PaperlayerView exact revision on arXiv

Source-linked AI summary

Control of Dynamics in Brain Networks

Evelyn Tang, Danielle S. Bassett

arXiv:1701.01531v3q-bio.QMcond-mat.dis-nncond-mat.softphysics.bio-phq-bio.NC

TL;DR

Controlling large-scale brain dynamics is difficult because neural systems contain many interacting components, while their control remains less understood than single-neuron control. This review synthesizes network-control models and applications spanning cognition, development, and medical intervention, and highlights extensions toward nonlinear dynamics and realistic control trajectories.

  • Problem

    Large-scale neural systems remain poorly understood from a control perspective because brain dynamics arise from complex interactions across many components.

  • Method

    The paper reviews network-control theory and related mathematical models for controlling local circuits and whole-brain connectomes.

  • Results

    The review covers control mechanisms and applications involving cognitive function, brain development, anesthesia, seizure suppression, and Parkinson’s disease.

  • Takeaways & Limitations

    Emerging nonlinear, time-dependent, and trajectory-based control methods provide directions for addressing open questions in neuroscience and medicine.

Abstract

from arXiv · show

The ability to effectively control brain dynamics holds great promise for the enhancement of cognitive function in humans, and the betterment of their quality of life. Yet, successfully controlling dynamics in neural systems is challenging, in part due to the immense complexity of the brain and the large set of interactions that can drive any single change. While we have gained some understanding of the control of single neurons, the control of large-scale neural systems -- networks of multiply interacting components -- remains poorly understood. Efforts to address this gap include the construction of tools for the control of brain networks, mostly adapted from control and dynamical systems theory. Informed by current opportunities for practical intervention, these theoretical contributions provide models that draw from a wide array of mathematical approaches. We present intriguing recent developments for effective strategies of control in dynamic brain networks, and we also describe potential mechanisms that underlie such processes. We review efforts in the control of general neurophysiological processes with implications for brain development and cognitive function, as well as the control of altered neurophysiological processes in medical contexts such as anesthesia administration, seizure suppression, and deep-brain stimulation for Parkinson's disease. We conclude with a forward-looking discussion regarding how emerging results from network control -- especially approaches that deal with nonlinear dynamics or more realistic trajectories for control transitions -- could be used to directly address pressing questions in neuroscience.

I. INTRODUCTION

The introduction frames brain-network control as an emerging theoretical field motivated by the brain’s complex interactions and new experimental, computational, and theoretical tools. It connects intrinsic cognitive control with network-control approaches aimed at understanding healthy function and therapeutic intervention.

  • Motivation: Brain dynamics span many spatial and temporal scales, with large numbers of interacting units that make control challenging.The cited context ranges from 302 neurons in C. elegans to roughly 86 billion neurons in the adult human brain.
  • Motivation: New neurotechnologies, computational tools, and theoretical models have enabled fundamentally new approaches to controlling neural activity and rhythms.The paper presents these developments as timely for building a broader theory of brain control.
  • Network perspective: The review focuses on systems-level control of local neural circuits and whole-brain connectomes using interacting-network models.Brain networks are represented as graphs whose nodes are functional units and whose edges encode structural links or synchronized dynamics.
  • Review aim: Network-control theory is introduced as a framework for explaining distributed cognitive control and probing interventions in brain networks.The review surveys existing work rather than presenting new data and connects computational models with open questions in neuroscience and medicine.
  • Intrinsic control: Cognitive control provides a model for understanding how internal brain mechanisms drive neural dynamics between states through flexible, distributed processing.The discussion links top-down control, conflict monitoring, adaptive gating, and reinforcement learning across interacting brain regions.

III. NETWORK CONTROL THEORY

Network-control theory models how external inputs can drive a brain network from an initial state to a desired target state. For linear dynamics, controllability is determined by whether the selected control nodes provide a full-rank controllability matrix.

  • Framework: Controllability asks whether an external input can drive a dynamical network from its current state to a specified target state.The framework represents brain networks as graphs with weighted adjacency matrix A and node-state vector x.
  • Control inputs: A subset of independently controlled nodes K injects control signals into the network through an input matrix.The input matrix is constructed from canonical vectors associated with the selected control nodes.
  • Controllability: The controlled linear network is controllable in T steps when an appropriate input can reach every final state from the zero initial state.The control signal is injected through nodes K, and the desired terminal condition is x(T) = x_f.
  • Controllability: The network is controllable in T steps if and only if its controllability matrix C_K,T has full row rank.The time horizon T is typically at least as large as the system size n.

