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White Matter Network Architecture Guides Direct Electrical Stimulation Through Optimal State Transitions

Jennifer Stiso, Ankit N. Khambhati, Tommaso Menara, Ari E. Kahn, Joel M. Stein, Sandihitsu R. Das, Richard Gorniak, Joseph Tracy, Brian Litt, Kathryn A. Davis, Fabio Pasqualetti, Timothy Lucas, Danielle S. Bassett

arXiv:1805.01260v1q-bio.NC

TL;DR

Focal stimulation’s effects on neighboring and distant brain activity remain insufficiently understood. The paper models stimulation through individual white-matter networks and tests predictions using direct stimulation data; true connectivity predicts transitions better than network null models, while farther target states require more energy.

  • Problem

    How focal stimulation changes activity in neighboring and distant brain areas remains insufficiently understood, limiting a first-principles account of stimulation effects.

    Source: Introduction · p. 3

  • Method

    The authors model stimulation with network control theory constrained by individual white-matter connectivity and use an optimal-control framework to estimate energy for state transitions.

    Source: Discussion · p. 15

  • Results

    True white-matter connectivity predicts stimulation-induced state transitions better than topological or spatial null models; transitions to more distant states require greater energy.

    Source: Discussion · p. 15

  • Takeaways & Limitations

    Individual white-matter architecture provides a basis for biologically grounded, individualized hypotheses about response to direct electrical stimulation.

    Source: Discussion · pp. 15–16

    Even simpler local cell-population simulations face challenges accounting for cell orientation and distance from the axon hillock.

    Source: Introduction · p. 3

Abstract

from arXiv · show

Electrical brain stimulation is currently being investigated as a therapy for neurological disease. However, opportunities to optimize such therapies are challenged by the fact that the beneficial impact of focal stimulation on both neighboring and distant regions is not well understood. Here, we use network control theory to build a model of brain network function that makes predictions about how stimulation spreads through the brain's white matter network and influences large-scale dynamics. We test these predictions using combined electrocorticography (ECoG) and diffusion weighted imaging (DWI) data who volunteered to participate in an extensive stimulation regimen. We posit a specific model-based manner in which white matter tracts constrain stimulation, defining its capacity to drive the brain to new states, including states associated with successful memory encoding. In a first validation of our model, we find that the true pattern of white matter tracts can be used to more accurately predict the state transitions induced by direct electrical stimulation than the artificial patterns of null models. We then use a targeted optimal control framework to solve for the optimal energy required to drive the brain to a given state. We show that, intuitively, our model predicts larger energy requirements when starting from states that are farther away from a target memory state. We then suggest testable hypotheses about which structural properties will lead to efficient stimulation for improving memory based on energy requirements. Our work demonstrates that individual white matter architecture plays a vital role in guiding the dynamics of direct electrical stimulation, more generally offering empirical support for the utility of network control theoretic models of brain response to stimulation.

Introduction

The paper addresses how stimulation at one site changes distributed brain activity, a gap that limits understanding of stimulation response. It proposes and tests a white-matter network-control model using ECoG and DWI data, including transitions toward memory-related states.

Introduction

  • How focal stimulation changes activity in neighboring and distant brain areas remains poorly understood, and network-control predictions had not been validated with direct electrical stimulation data.The paper also identifies a missing first-principles account of how an individual brain’s white-matter connectivity shapes stimulation effects.Source: Introduction · pp. 3, 5
  • Prior work links electrode location and individual white-matter connectivity to stimulation outcomes, with connectivity reported to predict successful outcomes.Small electrode-location differences can change which white-matter pathways are directly activated.Source: Introduction · p. 3
  • The study builds subject-specific structural networks from atlas regions and quantitative-anisotropy edges, then models activity changes under stimulation as linear network dynamics.It represents regional activity using electrode band power before and after stimulation and estimates energy to reach a classifier-defined good-memory state.Source: Introduction · p. 5
  • The authors use the model to estimate energy for transitions toward good-memory states and examine how network topology, spatial properties, and stimulation-site control properties relate to response.Good-memory biomarkers are derived from a multivariate classifier applied to ECoG data collected during a verbal memory task.Source: Introduction · p. 5
  • The authors report that stimulation responses are consistent with a simple linear model of network dynamics, supporting a control-theoretic account of direct stimulation.The paper tests this account against ECoG responses and uses it to model transitions toward a good-memory state.Source: Introduction · pp. 5–6

Results

The Results assess how white-matter network structure informs predictions of stimulation-induced brain-state changes and the energy required for transitions toward memory-related targets. They compare empirical and null-network predictions and examine how transition distance, network topology, controllability, and starting state relate to predicted stimulation energy.

