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. 2019 Feb 20;13:154–162. doi: 10.1016/j.isci.2019.02.017

The Hidden Control Architecture of Complex Brain Networks

Byeongwook Lee 1, Uiryong Kang 1, Hongjun Chang 1, Kwang-Hyun Cho 1,2,∗
PMCID: PMC6402303  PMID: 30844695

Summary

The brain controls various cognitive functions in a robust and efficient way. What is the control architecture of brain networks that enables such robust and optimal control? Is this brain control architecture distinct from that of other complex networks? Here, we developed a framework to delineate a control architecture of a complex network that is compatible with the behavior of the network and applied the framework to structural brain networks and other complex networks. As a result, we revealed that the brain networks have a distributed and overlapping control architecture governed by a small number of control nodes, which may be responsible for the robust and efficient brain functions. Moreover, our artificial network evolution analysis showed that the distributed and overlapping control architecture of the brain network emerges when it evolves toward increasing both robustness and efficiency.

Subject Areas: Neuroscience, Systems Neuroscience, Systems Biology, Control Theory

Graphical Abstract

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Highlights

  • •

    We develop a framework to delineate the control architecture of brain networks

  • •

    The control architecture of brain networks is compared with other complex networks

  • •

    Brain networks have a distributed and overlapping control architecture

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    Robust and efficient brain functions might be rooted in its control architecture

Neuroscience; Systems Neuroscience; Systems Biology; Control Theory

Introduction

Cognitive functions are performed by the coordinated control of multiple brain regions (Cocchi et al., 2013, Ghazanfar and Schroeder, 2006, Koh et al., 2011). Such control is evolutionarily optimized for efficiency (Tang et al., 2017) and robustness (Aerts et al., 2016). Although such optimality is rooted in the complex interconnectivity of brain regions (Tang et al., 2017), our understanding of fundamental control architecture of brain networks that determines the specific coordinated control of interconnected brain regions is still lacking. Here, we investigated the control architecture of structural brain networks (hereafter, brain networks) by analyzing high-resolution brain networks reconstructed from multiple species, including fruit fly, nematode worm, mouse, cat, macaque, and human (Jarrell et al., 2012, Rubinov et al., 2015, Shih et al., 2015), and comparing them with 26 real-world complex networks. Our analysis included the reconstruction of brain networks of 100 healthy human adults (Figure 1A) to identify the control architecture of human brain networks. Using structural and diffusion magnetic resonance imaging (MRI) data obtained from Human Connectome Project, we performed whole-brain parcellation and diffusion tractography to identify the anatomical connections (i.e., edges) between 164 brain regions (i.e., nodes) extracted from the Destrieux atlas (Fischl et al., 2004) (Figure 1A, see Transparent Methods for details).

Figure 1.

Figure 1

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Comparison of Brain Networks and Other Real-World Complex Networks

(A) Schematic depicting the basic procedure of reconstructing a structural brain network (hereafter, brain network. See Transparent Methods for details).

(B) A minimum dominating set (MDSet) is defined as a minimal subset of nodes (yellow circle) with which all other remaining nodes (black circle) can be reached by one-step interactions.

(C) Schematic describing the overall framework of analysis. The control architectures of brain networks (Caenorhabditis elegans, Drosophila, mouse, cat, macaque, and human) and other real-world complex networks (26 networks of 7 categories) were characterized on the basis of their MDSet. By examining the composition of MD-nodes in each network, the control architecture of each network was categorized into a particular type. The attributes of each control architecture were then analyzed from various perspectives.

(D) Scatterplot comparing the fraction of MD-nodes, MD, of various networks.

(E) Scatterplot showing the relationship between MD and the proportion of high-degree nodes in the MDSet for each network (left, Pearson correlation r = 0.572, p = 4.05 × 10−4).

See also Figure S1.

