Internal control of brain networks via sparse feedback

Abstract

The human brain is a complex system whose function depends on interactions between neurons and their ensembles across scales of organization. These interactions are restricted by anatomical and energetic constraints, and facilitate information processing and integration in response to cognitive demands. In this work, we considered the brain as a closed loop dynamic system under sparse feedback control. This controller design considered simultaneously control performance and feedback (communication) cost. As proof of principle, we applied this framework to structural and functional brain networks. Under high feedback cost only a small number of highly connected network nodes were controlled, which suggests that a small subset of brain regions may play a central role in the control of neural circuits, through a tradeoff between performance and communication cost.

1 |. INTRODUCTION

The representation of complex systems as networks spans a wide range of fields, from biology and social sciences to physics and engineering. Examples of such systems include the brain,¹ connections between airports,² cellular metabolisms,³ process systems,⁴ and power grids.⁵ In all cases, these systems can be distilled down to sets of entities (nodes) and interactions between them (edges) that can take the form of a graph. Network science provides a systematic framework for the representation and analysis of these systems and their interaction patterns. In the human brain, this framework has been used to study both structural and functional circuits (the connectome).¹ Anatomical connections between brain regions form the structural connectome, while functional connections, represent spatiotemporally coordinated spontaneous or task-related brain activity. In the human brain, these are measured with different neuroimaging modalities, including electroencephalography (EEG), magnetoencephalography (MEG), and functional MRI (fMRI), the latter measuring blood oxygenation rather than direct electromagnetic activity. Each of these modalities measures macroscale brain activity in discrete regions–the graph nodes. Edges represent the strength of coordination between these regions.

Functional interactions between brain regions dependent on and are constrained by the anatomical scaffolding that supports them, that is, the structural connectome.^6–15 Similarly to many biological, physical and engineered networks, the healthy adult connectome has hallmark characteristics that maximize its resilience but also optimize information processing and integration. These characteristics include modularity,¹⁶ core periphery,^17–19 hierarchy,^20,21 the presence of highly-connected regions (hubs),^22–24 and a small-world and rich-club organization.^14,25–28 The exact role of these topological properties in the internal control mechanisms governing the connectome remains unclear.^6,29

Prior Network Science studies have examined the relationship between topological properties, network controllability, and the selection of driver nodes (nodes on which control inputs act), assuming an open-loop control perspective.^30,31 In the context of the brain, a state-space framework has been used to describe brain dynamics, assuming that the human brain is theoretically controllable by appropriate (open-loop) control signals,^32,33 and that a small set of nodes, presumably anatomically and/or functionally important to the network, is critical to this control.³⁴ Yet, this perspective does not account for feedback mechanisms that govern internal network control. Thus, we propose a closed-loop perspective and, more specifically, sparsity promoting optimal feedback control,^35,36 as a framework for describing internal control mechanisms in the brain. This approach accounts for the fact that there is a cost associated with feedback information and posits that a controller that optimizes processing efficiency seeks a trade-off between control performance and controller sparsity.

We have recently applied sparsity promoting optimal feedback control to the macaque structural network.³⁷ In the present study, we apply it to human structural and functional brain networks. The overarching goal of this article is to demonstrate the potential of sparsity promoting control as a tool for characterizing control efficiency in brain networks and, in doing so, introduce the topic of closed-loop internal control of brain networks to the chemical engineering systems and control communities.

2 |. DATA AND METHODS

2.1 |. Sparsity promoting optimal controller synthesis

We consider a system for which the evolution of its state variables (states) follows linear time invariant dynamics

$$\dot{x}=Ax+B_{1}u+B_{2}d,$$ (1)

where $x\in {\mathbf{R}}^n$ are the states of the system, $u\in {\mathbf{R}}^{\mathrm{n}}$ are the control inputs, $d\in {\mathbf{R}}^{\mathrm{n}}$ are exogenous input signals, $A\in {\mathbf{R}}^{n\times n}$ captures the internal dynamic behavior of the system, and $B_1\in {\mathbf{R}}^{n\times n},B_2\in {\mathbf{R}}^{n\times n}$ are matrices capturing the effect of control inputs and state disturbances. We assume that at any time point $t$, the system state is known, and that the input signals $u,d$ affect all states, with $B_1$ and $B_2$ being identity matrices. In order to regulate the system, a static state feedback control law $u=-Kx$ is considered, where $K\in {\mathbf{R}}^{n\times n}$ is the feedback gain matrix, and the dynamics of the closed-loop system becomes

$$\dot{x}=\left(A-B_{1}K\right)x+B_{2}d.$$ (2)