B. Key driver nodes

Network control theory identifies driver nodes and controllability metrics for steering complex networks, while accounting for structural uncertainty and control-energy costs. Average, modal, and boundary controllability describe distinct abilities to reach nearby, difficult, or community-integrating states.

  • Under certain conditions, driver nodes capable of guiding an entire weighted, directed network can be estimated directly from its degree distribution.
  • Degree distribution alone may be insufficient to identify driver nodes, requiring network-structure information to be complemented by dynamics or approximations of node dynamics.
  • Structural controllability evaluates binary networks when edge weights are uncertain by distinguishing absent from present connections.
  • Control energy and metrics: Small Gramian eigenvalues indicate that certain target states require control energy beyond practical limits, motivating energy-based controllability metrics.
  • Control energy and metrics: Average controllability describes nodes that steer systems into many nearby states with little input energy, whereas modal controllability targets difficult-to-reach states requiring substantial energy.
  • Control energy and metrics: Boundary controllability identifies nodes between network communities and measures the ability to control integration and segregation across network modules.
  • Control energy and metrics: The three metrics provide useful whole-network estimates, but their behavior for partial-network dynamics and changing community structures remains open.

D. Application to brain networks

Brain network control applies controllability theory to structural connectomes constructed from neuroimaging and related connectivity measurements. Human studies have used these networks to evaluate average, modal, and boundary controllability, while incomplete observation introduces uncertainty.

  • Brain structural networks can be constructed by measuring water diffusion, reconstructing white-matter streamlines, and encoding estimated connection strengths in an adjacency matrix.
  • The resulting graph represents brain regions as nodes and inter-regional connection strengths as edges.
  • Gu et al. applied network control theory to human whole-brain structural networks containing between 83 and 1015 nodes.
  • That study evaluated average, modal, and boundary controllability in whole-brain structural networks.
  • Because observability is limited in living, behaving systems, non-invasive neuroimaging introduces uncertainty into both data and models relevant to control.

IV. UNDERSTANDING HEALTHY BRAIN FUNCTION THROUGH CONTROL THEORY

Network control theory offers testable mechanisms for cognitive control by linking controllability profiles to brain systems and their functions. Structural predictions distinguish systems associated with nearby-state transitions, difficult transitions, and attentional control, although broader validation remains necessary.

  • Gu et al. tested whether control strategies from dynamical-systems theory predict how the brain controls its intrinsic dynamics by calculating regional controllability strengths and control preferences.
  • Boundary-controllability hubs are distributed across systems, with ventral and dorsal attention systems predominant.
  • Average-controllability hubs are preferentially located in the default mode system, whereas modal-controllability hubs are concentrated in frontoparietal and cingulo-opercular control systems.
  • These controller types map onto the functions associated with their brain regions, linking nearby-state control, distant-state control, and cognitive or attentional demands.
  • The findings offer a possible mechanistic explanation based on white-matter microstructure for movement between cognitive states.
  • The broad trends support the relevance of control theory for capturing canonical concepts in cognitive control, but require validation in other species and data sets.

B. Network control and cognitive performance

Controllability metrics are linked to cognitive performance and development, while emerging nonlinear approaches may better capture transitions between neural dynamical regimes.

  • Cognitive performance: Relative average controllability in subcortical versus cortical regions predicts improved cognitive performance independently of age.A follow-up study connected these differences specifically to individual variation in cognitive control.
  • Development: Developmental network evolution increasingly structures brain networks in ways optimized for controlling a broader range of dynamics.Simulations with growth rules suggested increasingly structured organization from childhood through adulthood.
  • Development: Brain networks support energetically easy and costly transitions, with both average and modal controllability increasing across ages 8 to 22.The study analyzed 882 healthy youth and found the supported range of dynamics increased with age.
  • Open questions: Linear network-control models provide intuition near an operating point but cannot represent transitions between distinct dynamical regimes.Additional nonlinear control methods may therefore be necessary for limit-cycle, fixed-point, or attractor transitions.
  • Open questions: Decision-making studies illustrate how synchronized regional activity and state-space transitions can characterize rational and irrational mental states.High-frequency activity of 70-100 Hz increased during irrational decisions in the precuneus.