We developed a framework that determines the control architecture of complex networks on the basis of the minimum dominating set (MDSet) (Haynes et al., 1998), which refers to a minimal subset of nodes (MD-nodes) that control the remaining nodes through a one-step direct interaction (Figure 1B, see Transparent Methods). The important role of MD-nodes in network control is recognized both in theory (Nacher and Akutsu, 2012) and in various real-world networks (Nacher and Akutsu, 2013, Wan et al., 2002, Wuchty, 2014). In particular, the concept of MDSet has recently been adopted to analyze various biological networks, and the results showed that MD-nodes not only occupy strategic locations to control the networks but are also associated with various biological functions (Nacher and Akutsu, 2013, Nacher and Akutsu, 2016, Sun, 2015, Wakai et al., 2017, Wuchty, 2014, Zhang et al., 2016). Thus, we postulated that the composition of MD-nodes in a network represents its hidden control architecture. By examining the MD-nodes in various complex networks, we delineated the distinct control architecture of each network, categorized the control architectures on the basis of the composition of MD-nodes in each network, and determined the attributes of each type of control architecture (Figure 1C). Through this process, we revealed the hidden control architecture of brain networks that is responsible for the efficient and robust control of brain functions.

Results

Composition of Minimum Dominating Set in Brain Networks

We determined the MDSet of each network and compared the fraction of MD-nodes (MD) of various networks. MD is defined as the proportion of the size of MDSet to the size of the network and represents the minimum effort required to control the entire network (Nacher and Akutsu, 2012). We found that the average MD of brain networks is lower than that of other networks (Figure 1D, see Table S1), implying that brain networks are optimized to minimize the control effort. High-degree nodes (nodes connected to many other nodes), especially highest-degree nodes, are more likely to be MD-nodes. Thus, we hypothesized that a network with low MD would have MD-nodes that were enriched in high-degree nodes, which would result in a negative correlation between MD and the proportion of high-degree nodes in the MDSet. To test this hypothesis, we investigated the relationship between the proportion of the top 5% high-degree nodes that are included in the MDSet and MD. Unexpectedly, MD and the proportion of high-degree nodes showed a positive correlation (Figure 1E, left, see Table S1). This positive correlation held when performed with the top 2%, 3%, 5%, 10%, 15%, and 20% of high-degree nodes (Figure S1), showing that this is a robust relationship. The proportion of high-degree nodes in the MDSets was more than 50% in most of the networks except brain networks, in which the proportion was much smaller (only 29% on average for eight brain networks, Figure 1E, right). This implied that the MD-nodes in brain networks are not solely determined by their degrees. Instead, they might be strategically placed and have a characteristic composition in the networks.

Identification of the Hidden Control Architecture of Brain Networks

To investigate the composition of MD-nodes in brain networks, we introduced two measures: distribution of control (DC) and overlap in control area (OCA), where we define DC as the ratio of control area dominated by the MD-nodes in the top 5% high-degree nodes to that dominated by the rest of the MD-nodes, and we define OCA as the degree of overlap across all control areas among different MD-nodes (Figure 2A, left, see Transparent Methods). We tested various percentages (N = 5%–20%) of the top high-degree nodes and found that we could capture the particular control architecture of the brain networks that is distinct from other networks at N = 5% (Figure S2). Thus, we set N = 5% for the definition of DC measure. We defined networks with DC > 1 as centralized and DC ≤ 1 as distributed. We confirmed the robustness of the classification by determining 100 MDSets for each network and computing the DC values of the corresponding MDSets (Figure S3A). We defined OCA > 1.5 as overlapping and OCA ≤ 1.5 as non-overlapping and confirmed the robustness of this classification (Figure S3A). Using these two measures, we categorized the control architectures of various networks into four distinct types (Figure 2A, right). We found that all brain networks are categorized into one type, the “distributed and overlapping” control architecture (Figure 2B, see Table S1).

Figure 2.

Figure 2

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The Characteristics of Different Control Architectures

(A) Categorization of control architectures. On the basis of two measures, distribution of control (DC) and overlap in control area (OCA), which determine the interdependence of control areas (measured by counting the number of nodes dominated by an MD-node, see Transparent Methods), the control architecture of a network can be categorized into four types. The background color indicates the type of different control architecture: blue for centralized and non-overlapping control architecture, pink for centralized and overlapping control architecture, yellow for distributed and non-overlapping control architecture, and purple for distributed and overlapping control architecture. The background color of the scatterplots in Figures 2B–2D indicates each type of control architecture described in Figure 2A.