Predicting Post-Stimulation States by Open Loop Control

Inducing State Transitions by Optimal Network Control

  • Greater Frobenius-norm distance between initial and post-stimulation states was associated with more transition energy (β = 8.3 × 10−3, p < 2 × 10−16), while higher initial memory-encoding probability was associated with less (β = −0.18, p < 2×10−16).Source: Figure 3 · p. 8, Inducing State Transitions by Optimal… · p. 10
  • Observed memory-state change was positively related to transition energy (β = 8.3 × 10−3, p = 2 × 10−16), consistent with nearby states requiring less energy than distant states.Source: Figure 3 · p. 8, Inducing State Transitions by Optimal… · p. 10
  • The energy–distance relationships were significantly attenuated in both null networks; the topological-null result had p = 6.1 × 10−4.Source: Inducing State Transitions by Optimal… · p. 10
  • In 2 of 3 experimental sessions that included sham stimulation, stimulation trials had significantly larger pre-to-post state distances than sham (Frobenius norm; permutation test, p < 6.8×10−3).Energy was consistently greater for stimulation trials than sham trials (paired t-test, p = 0.01).Source: Inducing State Transitions by Optimal… · p. 11

The Role of Network Topology on Stimulation-Based Control

  • Optimal control energy differed by network type (F(2, 75) = 4.00, p = 0.03): empirical and topological null networks differed (t = −2.6, p = 0.01), but empirical and spatial null networks did not (p > 0.05).The reported comparisons suggest spatial embedding shared by empirical and spatial null networks may affect control difficulty.Source: The Role of Network Topology on… · p. 11, Figure 4 · p. 12
  • Determinant ratio explained energy variance after accounting for network type (χ2 = 12.2, p = 4.7 × 10−4), consistent with spatial embedding imposing energy barriers via connection strength and homogeneity.The determinant ratio captures the trade-off between connection strength and homogeneity.Source: Figure 4 · p. 12, The Role of Network Topology on… · p. 12

Characteristics of Efficient Regional Controllers

Effective Prediction of Energy Requirements

  • OOB mean squared error was 9.28×10−3 versus null µ = 9.62×10−3±2.97×10−5, and the model explained 93.2% of predicted-energy variance.The model used determinant ratio, persistent controllability, and initial memory-encoding probability; their variable importance values were 627, 320, and 23.0, respectively.Source: Effective Prediction of Energy… · p. 14, Figure 6 · p. 15

Discussion

The Discussion considers how individual white matter connectivity constrains stimulation-driven brain-state transitions and how network control theory can generate biologically grounded hypotheses about direct electrical stimulation outcomes. It discusses support for memory-targeted control alongside limitations in the data and model assumptions.

Discussion

  • An individual’s true white matter connectivity predicted time-varying ECoG power changes better than topological or spatial network null models.The model stipulates simplified, noise-free, linear, continuous-time, time-invariant neural dynamics.Source: Discussion · p. 15
  • Network topology and initial brain state together explained 93% of variance in required energy, and transitions to more distant memory-related states required more energy.The distance effect was especially salient when distance reflected differences in the probability that cross-regional ECoG power patterns supported memory encoding.Source: Discussion · pp. 15–16

A Role for Control Theory in Modern Neuroscience

  • Network control theory addresses the gap between human predictions and perturbation validation by testing whether individual white matter connectivity constrains empirically recorded state transitions.The work provides theoretical support for observations that structural connectivity can predict behavioral effects of stimulation.Source: A Role for Control Theory in Modern… · p. 16