V. TARGETING THERAPEUTIC INTERVENTIONS TO MAXIMIZE BENEFICIAL OUTCOMES TO PATIENTS

Clinical applications use control theory to modulate anesthesia, model Parkinsonian circuitry, and develop closed-loop stimulation strategies. These approaches aim to improve intervention precision while accounting for distributed network mechanisms.

  • Anesthesia titration: Deeper propofol anesthesia produces burst suppression, marked by concurrent bursts and epochs of reduced electrical and metabolic activity.Figure 7 contrasts ordinary deep anesthesia with the quiescent epochs characteristic of burst suppression.
  • Anesthesia titration: Closed-loop anesthesia systems adapt propofol delivery over time to maintain medically induced coma while potentially reducing overdose-related side effects.Real-time burst-suppression monitoring provides a control target for tracking consciousness.
  • Deep-brain stimulation for Parkinson’s disease: Parkinson’s DBS may work through network-level effects that regularize basal-ganglia firing across the cortico-basal ganglia-thalamo-cortical loop.Simulations suggest 130 Hz stimulation resonates with the overall loop and that distributed circuit effects matter more than one stimulation site alone.
  • Deep-brain stimulation for Parkinson’s disease: Clinical control strategies may need to target multiple mechanisms, including circuit resonance, peripheral tremor coupling, and motor-area phase locking.The review presents these as candidate mechanisms rather than a comprehensive account of therapeutic control.
  • Deep-brain stimulation for Parkinson’s disease: Closed-loop DBS models use mean-field descriptions of basal-ganglia physiology to tune stimulation parameters to patient physiology.If empirically validated, this approach could improve on clinician experience-based trial-and-error tuning.

C. Non-invasive transcranial stimulation

Non-invasive stimulation approaches use externally applied fields to influence brain dynamics, including closed-loop seizure suppression and connectivity-informed control. Models and clinical contrasts emphasize that intervention strategies must be matched to seizure dynamics and validated experimentally.

  • Transcranial magnetic or electric stimulation applies a brief field through the scalp and has shown utility for depression and other neurological and psychiatric disorders.
  • Closed-loop transcranial electrical stimulation reduced seizure duration by 60% on average in a rat epilepsy model.
  • Wilson-Cowan models represent distributed seizure control with stimulating-electrode grids that help stem and direct propagating electrical activity.
  • Patient-MRI connectivity can personalize stimulation models covering larger spatial areas, while other approaches target local seizure-related regions.
  • Because fully synchronized states occur in only some seizure types, control strategies likely need to differ across seizure etiologies and require experimental validation.

A. Synchrony of neural populations

Synchrony describes shared neural dynamics, and network structure helps determine both its stability and controllability. Studies link synchronizability to brain maturation and show that topology, coupling, motifs, and symmetries shape possible control strategies.

  • Synchrony occurs when neuronal populations or brain regions exhibit identical dynamics, and transitions between synchrony and desynchrony matter for epilepsy and Parkinson’s disease.
  • Structural drivers of synchrony: The master stability function assesses perturbative stability of synchronous oscillator states through the network Laplacian’s positive eigenvalues.
  • Structural drivers of synchrony: Inverse Laplacian-eigenvalue variance provides a global synchronizability measure, with smaller eigenvalue spread typically indicating easier synchronization.
  • Structural drivers of synchrony: In 882 youths aged 8–22, greater synchronizability was associated with lower average and modal controllability, while synchronizability decreased with age.
  • Structural drivers of synchrony: Network coupling strength, topology, structural symmetries, and three-node motifs influence transitions toward synchrony and the controllability of nonlinear neuronal dynamics.

B. The cost of controlling specific trajectories

Trajectory-control frameworks model movement between brain activation states by balancing control energy with distance from a target. Simulations connect efficient transitions to network communicability, while highlighting limits imposed by trajectory selection.

  • Control metrics distinguish average controllability for nearby states from modal controllability for distant states, while coarse-graining over many transitions.
  • Trajectory models seek an input that moves the system from an initial state x0 to a target state xT while minimizing transition energy and target distance.
  • Simulations of transitions from baseline to visual, auditory, and motor states found that efficient drivers had high communicability to the target.
  • Communicability weights walks of all lengths, indicating that long-distance network walks contribute to efficient control.
  • Mild traumatic brain injury was associated with reduced specificity in the putative control processes supported by brain networks.
  • The modeling framework requires selecting trajectories, and further work is needed to identify trajectories that better reveal actual brain dynamics.