(B) Scatterplot of DC versus OCA for each network. Using these two measures, each network can be categorized into a different type of control architecture. See also Figures S2 and S3.

(C) Domination stability of each control architecture under targeted attack. Disruption of control area along with the accumulated network damage was investigated for various networks (represented by each thin line). See also Figure S4.

(D) Scatterplot showing the mean modal controllability versus mean average controllability of various networks (Pearson correlation r = −0.83, p = 1.57 × 10−9) and the underlying control architectures associated with controllability preference.

To further investigate whether this particular architecture is related to the robustness and efficiency of brain functioning, we examined the domination stability (Molnár et al., 2015) (see Transparent Methods), which reflects the robustness of the network to loss of nodes, of brain networks. Brain functions are robust against random attack (Hillary and Grafman, 2017), in biological terms, small, randomly occurring instances of neural injury or death or disruption in synaptic activity. Consequently, not all brain lesions have observable functional consequences. In contrast, damage of high-degree nodes, such as that caused by increased metabolic stress accompanied by high network burden, is associated with neurological disorders (Fornito et al., 2015), indicating that high-degree nodes represent vulnerabilities in the network with functional consequences to their disruption. Therefore, to measure domination stability we monitored the disruption of control area along with the network damage resulting from random attacks (Figure S4A) and targeted attacks that preferentially removed the highest-degree nodes (Figure S4B) in sequence to represent the worst-case deterioration of network. In contrast to random attacks for which each type of network was robust (Figure S4A), each category of control architecture showed distinct domination stability to targeted attack of the highest-degree nodes (Figures 2C, S4B, and S4C). The centralized control architecture was the most fragile, showing a reduction in domination stability after removal of the fewest high-degree nodes (Figure 2C, left), whereas the distributed and overlapping control architecture showed the most robust stability and exhibited stair-like decrement against the sequential targeted attack (Figure 2C, right). We postulate that this property of distributed and overlapping control architecture might represent the robustness of brain functions despite deterioration of brain networks.

We also examined the controllability of different control architectures (see Transparent Methods). Here, controllability means the capability of driving a network state defined by a set of node activities into another one (Klamka, 1963). In particular, we considered mean average controllability and mean modal controllability, which describes the average ability to drive a network state into the easy-to-reach nearby state and that to drive into difficult-to-reach distant state, respectively (Gu et al., 2015) (see Transparent Methods). Under the previously known trade-off relationship between these two controllability measures (Tang et al., 2017), most of the networks showed a clear preference for either average or modal controllability (Figure 2D, see Table S1). Furthermore, such preference was associated with the underlying control architecture. For instance, networks with the centralized control architecture showed high mean modal controllability and low mean average controllability, whereas the networks with the distributed and non-overlapping control architecture showed the opposite. Notably, only the networks with the distributed and overlapping control architecture showed balanced controllability with respect to these two measures, indicating that the brain networks are optimized for both types of controllability.

The Structural Principle Underlying the Composition of MDSet in Brain Networks

Modern network science has revealed fundamental aspects of brain network organization, such as hierarchical modularity, hub nodes, and rich-club (RC)-nodes (Park and Friston, 2013, Sporns and Betzel, 2016, Van Den Heuvel and Sporns, 2011). RC-nodes function as connector hubs that determine inter-modular connectivity and are responsible for global integration (Van Den Heuvel and Sporns, 2011) (Figure 3A). We investigated whether MD-nodes have similar or distinct roles from RC-nodes and what kind of structural principles underlie the composition of MDSet. Therefore, we explored the MDSet of human brain networks with respect to modularity and RC organization of the networks (Figure 3A, see Transparent Methods). We found that the degree distribution of MD-nodes differs from that of RC-nodes (Figure 3B, left) and that only 38% of MD-nodes correspond to RC-nodes (Figure 3B, right).

Figure 3.

Figure 3

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The Structural Principle Underlying the Composition of MDSet in Human Brain Networks

(A) Illustration of identifying MDSets from brain networks of 100 healthy adult subjects, composed of 164 cortical and subcortical regions extracted from the Destrieux atlas. The structural principle underlying the composition of MDSet is explored with respect to modularity and rich-club (RC) organization of the network.