The Principle of Optimal Control in Brain State Transitions

  • The optimal-control model links expected energy to changes in the probability that ECoG power patterns support good memory encoding.Relaxing the input matrix allows large input at stimulated regions and small random input elsewhere, reducing calculation error and producing narrowly distributed trajectories.Source: The Principle of Optimal Control in… · pp. 16–17

Topological Influencers of Control

  • Lower determinant ratios and higher persistent controllability were associated with lower control energy; persistent controllability predicted energy across frequency bands, particularly in the α-band.The α-band effect is notable given its known role in memory encoding.Source: Topological Influencers of Control · p. 17

Methodological Considerations

  • Interpretation is limited by incomplete DWI tractography, an epilepsy patient sample, and simplified linear, time-invariant model assumptions.The modeled state uses band-limited ECoG power, which misses non-sinusoidal and phase-dependent signal features; future work should test other state spaces and non-invasive stimulation.Source: Primary Data · pp. 17–18, Modelling Assumptions · p. 18

Conclusions and Future Directions

  • The study’s targeted direct-stimulation model tracks biological intuitions and is influenced by regional and global properties of white matter connectivity.The authors present control-theoretic modeling as a promising approach for guiding stimulation toward brain states associated with memory encoding.Source: Conclusions and Future Directions · p. 19
  • The discussion identifies hypotheses about the outcome of direct electrical stimulation.Source: Conclusions and Future Directions · p. 19

Data Acquisition and Preprocessing

The study combined ECoG and diffusion imaging from 11 participants enrolled in a multicenter project on stimulation and memory-related brain function. ECoG recordings and diffusion scans were processed to represent brain activity and white-matter connectivity.

Data Acquisition and Preprocessing

  • Eleven participants were recruited at Thomas Jefferson University Hospital and the Hospital of the University of Pennsylvania for a multicenter study of stimulation and memory-related brain function.The protocol was approved by each hospital’s institutional review board, and participants provided written informed consent.Source: Data Acquisition and Preprocessing · p. 19
  • ECoG came from subdural and depth electrodes whose clinical placement targeted epileptogenic regions.Subdural contacts were arranged in strips and grids; depth electrodes had 8–12 contacts each.Source: Data Acquisition and Preprocessing · p. 19
  • Brain states were represented by z-scored electrode power in eight logarithmically spaced frequency bands from 1 to 200 Hz.Source: Data Acquisition and Preprocessing · p. 19
  • Diffusion scans were corrected for motion and eddy currents, parcellated using the Lausanne atlas, and processed with whole-brain deterministic tractography.The pipeline generated 1,000,000 streamlines per subject and held the number of streamlines constant across participants.Source: Data Acquisition and Preprocessing · p. 19
  • Tractography used reconstructed orientation density functions and derived QA values, with a 35-degree maximum turning angle and 500 mm maximum streamline length.Source: Data Acquisition and Preprocessing · p. 19

Memory State Classification

Non-stimulation ECoG recorded during a verbal memory task was used to train a personalized classifier for each participant. The simulations’ memory target was defined from states receiving the highest classifier probabilities.

Memory State Classification

  • A personalized memory classifier was trained on spectral power averaged over time within each word-encoding epoch from non-stimulation ECoG collected during a verbal memory task.Source: Memory State Classification · p. 20
  • Each classifier estimated the probability of a good-memory state for pre- and post-stimulation recordings, and the target was the average of the top 5% of states by probability.Source: Memory State Classification · p. 20
  • The 5% threshold was chosen to retain enough trials while selecting memory-relevant features rather than noise; target-state probabilities ranged from 0.61 to 0.74.Source: Memory State Classification · p. 20

Post-Stimulation State Correlations

The analysis compared simulated post-stimulation trajectories with empirically observed post-stimulation states. It measured both the strongest correlation and when it occurred, excluding the earliest simulated time points.