In this setting, the effect of input signals $d$ on the output $z={\left[\begin{array}{ll}x& u\end{array}\right]}^T$ can be captured by the ${\mathscr{H}}_2$ norm of the closed loop system and is equal to $J\left(K\right)=\text{trace}\left(P\right(K\left)\right)$, where $P\in {\mathbf{R}}^{n\times n},P=P^T\succ 0$. This matrix depends on the feedback gain matrix via the following equation

$$(A-K)P+P(A-K{)}^T+\left(I+K^{T}K\right)=0.$$ (3)

We aim to design the feedback gain matrix $K$ so that $J\left(K\right)$ is minimized subject to Equation 3. The solution to this problem generally leads to a dense matrix $K$, with all nonzero elements. Although this matrix will provide optimal control performance, such that the effect of the signal $d$ is minimized due to minimization of the ${\mathscr{H}}_2$ norm, it does not account for the cost associated with feedback. To account for this cost,³⁵ proposed to penalize the nonzero entries in the feedback gain matrix; the resulting problem is

$${\text{minimize}}_K\text{trace}\left(P\right(K\left)\right)+p\text{card}\left(K\right)$$ (4)

where $\text{card}\left(K\right)$ is equal to the number of nonzero entries of the $K$ matrix and $p$ is the feedback cost. The first term of the objective, $J\left(K\right)$, represents the control performance and the second term $\text{card}\left(K\right)$ the control cost. The value of the feedback cost $p$ affects the structure of the feedback gain matrix, since for $p=0$ we obtain a dense feedback gain matrix as discussed above, whereas for a stable system as $p\to \infty$ the feedback gain matrix becomes zero. This problem can be reformulated as

$$\begin{array}{ll}{\text{minimize}}_{K,G}& \text{trace}\left(P\right(K\left)\right)+p\text{card}\left(G\right)\\ \text{subject}\text{to}& K=G.\end{array}$$ (5)

and solved using the alternating direction method of multipliers.¹ We refer the reader to Reference 35 for a detailed description of the algorithm.

2.2 |. Sparse feedback control of brain networks

In the rest of the article, we adopt the aforementioned framework to analyze the internal control of structural and functional brain networks. These are represented by a graph $𝒢(V,E)$ where $V,E$ is the set of $n$ nodes (brain regions) and $m$ edges (structural or functional connections). Associations between nodes in this graph that exceed a threshold for strength of pairwise interactions are represented by an adjacency matrix A.

For simplicity, we model the brain network as a linear time invariant dynamic system, which we acknowledge that it may be an oversimplified approximation of true neural dynamics. The system described by Equation 1, where a state $x_i$ is associated with every brain region $i$. Dynamic interactions between brain regions are assumed to be governed by a sparsity promoting feedback controller (Figure 1).

FIGURE 1.

The brain as a closed loop control system.

Under this assumption the control action at node $i$ is given by

$$u_i\left(t\right)=\sum _{j\in V}-K_{ij}{x}_j\left(t\right).$$ (6)

Thus, computation of the control action at node $i$ requires knowledge of the dynamic state of the other nodes. The interaction pattern between nodes is captured by the entries of the feedback gain matrix $K$. For example, if $K_{ij}$ is zero then the state of node $j$ does not affect the control action at node $i$ and no information is exchanged between node $i$ and $j$.

The problem formulation assumes that all network nodes are subject to control action. While this may not necessarily be the case, we do not know a priori which nodes are activated by feedback control action. This will be determined from the solution of the control problem (i.e., the form of the feedback gain matrix $K$ and specifically its nonzero elements).

2.3 |. Structural connectome data and modeling

A dataset consisting of 30 adjacency matrices for adult structural brain networks is available through the Complex Systems Lab at UPenn and is associated with Reference 33, wherein it was analyzed. In that work, diffusion spectrum magnetic resonance imaging (DSI) data were used to construct adjacency matrices for each subject, using state-of-the-art DSI reconstruction and fiber tracking tools. Each brain was parcellated using the Lausanne 2008 atlas, resulting in a 129 × 129 structural connectivity matrix, $A_s$.³⁸ Each element of $A_{s,ij}$ corresponds to the number of fibers between region $i$ and region $j$ and was normalized by the sum of volumes for $i$ and $j$. The resulting matrix was used as an adjacency matrix in our work. A detailed description of the data acquisition and processing is provided in References 32,33.