(B) Degree distributions of all nodes, MD-nodes, and RC-nodes (left). The thin lines of black, yellow, and purple indicate the degree distributions of all nodes, MD-nodes, and RC-nodes of each network, respectively. The thick lines of black, yellow, and purple indicate the average of all thin lines of the same color. Comparison of MD-nodes and RC-nodes. Data are represented as box-and-whisker plot (right).

(C) Contribution of MD-nodes to the clustering capacity of each network is examined by the measure of transitivity. Data are represented as box-and-whisker plots.

(D) Participation coefficient (PC) values of MD-nodes are computed. MD-nodes are classified into either provincial or connector MD-nodes by their PC value (provincial if PC ≤ 0.5 and connector otherwise).

(E) Ratio of provincial to connector MD-nodes is computed. Data are represented as box-and-whisker plots.

(F) Illustration of the distributed and overlapping control architecture of brain networks that is formed by provincial and connector MD-nodes.

(G) Contribution of provincial- and connector-MD-nodes to determining the optimal (i.e., balanced) controllability of brain networks (left). Effect of elimination of provincial- and connector-MD-nodes on mean average controllability and mean modal controllability (right).

(H) Identification of the highly selected MD-nodes across 100 subjects. MD-nodes that are selected from more than 60% of the 100 subjects were chosen as the highly selected MD-nodes. Twelve nodes were chosen as the highly selected MD-nodes. FFG, fusiform gyrus; IFG, inferior frontal gyrus; PUT; putamen; S.CC, sulcus of corpus callosum; SFG, superior frontal gyrus; SPG, superior parietal gyrus; STG, superior temporal gyrus. Association of the highly selected MD nodes with the specific cognitive system (right).

Another type of hub node is the provincial hub node, which tends to have intra-modular connectivity (Sporns et al., 2007). Deletion of provincial hubs decreases network transitivity, measure reflecting the prevalence of clustered connectivity in the network (Sporns et al., 2007). We examined whether MD-nodes are provincial hubs by exploring the relationship between MD-nodes and network transitivity. We found that deletion of MD-nodes decreases the transitivity of networks (Figure 3C), indicating that many MD-nodes are provincial hub nodes. We further investigated the overall structural principle underlying the composition of MDSet by classifying MD-nodes into provincial nodes or connector nodes on the basis of their participation coefficients (Figure 3D, see Transparent Methods), a measure of a node's contribution to intra- or inter-modular connectivity. We found that the ratio of provincial to connector nodes is about 6:4 on average (Figure 3E). This means that about 60% MD-nodes are provincial nodes, which function as local controllers for segregated modules (that is execute distributed control), and the remaining MD-nodes function as global controllers for multiple modules (that is execute overlapping control). Thus, our results indicated that the MD-nodes constitute a distributed and overlapping control architecture of brain networks (Figure 3F). We explored whether this particular control architecture contributes to the optimal (i.e., balanced) controllability of brain networks with both effective average and modal controllability (Figure 2D). We selectively eliminated either provincial MD-nodes or connector MD-nodes from the network and assessed the effect on controllability (Figure 3G). We found that deletion of provincial MD-nodes reduces mean average controllability, whereas deletion of connector MD-nodes reduces mean modal controllability (Figure 3G, right). These results indicated that the combination of provincial and connector MD-nodes is a key structural characteristic resulting in the optimal controllability of brain networks.

