • The analysis used diffusion spectrum images from 30 participants and constructed streamline adjacency matrices from the reconstructed and parcellated data.The atlas contained 129 regions, and each matrix entry represented streamline count divided by the summed volumes of its two regions.Source: Data acquisition and processing · p. 11

Cognitive systems

The methods assign brain regions to intrinsic connectivity systems and model transitions between system-defined states. They also introduce communicability to characterize indirect connections between regions.

Cognitive systems

  • Regions were assigned to one of eight intrinsic connectivity networks according to the system with which each region overlapped most.The systems included default mode, control, dorsal attention, saliency/ventral attention, somatomotor, visual, limbic, and sub-cortical networks.Source: Cognitive systems · p. 11

Control tasks

  • The control tasks set one cognitive system active initially and another active at the target, with a control horizon of T = 1.The study considered a default-mode-to-visual example and calculated optimal inputs for the specified boundary states.Source: Control tasks · p. 12

Network communicability

  • Communicability represents weighted direct and indirect connections between pairs of regions, with longer walks contributing less than shorter ones.Source: Network communicability · p. 12

Optimal class assignments and simulated annealing

The authors search over class sizes and node assignments to reduce transition energy, then define and select a rich club for analysis.

Optimal class assignments and simulated annealing

  • Simulated annealing reduced energy across 44 class compositions, with 50 runs per composition and 100 lowest-energy assignments retained for analysis.Each run used 100 stages of 5,000 iterations, with the temperature decreased by a factor of 0.925 per stage.Source: Optimal class assignments and… · p. 12
  • The number of bulk nodes was varied from 48 to 104, while the remaining nodes were split among initial and target classes in different proportions.The number of nodes assigned to each class was treated as a free parameter.Source: Optimal class assignments and… · p. 12

Rich club detection

  • A rich club is a densely interconnected subset of nodes with degree ≥k; the analysis focuses on k=84, where mean normalized rich-club coefficient is greatest.Coefficients were normalized against randomized networks preserving the degree sequence.Source: Rich club detection · p. 12

Appendix: Optimal control

The appendix formulates optimal control as minimizing the input cost for transitions between states. It derives optimal inputs using a stable system matrix and the Pontryagin minimization principle.

  • Optimal control selects inputs uK that minimize a cost for driving the system from x0 to xT, despite infinitely many possible inputs.Source: Appendix: Optimal control · p. 13
  • With ρ=100, the Hamiltonian combines input cost with system dynamics to derive the optimal inputs.Source: Appendix: Optimal control · p. 13
  • Scaling the weighted-connectivity matrix by its largest eigenvalue and subtracting 1 from its diagonal makes all eigenvalues negative and stabilizes the system.The authors then apply the Pontryagin minimization principle to obtain optimality conditions.Source: Appendix: Optimal control · p. 13
  • Given p*(0) and x0, integrating the trajectory forward yields xT and the subsequent optimal inputs u*K.Source: Appendix: Optimal control · p. 13
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