We assume that the adjacency matrix $A_s$ captures the open-loop dynamic behavior of the brain, that is, it corresponds to the matrix A in Equation 1. We normalize the adjacency matrix as follows³⁹:

$$A_{s,\mathit{\text{norm}}}=A_s\frac{1}{{\lambda }_{\text{max}}\left(A_s\right)+1}-I,$$ (7)

where ${\lambda }_{\text{max}}\left(A_s\right)$ is the maximum eigenvalue of $A_s$. This guarantees that the maximum eigenvalue of $A_{s,\mathit{\text{norm}}}$ is negative, that is, the open loop system is stable. This scaling is necessary to avoid large values of the states which are biologically implausible. Also, under this scaling the diagonal entries $A_{s,\mathit{\text{norm}}}(i,i)$ are equal to −1 denoting that all the brain regions have similar internal dynamics, and the off diagonal terms are equal to $A_{s,\mathit{\text{norm}}}(i,j)=A_s(i,j)/\left({\lambda }_{\text{max}}\left(A_s\right)+1\right)$.

2.4 |. Functional connectome data and modeling

The functional connectome data analyzed in this work from resting-state fMRI (rs-fMRI) collected in the ABCD study, which measures brain development in almost 12,000 children from pre/early adolescence to young adulthood.⁴⁰ The data preprocessing and connectivity estimation are described in detail in Reference 41. Briefly, following structural segmentation, coregistration of fMRI to structural MRI, slice-time correction, removal of frames that do not meet the displacement threshold, and normalization to MNI152 space, each participant’s voxel-level time series were spatially downsampled to parcel-level time series using a combination of high-resolution cortical (1000 parcels⁴²) and subcortical and cebellar atlases.^43,44 Although a substantial number of participants had data of adequate quality for more than one 5-in resting-state fMRI run, in this study data from each participant’s best run (selected based on connectivity and lowest number of frames censored for motion) were analyzed. Large-scale networks considered in this study are those identified in Reference 45. The data were further downsampled to 90 regions of interest (ROI) based on anatomical considerations. Primary analyzed were conducted at the level of these ROIs. For comparison, additional analyses at the higher resolution of the 1088 parcels were also performed.

Given the computational cost of all analyses, in this preliminary investigation, a sample of 30 functional connectomes was selected at random from a sample of over 5000 brains. Connectivity matrices were estimated using peak cross-correlation between pairs of fMRI time series. Each connectivity matrix was thresholded using a cohort-wide threshold (estimated from >5000 brains), corresponding to the moderate connectivity outlier. Any entries below this correlation threshold ($r=0.59$) were set equal to zero, resulting in a 90 × 90 adjacency matrix $A_f$ for each brain. In every adjacency matrix the diagonal terms are set equal to zero and the matrix is normalized using Equation 7.

3 |. RESULTS AND DISCUSSION

In this section, we solve the sparsity promoting optimal controller synthesis problem for the analyzed structural and functional connectomes, for different values of the feedback cost parameter $p$. As discussed earlier, the feedback gain matrix $K$ captures the pattern of interactions between brain regions. Communication between brain areas regions in response to cognitive demands and sensory processing can be energy-expensive.^46,47 For this reason, we focus our analysis on the case of high feedback cost.

3.1 |. Application to structural brain networks

We applied sparsity promoting control to the human structural networks described in Section 2.3 for different values of feedback cost $p$. We solved the problem for $p\in \left[{6.510}^{-6},31.6\right]$. The value of ${6.510}^{-6}$ was selected empirically. For $p>31.6$ all the entries in the feedback gain matrix were equal to zero. The number of nonzero entries in the adjacency matrix varied for each brain in the dataset. Median number of nonzero entries across brains was 9758 (IQR = 768). Structural brain network 1 is shown in Figure 2 as an example. Highly connected nodes that are controlled under high feedback cost are highlighted. Networks were visualized using graphtool.⁴⁸

FIGURE 2.

Network representation of a structural brain network in the analyzed dataset. The yellow nodes correspond to highly connected nodes that are controlled under high feedback cost.

The adjacency matrix for structural brain network 1 (the first in the dataset) and multiple feedback gain matrices $K$ are shown in Figure 3. Each nonzero element $(i,j)$ of $K$, represents a node $i$ that is controlled by the action of node $j$. For low feedback cost, $p\in \left[{6.510}^{-6},{2.310}^{-4}\right]$, the structure of the feedback gain matrix is similar to the structure of the adjacency matrix. This implies that the control of the structural brain network is constrained anatomically. In other words, a physical connection must exist in order for the control of node $i$ to be directly influenced by the state of node $j$.