Post-Stimulation State Correlations

  • Simulations began from the observed pre-stimulation state and modeled stimulation to a Lausanne atlas region.For a region with an electrode, its initial state value was z-scored power; other regions were assigned 1.Source: Post-Stimulation State Correlations · p. 20
  • Predicted and observed post-stimulation states were compared using two-dimensional Pearson correlation from simulated time t = 5 through T.Times before t = 5 were excluded so the initial state would not be counted as the peak.Source: Post-Stimulation State Correlations · p. 20
  • The analysis recorded the maximum correlation and the time at which the largest-magnitude correlation, positive or negative, occurred.Source: Post-Stimulation State Correlations · p. 20

Null Models

The observed correlation measures were compared with two kinds of randomized structural networks. The null models preserved different structural properties, and new random graphs were generated for each trial.

Null Models

  • The observed maximum-correlation and timing statistics were compared with those expected under the null networks.Source: Null Models · p. 20
  • The topological null model preserved edge count and total strength, while the spatially embedded null model also preserved the relationship between edge strength and distance.Source: Null Models · p. 20
  • For the spatial null model, distance effects were removed from edge-weight means and variances before rewiring, then added back to the rewired graph.Source: Null Models · pp. 20–21
  • A new random graph was generated for every trial, with at least 192 trials; null models used stabilized structural matrices and were scaled by γ in optimal-control analyses.Source: Null Models · p. 21

Optimal Network Control

The section defines how the model measures brain-state similarity and optimal-control energy. Energy accounts for graded input weights, and a trial-level measure integrates energy over time.

Optimal Network Control

  • The total input-energy measure incorporates both the input u and the graded weights of B.Regions without electrodes receive approximately 5 × 10−5 on the corresponding diagonal of B, making optimal-energy calculations tractable and motivating inclusion of B in the energy measure.Source: Optimal Network Control · p. 21
  • Simulated and empirical brain states are compared using two-dimensional Pearson correlation between region-by-frequency matrices, including only regions with electrodes.The same state matrix is used across analyses unless otherwise specified.Source: Optimal Network Control · p. 21
  • A trial’s energy is the time-integral of the Frobenius norm of its energy matrix, with energy calculated separately by frequency band.Band-specific energies are summarized in region-by-frequency matrices at each time point.Source: Optimal Network Control · p. 21

Network Statistics

The section defines network statistics for assessing controllability and stimulation-related dynamics. These include a driver-to-non-driver connection statistic and modal controllability measures focused on slow or fast system modes.

Network Statistics

  • The determinant ratio compares the Gram matrix of driver-to-non-driver connections with matrices formed by iteratively removing each non-driver node.It captures the trade-off between connection strength and homogeneity; the stated assumptions—more drivers than non-drivers and initial and final states distributed around zero—hold for these simulations.Source: Network Statistics · pp. 21–22
  • Trace(C−1) is used as an equivalent estimate of the average determinant ratio because determinants of large matrices can be computationally challenging.C is the Gram matrix of all driver-to-non-driver connections, and Ck represents connections from driver nodes to all non-driver nodes except the kth.Source: Network Statistics · p. 22
  • Persistent and transient modal controllability measure how adding energy at a node perturbs slow and fast system modes, respectively.The measures use the 10% largest or smallest eigenvalues, respectively; the section notes that the effect of this cutoff is explored in the Supplement.Source: Network Statistics · p. 22
  • Modal controllability is adapted from a discrete-time estimate to continuous time using the system eigenvalues and the trajectory time step.A small eigenvector entry vij indicates that the jth mode is poorly controllable from node i.Source: Network Statistics · p. 22

Random Forest Model

The section describes random forest models that average predictions across decision trees and use bootstrapping and randomized feature selection to reduce overfitting.

Random Forest Model

  • The random forest averages predictions from 500 decision trees, whose branches split predictors to reduce prediction error.Each branch predicts the mean value for observations within that branch.Source: Random Forest Model · p. 22
  • Bootstrapping and random selection of the split variable are used to avoid overfitting, and out-of-bag mean squared error evaluates samples omitted from each tree’s training data.The section directs readers to the Supplemental Methods for further details.Source: Random Forest Model · p. 22

Data Sharing

The section provides a code repository and states that data will be made available.

Data Sharing

  • Simulation and selected-metric code is available at the named GitHub repository, and data will be made available.The passage does not specify when the data will be available.Source: Data Sharing · p. 23
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