The human ability to perform complex cognitive functions are rooted in the inter-regional brain network. In particular, previous neuroimaging studies revealed that subnetworks composed of different sets of brain regions are involved in carrying out distinct cognitive functions (Dosenbach et al., 2007, Power et al., 2011). These subnetworks are referred to as cognitive control networks or systems and commonly classified into visual, auditory, sensorimotor, attention, subcortical, frontoparietal, cingulo-opercular, and default-mode system (Power et al., 2011). From a cognitive perspective, the control of each cognitive system is important for implementing distinct cognitive functions and for a smooth transition between them. To explore the relationship between MD-nodes and cognitive functions, we chose the 12 MD-nodes, representing specific regions of the brain that are highly selected across subjects (Figure 3H, left), and examined whether the selected 12 bihemispheric regions are differently located in or between the cognitive systems. We found that each MD-node is differently associated with the cognitive system, suggesting that MD-nodes play a central role in controlling distinct cognitive functions (Figure 3H, right. See Table S2 for details). In particular, by classifying the MD-nodes into provincial or connector nodes, we found that provincial MD-nodes are associated with specialized cognitive control systems, such as the visual, sensorimotor, attention system, whereas the connector MD-nodes are associated with the default-mode system that enables the brain to move smoothly between different cognitive functions. These results indicate that MD-nodes might occupy strategic locations in the brain networks to control the cognitive functions.

Exploring the Development of Control Architecture by Artificial Network Evolution

We found that the brain networks have a distributed and overlapping control architecture and such a control architecture provides enhanced robustness (Figure 2C) and optimal (i.e., balanced) controllability (Figure 2D). To examine the relationship between structure and functional characteristics, we performed artificial network evolution starting from random null networks derived from the human brain networks and investigated robustness, controllability, and the corresponding control architectures along with evolution trajectories (Figure S5, see Transparent Methods for details). For this purpose, we first compared the following three features of the random null networks and the brain networks: domination stability against targeted attack of top 20% high-degree nodes, mean average controllability, and mean modal controllability (see Transparent Methods). As a result, we found that random null networks have much lower robustness and controllability than the brain networks (Figures 4A and 4B, and see Figure S5 for details). Moreover, random null networks have a centralized and overlapping control architecture in contrast with brain networks (Figures 4C and S5).

Figure 4.

Figure 4

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Exploring the Development of Control Architecture by Artificial Network Evolution

(A–C) We generated 100 degree-preserved random null networks (blue circles) derived from 100 structural human brain networks composed of 164 nodes (gray circles). Each colored circle represents an individual network. Starting from the generated random null networks, we performed artificial network evolution on each network by employing Pareto optimization to advance the following objective functions: (A) domination stability against targeted attack of top 20% high-degree nodes; (B) mean average controllability and mean modal controllability. Pink line in each figure indicates an evolution trajectory of each random null network, and red circles represent the pareto-optimal networks after the artificial evolution. (C) As the evolution proceeds, the control architecture of networks was transformed from a centralized and overlapping control architecture to the distributed and overlapping control architecture. See also Figure S5.

Starting from the random null networks, we performed artificial network evolution by employing Pareto optimization (Holland and Goldberg, 1989) in the direction of increasing the aforementioned three features (domination stability, mean average controllability, and mean modal controllability). Each evolutionary epoch was performed by carrying out two steps: network variation and network selection (see Transparent Methods for details). We generated randomly rewired networks, then we chose one that most advances the three features and used it in the next evolutionary epoch. Intriguingly, the artificial evolution trajectories that increase robustness and controllability (Figures 4A and 4B) lead to the distributed and overlapping control architecture (Figure 4C), implying that such a control architecture of the brain network might have emerged during evolution toward increasing both robustness and efficiency.

Discussion

In many cases, efficiency and robustness are often regarded as having a trade-off relationship. However, the brain unusually exhibits both attributes when it performs complex cognitive functions. Such optimality must be rooted in a specific coordinated control of interconnected brain regions, but our understanding of the intrinsic control architecture of complex brain networks is still lacking. In this study, we investigated the intrinsic control architecture of structural brain networks of various species and compared them with the control architectures of other biological and man-made (e.g., social, infrastructural, technological) complex networks. In particular, we developed a framework for analyzing the control architecture of complex networks based on the minimum dominating set (MDSet), which refers to a minimal subset of nodes (MD-nodes) that control remaining nodes with one-step direct interaction.