FIGURE 3.

Structure of adjacency matrix and feedback gain matrix $K$ for different feedback cost $p$ for the human structural brain network in Figure 2; nz is the number of nonzero terms.

Increasing the feedback cost $p$ leads to a sparser feedback gain matrix (Figure 3). Moreover, in the region of high feedback cost ($p\in [1,31.6]$), the $K$ matrix becomes diagonal and subsequently subdiagonal; that is, only a subset of diagonal terms are nonzero. If the only nonzero entry in a row is $K_{ii}$, then the control action on that node is entirely self-dependent. In the context of high feedback cost, any node $i$ whose row contains a nonzero $K_{ii}$ term is referred to as a “controlled node.” Figure 3 shows that there is a regime of $p$ where all the nodes are self-controlled (for $p=1$ the feedback gain matrix becomes perfectly diagonal) and there is a second regime of higher p values where only a subset of the nodes are self-controlled. The value $p=18.3$ is in the latter regime, and results in a $K$ matrix with a sub-diagonal structure (in this case with only six controlled nodes). These nodes are shown in yellow in Figure 2.

Figure 4 shows boxplots reflecting the distribution and statistics of controlled nodes across all 30 brains, as a function of increasing feedback cost $p$ on $K$. For each $p$ value, the figure shows the median number of controlled nodes, 25th and 75th quartiles, and outliers. The median consistently decreases with increasing $p$, presumably approaching zero controlled nodes as the feedback cost goes to infinity. Across brains, few nodes remain controlled under high feedback cost; by $p=18.3$, on average only ~5% of the nodes (7 nodes) are controlled. To investigate the consistency of controlled nodes across brains, the controlled nodes in each brain were plotted for $p=18.3$ (a value at which the feedback gain matrix is sub-diagonal). Figure 5 shows that some nodes were consistently controlled across brains, suggesting a potentially critical role of some brain regions that remain controlled at high feedback cost.

FIGURE 4.

Boxplot of the number of controlled nodes at different feedback cost $p$ for the 30 structural brain networks.

FIGURE 5.

Controlled nodes for the 30 structural brain networks for $p=18.3$.

We further analyzed the topological characteristics of nodes controlled under high feedback cost ($p=18.3$) by estimating four different measures of the controlled nodes’ topological role in the network: eigenvector, closeness, betweenness, and degree centralities.⁴⁹ These measures are used to quantify the importance of a node in a network in terms of facilitating the propagation of information through it. An example of how median centrality of controlled nodes varies with increasing p is shown for the structural brain in Figure 2 is shown in Figure 6. The plots suggest that nodes controlled under high feedback cost also have high centrality, which is higher than the average centrality of the network. These observations hold true across centrality measures.

FIGURE 6.

Median centrality measures of nodes that are controlled for high feedback cost for structural brain 1. The dashed line indicates the average centrality of the nodes in the structural brain network.

Moreover, Table 1 provides summary statistics across analyzed brains, providing evidence that the centrality of controlled nodes is higher than the median centrality of all nodes in the network for $p=18.3$. These results imply that the controlled nodes are highly connected compared to other nodes in the network, and thus are potentially network hubs.⁵⁰ Highly connected regions may be associated with lower control energy when open loop transitions between dynamic states are considered.³³ Our results suggest that under high feedback cost, these are the nodes being controlled, and thus, they are critical in achieving a balance between control performance and feedback cost.

TABLE 1.

Median and IQR values of centrality measures for the 30 brains and the nodes with nonzero diagonal entries (controlled nodes) in the feedback gain matrix for $p=18.3$. Controlled nodes are referred to as “active” here.

