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

Abstract

Optimizing direct electrical stimulation for the treatment of neurological disease remains difficult due to an incomplete understanding of its physical propagation through brain tissue. Here, we use network control theory to predict how stimulation spreads through white matter to influence spatially distributed dynamics. We test the theory's predictions using a unique dataset comprising diffusion weighted imaging and electrocorticography in epilepsy patients undergoing grid stimulation. We find statistically significant shared variance between the predicted activity state transitions and the observed activity state transitions. We then use an optimal control framework to posit testable hypotheses regarding which brain states and structural properties will efficiently improve memory encoding when stimulated. Our work quantifies the role that white matter architecture plays in guiding the dynamics of direct electrical stimulation and offers empirical support for the utility of network control theory in explaining the brain's response to stimulation.

Keywords: brain network; brain stimulation; network control theory.

Figure 1.. Schematic of Methods

(A) Depiction of network construction and definition of brain state. (Left) We segment subjects’ diffusion weighted imaging data into N = 234 regions of interest using a Lausanne atlas (Cammoun et al., 2012). We treat each region as a node in a whole-brain network, irrespective of whether the region contains an electrode. Edges between nodes represent mean quantitative anisotropy (Yeh et al., 2013) along the streamlines connecting them. (Right, top) We summarize the network in an N×N adjacency matrix. (Right, bottom) A brain state is defined as the N×1 vector comprising activity across the N regions. Any element of the vector corresponding to a region with an electrode is defined as the band-limited power of ECoG activity measured by that electrode. Each brain state is also associated with an estimated probability of being in a good memory state, using a previously validated machine learning classifier approach (Ezzyat et al., 2017). (B) Schematic of a single stimulation trial. First, ECoG data are collected for 500 ms. Then, stimulation is applied to a given electrode for 250–1,000 ms. Finally, ECoG data are again collected after the stimulation. (C) Schematic of the open loop and optimal control paradigms. In the open loop design, energy u(t) is applied in silico at the stimulation site to the initial, prestimulation brain state x(0). The system will travel to some other state x(T), as stipulated by our model of neural dynamics, and we will measure the similarity between that predicted state and the empirically observed post-stimulation state. In the optimal control design, the initial brain state x(0) has some position in space that evolves over time toward a predefined target state x(T). At every time point, we calculate the optimal energy (u(t)) required at the stimulating electrode to propel the system to the target state.

Figure 2.. Post-stimulation Brain State Depends on White Matter Network Architecture

(A) Boxplots depicting the average maximum correlation between the empirically observed post-stimulation state and the predicted post-stimulation state at everytime point in the simulated trajectory x(t) for N = 11 subjects. Boxplots indicatethe median (solid horizontal black line) and quartiles ofthe data. Each data point represents a single subject, averaged over all of the trials (with different stimulation parameters). (B) Boxplots depicting the average time to reach the peak magnitude (positive or negative) correlation between the empirically observed post-stimulation state and the theoretically predicted post-stimulation state at every time point in the simulated trajectory x(t). Time is measured in a.u. Color indicates theoretical predictions from Equation 1, where A is (1) the empirical network (purple) estimated from the diffusion imaging data, (2) the topological null network (dark charcoal), and (3) the spatial null network (light charcoal). See also Figures S1 and S2.

Figure 3.. Longer-Distance Trajectories Require More Stimulation Energy

(A) The normalized energy required to transition between the initial state and the post-stimulation state, as a function of the Frobenius norm between the initial state and the post-stimulation state. The black solid line represents the best linear fit (with gray representing standard error) and is provided simply as a guide to the eye (β = 8.3 × 10⁻³, t = 18.11, p < 2 × 10⁻¹⁶). Normalization is also performed to enhance visual clarity. (B) The energy required to transition to a good memory state, as a function of the initial probability of being in a good memory state (β = −0.18, t = −14.4, p < 2 × 10⁻¹⁶). (C) The energy required to transition to a good memory state as a function of the empirical change in memory state resulting from stimulation (β = 9.5 × 10⁻², t = 8.43, p < 2 × 10⁻¹⁶). (D) In N = 3 experimental sessions that included both sham and stimulation trials, we calculated the energy required to reach the post-stimulation state or the post-sham state, rather than a target good memory state. Here, we show the difference in energy required for sham state transitions in comparison to stimulation state transitions (paired t test, N = 3, p = 0.01). Error bars indicate SEMs across trials. Across all four panels, different shades of blue indicate different experimental sessions and subjects (N = 16). See also Figure S4.

Figure 4.. Topological and Spatial Constraints on the Energy Required for Stimulation-Based Control

(A) Average input energy required for each transition from the pre-stimulation state to a good memory state, as theoretically predicted from Equation 1, where A is (1) the empirical network (purple) estimated from the diffusion imaging data, (2) the topological null network (dark charcoal), and (3) the spatial null network (light charcoal) for N = 11 subjects. (B) The relation between the determinant ratio and the energy required for the transition from the pre-stimulation state to a good memory state. The color scheme is identical to that used in (A). The p value is from a paired t test: N = 11, t = 3.64, p = 4.6 × 10⁻³. See also Figure S5.

Figure 5.. Role of Local Topology Around the Region Being Stimulated

(A) Transitions from the observed initial state to a good memory state required significantly greater energy when affected by the middle temporal sensors than when affected by the inferior temporal sensors. (B) Relation between persistent (χ² = 3.89, p = 0.049) (top) or transient (χ² = 1.69, p = 0.19) (bottom) controllability of the stimulated region and the energy predicted from optimal transitions from the initial state to a good memory state. We only allow energy to be injected into a single electrode-containing region, and we consider a broadband state matrix. Every color is a subject (N = 11) and every dot is a different simulated stimulation site. (C) As in (B), but when considering the α band state vector only (persistent controllability: χ² = 13.8, p = 2.00 × 10⁻⁴; transient controllability: χ² = 11.4, p = 7.5 × 10⁻⁴). See also Figure S6.

Figure 6.. Network Topology and Brain State Predict Energy Requirements

(A) Schematic of the three topology and state features included in the random forest model that we built to predict energy requirements. Network-level effects (tan) are captured by the determinant ratio, regional effects (brown) are captured by persistent controllability, and state-dependent effects (red) are captured by the initial memory state. (B) Comparison of the out-of-bag mean squared error for a model in which each subject’s (N = 11) determinant ratio, persistent controllability, and initial memory state are used to predict their required energy. We compared the performance of this model to the performance of a distribution of N = 1,000 models in which the association between energy values and predictors was permuted uniformly at random.