Here, by exploring and comparing the structural principles underlying the composition of MDSets of various complex networks, we delineated their distinct control architectures. Interestingly, we found that the proportion of MDSet in brain networks is remarkably smaller compared with other complex networks (Figure 1D), implying that brain networks may have been optimized to minimize the control cost. Furthermore, we found that the MDSet of brain networks is not solely determined by the degree of nodes but rather strategically placed to form a particular control architecture (Figure 1E). Consequently, we revealed the hidden control architecture of brain networks, namely, distributed and overlapping control architecture that is distinct from other complex networks (Figure 2B). We found that such a particular control architecture brings about robustness against targeted attack (i.e., preferential attack on high-degree nodes, Figure 2C), which might be a fundamental basis of the robust brain functions against preferential damages of high-degree nodes (i.e., brain regions) (Fornito et al., 2015). Moreover, we found that the particular control architecture of brain networks also enables high efficiency in switching from one network state (defined by a set of node activities) to another (Figure 2D), a capability that is crucial for traversing diverse cognitive states. Lastly, our artificial network evolution analysis showed that the distributed and overlapping control architecture of the brain network emerges when it evolves toward increasing both robustness and efficiency (Figure 4). Taken together, our results suggest that the distributed and overlapping control architecture of brain networks might be responsible for the robust and efficient control of brain functions.

A variety of biological processes are determined by the underlying regulatory networks (Assmus et al., 2006, Dubitzky et al., 2013, Eshaghi et al., 2010, Kim and Cho, 2006, Kim et al., 2011, Kwon and Cho, 2007, Lee et al., 2018, Murray et al., 2010, Park et al., 2006, Shin et al., 2006, Sreenath et al., 2008) and disruption of the networks can lead to diverse biological disorders (Shin et al., 2014, Shin et al., 2017, Yeo et al., 2018). Hence, control of a biological network has become an important issue to systematically regulate or modulate biological processes at a network level (Kim et al., 2013, Wolkenhauer et al., 2004). Likewise, there is a growing interest in brain network control (Bassett and Sporns, 2017), and various control strategies were suggested (Betzel et al., 2016, Gu et al., 2015, Khambhati et al., 2016, Yan et al., 2017) with the aim of developing therapeutics (Braun et al., 2018) and methods for cognitive enhancement (Kenett et al., 2018). It is, however, essential to understand the inherent control architecture of brain networks before applying any external interventions, because the brain itself is already an autonomously controlled system. Our study revealed an intrinsic control architecture of brain networks that not only sheds light on the intrinsic control properties of a normal brain but also provides a basis for exogenous control of brain networks to address altered control architectures of various neurological disorders.

Limitations of the Study

In this study, we performed tractography and constructed structural brain networks (i.e., connectome) of 100 healthy human adults to investigate the control architecture of the human brain networks. Yet, there exist inherent limitations in the current tractography algorithms (Daducci et al., 2016, Jbabdi and Johansen-Berg, 2011, Jones and Cercignani, 2010). Briefly, current tractography algorithms can yield false-positive and false-negative connections, and in consequence, genuine connections might be detected invalid and spurious ones as plausible. Accordingly, biased or inaccurate conclusions can be drawn from missing or duplicated connections. These limitations have motivated ongoing technical improvement of tractography algorithms and development of connectome validation standards.

Methods

All methods can be found in the accompanying Transparent Methods supplemental file.

Acknowledgments

We thank Nancy R. Gough (BioSerendipity, LLC) for editorial assistance. This work was supported by the National Research Foundation of Korea (NRF) grants funded by the Korea Government, the Ministry of Science, and ICT (2017R1A2A1A17069642 and 2015M3A9A7067220).

Author Contributions

K.-H.C. designed the project and supervised the study. B. L. and K.-H.C. designed experiments and wrote the paper; B. L., U.K., and H.C. performed experiments and analyzed data; K.-H.C. obtained funding.

Declaration of Interests

The authors declare no competing interests.

Published: March 29, 2019

Footnotes

Supplemental Information can be found online at https://doi.org/10.1016/j.isci.2019.02.017.

Supplemental Information

Document S1. Transparent Methods, Figures S1–S5, and Tables S1 and S2
mmc1.pdf (3.7MB, pdf)

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Associated Data

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Supplementary Materials

Document S1. Transparent Methods, Figures S1–S5, and Tables S1 and S2
mmc1.pdf (3.7MB, pdf)

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