Median	Eigenvector	Closeness	Betweenness	Degree
Active nodes	0.106	0.796	0.005	0.744
Inactive	0.083	0.702	0.002	0.576
All nodes	0.085	0.707	0.002	0.585
IQR
Active nodes	0.017	0.088	0.0033	0.14
Inactive	0.023	0.085	0.0023	0.17
All nodes	0.024	0.088	0.0024	0.18

3.2 |. Application to functional brain networks

In this section, we solve the sparsity promoting problem for functional connectomes, for values of feedback cost $p\in \left[{10}^{-5},11\right]$. This range of $p$ was empirically chosen so that it spans the range from low to very high feedback cost. Median number of nonzero entries in the adjacency matrix over the 30 brains was 144 (IQR = 64). The adjacency matrix and feedback gain matrices (corresponding to different values of $p$) for a representative functional network are shown in Figure 7. Resting-state functional networks estimated from fMRI run a few minutes long represent snapshots in time of spontaneously coordinated activity in the brain. Although the associated adjacency matrix is anatomically constrained, some regions may not be functionally (statistically) connected at a particular 5-min snapshot. The feedback gain matrix, which is also anatomically constrained and is associated with an endogenous mechanism of control and not functional connectivity, can thus have more nonzero elements than the functional adjacency matrix. This is indeed the case for the feedback gain matrix under low feedback control in Figure 7. Furthermore, we observe that for $p=0.20$ the feedback gain matrix becomes diagonal (this value represents the first $p$ at which $K$ becomes diagonal) and for $p=9.25$, an empirically chosen value in the regime of very high feedback cost, there are only 13 controlled nodes (ROIs) (14% of total nodes) and the feedback gain matrix is sub-diagonal. For higher values of $p$, there are even fewer controlled nodes.

FIGURE 7.

Topology of the adjacency matrix and the feedback gain matrix $K$ for different feedback cost $p$ for a representative functional network.

We solved the sparsity-promoting problem for all 30 brains in the dataset for $p\in \left[\mathrm{8.5,11}\right]$. The median number of nonzero entries in the feedback gain matrix decreased as the feedback cost $p$ increased (see Figure 8). For example, for $p=9.25$, the average percentage of controlled nodes is 9.6% (the median percentage is 8.8%) across all brains. The active nodes for this $p$ are shown in Figure 10.

FIGURE 8.

Boxplot of the number of controlled ROIs (90 nodes) at different feedback cost $p$ for the 30 functional brain networks.

FIGURE 10.

Controlled ROIs (90 nodes) for the 30 functional brain networks for $p=9.25$. The nodes highlighted in red color correspond to the nodes highlighted in red in Figure 11.

Similar results were obtained at the higher parcel-level resolution. The sparsity-promoting problem was solved for the same values of the parameter $p$. The number of nonzero entries as a function of the parameter $p$ is presented in Figure 9 and the active nodes are presented in Figure 11. From Figures 8 and 9 we observe that at both spatial scales, the number of nonzero entries decreases as the feedback cost increases. Furthermore, from Figures 10 and 11 we observe that for $p=9.25$ the same brain regions are controlled. Specifically, the highlighted red ROIs in Figure 10 correspond to the higher-resolution red parcels in Figure 11, and at both scales constitute elements of visual networks. These results suggest that, at least in a subset of brains control nodes can be localized in parts of the same network at more than one scale.

FIGURE 9.

Boxplot of the number of controlled parcels (1088 nodes) at different feedback cost $p$ for the 30 functional brain networks.

FIGURE 11.

Controlled parcels (1088 nodes) for the 30 functional brain networks for $p=9.25$. The nodes highlighted in red color correspond to the nodes highlighted in red in Figure 10.

For the representative functional connectome in Figure 7, we computed average measures of centrality for controlled nodes and compared them to those of the entire network. These results are shown in Figure 12. Similarly to structural networks, overall centrality of controlled nodes was higher than those of the entire network. Summary statistics for the 30 brains and $p=9.25$ are provided in Table 2. Centrality of controlled nodes under this high feedback cost was consistently higher than that of noncontrolled (inactive) nodes, or the entire network.

FIGURE 12.

Median measures of centrality of active (controlled) ROIs in one human functional brain network as function of the feedback cost parameter. The dashed line denotes the median centrality of the nodes in the network.

TABLE 2.

Median and IQR values of centrality measures for the 30 functional brains and controlled ROIs in the feedback gain matrix for $p=9.25$.