C. Empirical tools for control of specific neural dynamics or pathways

Empirical control tools connect theoretical models with interventions targeting specific neural cells, circuits, and pathways. Optogenetics offers precise closed-loop manipulation, but stimulation effects depend on cell type, opsin kinetics, and intended outcomes.

  • Empirical data and experimental control tools are needed to inform and validate models of neural synchrony and brain-state transitions.
  • Optogenetics provides millisecond-scale optical control of defined cell types, sometimes at single-cell resolution, enabling causal investigation of neural circuitry.
  • Simultaneous recording and targeted stimulation enable closed-loop control in animals, but stimulation outcomes require specific design choices.
  • ChR2(H134R) can evoke circuit-level gamma oscillations above 60 Hz without reliably driving individual pyramidal cells at those frequencies.
  • Controlled pathway and circuit manipulation can clarify their contributions to brain function, while their concerted effects on cognition remain an area for further study.

VII. EMERGING CONTROL METHODS WITH POTENTIAL UTILITY IN NEUROSCIENCE

Emerging control methods extend brain-network control beyond simplified linear models by addressing nonlinear dynamics, practical trajectory feasibility, incomplete information, changing topology, and attractor switching.

  • VII. EMERGING CONTROL METHODS WITH POTENTIAL UTILITY IN NEUROSCIENCE: Linear and simplified models offer useful insights, but brain dynamics require methods that capture nonlinear regimes, time dependence, and problem-specific control strategies.Proposed extensions include perturbations, stochasticity, and network-topology properties.
  • VII. EMERGING CONTROL METHODS WITH POTENTIAL UTILITY IN NEUROSCIENCE: Feedback vertex sets identify nodes whose open-loop control can switch a system between dynamical attractors under broad nonlinear conditions.The formalism requires conditions such as continuity, dissipation, and decay, which are often satisfied by real systems.
  • VII. EMERGING CONTROL METHODS WITH POTENTIAL UTILITY IN NEUROSCIENCE: Comparisons between feedback vertex set control and structural controllability reveal topological differences underlying their distinct control predictions.The comparison has also been applied to directed gene-regulation networks.
  • VII. EMERGING CONTROL METHODS WITH POTENTIAL UTILITY IN NEUROSCIENCE: Stable motifs can transiently control logical network dynamics so the system reaches and remains in a desired state when network structure and function are known.This approach has been illustrated in leukemia-signaling and other biological networks.
  • VII. EMERGING CONTROL METHODS WITH POTENTIAL UTILITY IN NEUROSCIENCE: Data-driven optimization can design interventions from partial time-series observations, while dynamically changing edges may enable faster or less energetically costly control.These approaches are relevant to limited-information settings and potentially to seizure-spread control.
  • VII. EMERGING CONTROL METHODS WITH POTENTIAL UTILITY IN NEUROSCIENCE: The controllability Gramian’s condition number is crucial for practical control because ill conditioning can cause numerical failure and nonlocal trajectories even in linear systems.The numerical controllability transition relates control success to the number of control inputs.

B. Exploiting system properties

System-property-based methods seek control strategies that exploit perturbations, quasipotential structure, topology, modularity, and symmetries rather than targeting only nodes or edges.

  • B. Exploiting system properties: Compensatory perturbations can steer a dynamical system toward desired states without restricting interventions to particular nodes or edges.The approach is presented as suitable for realistic nonlinear or stochastic regimes.
  • B. Exploiting system properties: Minimum-action paths and transition rates can identify interventions that reshape quasipotential barriers between stable states.Optimizing transition rates alters the likely noise-induced transitions between stable states.
  • B. Exploiting system properties: Network diameter and edge weights influence the control degree and energy required for groups of nodes to drive a network toward a desired state.This work examines controllability as a function of these topological and weighted properties.
  • B. Exploiting system properties: Sources, sinks, degree distributions, and local complexity such as cycles shape control profiles in empirical networks.These features correlate with control properties across real-network analyses.
  • B. Exploiting system properties: Modularity and symmetries can decompose continuous-time network dynamics into interacting local control systems and generate conjugate dynamical systems through graph fibrations.This formalism generalizes relationships between network structure, synchrony, and dynamics.
  • B. Exploiting system properties: Rapid theoretical and technological progress has expanded brain-network control methods while leaving many open questions for future work.The review frames the field as an emerging area with potential implications for health and cognitive function.
Loading 1701.01531v3…