Median	Eigenvector	Closeness	Betweenness (<10−3)	Degree
Active nodes	0.301	0.068	2.5 <10−4	0.078
Inactive	<10−4	<10−4	<10−4	0.022
All nodes	<10−4	<10−4	<10−4	0.022
IQR
Active nodes	0.240	0.044	0.002	0.033
Inactive	<10−4	<10−4	<10−4	<10−4
All nodes	<10−4	<10−4	<10−4	<10−4

3.3 |. Analysis of controlled regions of interest in resting-state networks

We also investigated membership of ROIs controlled at high feedback cost in large-scale resting-state networks. For the purpose of this preliminary investigation, ROIs were grouped into 8 cortical networks, as well as the hippocampus, basal ganglia, amygdala, thalamus, and the cerebellum. In individual networks, ROIs in the left and right hemispheres were examined together $(p\in [8.5,10.75\left]\right)$.⁴⁵ The number of controlled nodes in each network was normalized by the total number of nodes in that network, given differential network size. The results were averaged across the 30 functional brains and are shown in Figure 13. In some networks, the number of controlled nodes decreased sharply within a small range of penalty parameter $p$. This could be attributed to the very narrow range of the eigenvalues of the corresponding normalized $A$ matrices. In addition, the visual network stands apart from the rest, in that the number of controlled nodes decreases at a slower rate as a function of increasing $p$.

FIGURE 13.

Median (over analyzed connectomes) fraction of controlled ROIs in each of 8 resting-state cortical networks, 4 subcortical networks and the cerebellum. The number of controlled nodes in each region divided by the number of total possible controlled nodes in each region is shown as a function of $p$.

The consistency of this finding was further examined in individual brains. Specifically, we examined brains for which the percentage of active ROIs within the visual network was above 50%. Out of 30 connectomes, 14 (46.7%) fell in this category. Figure 14, shows percent of controlled nodes as a function of p averaged for this subset of 14 brains. In addition, Figure 15 shows the anatomical locations of controlled ROIs in 5 brains of this subset, for 3 values of $p$ (9.25, 9.5, and 10.0). Primarily posterior areas of the visual network (ROIs overlapping primary and secondary visual areas and inferior temporal areas) were controlled at high feedback cost. In early adolescence, during which maturating neural circuits undergo significant reorganization, the visual network is one of the most anatomically and functionally developed networks. It is, therefore, possible that regions consistently controlled at high feedback cost across brains may correspond to those that are better developed than others at these ages. Thus, given that functional connections are anatomically constrained, controllability estimated from resting-state (spontaneous) functional activity in the brain may reflect an anatomical property of the connectome. Evidently, these observations are preliminary and speculative and need to be confirmed in a much larger cohort.

FIGURE 14.

Controlled ROI prevalence for a subset of 14 brains with a consistent higher proportion of controlled nodes in the visual network.

FIGURE 15.

Controlled ROIs for $p$ values of 9.25, 9.5, and 10.0 visualized for 5 subjects which exhibit consistent control in the visual network. Solid circles correspond to controlled ROIs on the left hemisphere and transparent circles to controlled ROIs on the right hemisphere.

4 |. CONCLUDING REMARKS

By applying a sparsity-promoting controller synthesis approach to both structural and functional brain networks, we have highlighted a potential role that highly connected nodes may play in the control of the human brain.

Specifically, at a particular high feedback cost a perfectly diagonal feedback gain matrix is obtained, for which every node is controlled separately, without input from other nodes, and beyond this value the feedback gain matrix becomes sub-diagonal. Analysis of this sub-diagonal structure suggests that controlled nodes that remain active at exceedingly high feedback cost have significantly higher connectedness compared to an average node within the network, that is, they may be highly connected hubs. These regions are critical to the rich-club organization of the human connectome and cognitive processing, as they may integrate information from distributed computations in domain-specific functional regions. The optimization of feedback cost thresholds at which nodes become self-controlled or are no longer controlled was beyond the scope of this study, but an important future direction of this work. Nevertheless, our findings provide an additional new perspective on the mechanistic role of brain hubs in optimizing information processing by balancing performance and feedback cost.

Abbreviations:

ABCD

adolescent brain cognitive development

BOLD

blood oxygen level dependent

ROI

region of interest

DATA AVAILABILITY STATEMENT

The structural connectomes were obtained from Bassett’s Complex Systems Lab and can be found at https://complexsystemsupenn.com/codedata. Functional connectomes were derived from the raw ABCD dataset, which may be accessed upon request at https://nda.nih.gov/abcd/. The preprocessed connectivity matrices were provided courtesy of the Computational Neuroscience Laboratory at Boston Children’s Hospital. The base code for the sparsity promoting controller can be obtained from http://www.ece.umn.edu/users/mihailo/software/lqrsp/ and the code for solving the sparsity promoting optimal controller synthesis problem for the structural and the functional brains can be obtained from https://github.com/DaoutidisLab/SparseControlBrains.git. Contact the authors for more information.