Ann. N.Y. Acad. Sci. ISSN 0077-8923

A N N A L S O F T H E N E W Y O R K A C A D E M Y O F SC I E N C E S Issue: The Year in Cognitive Neuroscience

The human connectome: a complex network Olaf Sporns Department of Psychological and Brain Sciences, Indiana University, Bloomington, Indiana Address for correspondence: Olaf Sporns, Department of Psychological and Brain Sciences, Indiana University, Bloomington, Indiana 47405. [email protected]

The human brain is a complex network. An important first step toward understanding the function of such a network is to map its elements and connections, to create a comprehensive structural description of the network architecture. This paper reviews current empirical efforts toward generating a network map of the human brain, the human connectome, and explores how the connectome can provide new insights into the organization of the brain’s structural connections and their role in shaping functional dynamics. Network studies of structural connectivity obtained from noninvasive neuroimaging have revealed a number of highly nonrandom network attributes, including high clustering and modularity combined with high efficiency and short path length. The combination of these attributes simultaneously promotes high specialization and high integration within a modular small-world architecture. Structural and functional networks share some of the same characteristics, although their relationship is complex and nonlinear. Future studies of the human connectome will greatly expand our knowledge of network topology and dynamics in the healthy, developing, aging, and diseased brain. Keywords: networks; brain anatomy; neural dynamics; diffusion imaging; resting state; complex systems

Introduction We live in the age of networks. Our social interactions, be they personal relationships, political associations, financial transactions, professional collaborations, the spreading of rumors or diseases, or physical transport and travel, increasingly occur within networks that are evolving across time. All of these networks are examples of complex systems, with highly structured connectivity patterns, multiscale organization, nonlinear dynamics, resilient responses to external challenges, and the capacity for self-organization that gives rise to collective or group phenomena. Modern developments in graph theory and complex systems have delivered important insights into the structure and function of these diverse networks, as well as quantitative models that can both explain and predict network phenomena.1–5 Over the past decade, many studies across a range of scientific disciplines have demonstrated that complex networks pervade not only the social sciences, but also biology, from the organization of single

cells6 to that of entire ecosystems.7 In neuroscience, the idea that the brain is a neural network has deep historical roots. Yet the application of quantitative network theory to the brain is a recent development.8 This review will focus on the application of network concepts and network thinking to the human brain. I will survey the contributions of network science to our understanding of how human brain function arises from the interactions of cells and systems. Throughout, I will attempt to chart a path into the future, extrapolating the current flow of ideas and their potential impact on models and theories of the brain. A major driving force of recent progress has been the development of innovative methods for mapping brain connectivity, at cellular and systems scales. In the human brain, the continued refinement of noninvasive diffusion imaging and computational tractography has begun to reveal large-scale anatomical connections in unprecedented detail. For the first time, these methods allow the acquisition of comprehensive whole-brain data sets from individual human subjects and their

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comparison with individual data on brain dynamics, cognition, behavior, and genetics. The mathematical and theoretical framework of complex networks is an indispensable ingredient in the quantification, analysis, and modeling of these challenging data sets. Driven by technological and theoretical developments a new picture of the human brain is taking shape—a picture that views cognitive processes as the result of collective and coordinate phenomena unfolding within a complex network.8–12 As neuronal activity gives rise to dynamic patterns and processes, anatomical or structural connections play an important role in shaping these dynamics. Connectivity creates a structural skeleton for a dynamic regime conducive to flexible and robust neural computation. Comprehensive information on structural connectivity is fundamental for the formulation of mechanistic models of network processes underlying human brain function. Hence, I will begin with a discussion of recent progress in obtaining a complete map of the brain’s connections, the human connectome. What is the human connectome? The human connectome is “a comprehensive structural description of the network of elements and connections forming the human brain.”13,14 Three aspects of the connectome are central to this definition, and they deserve to be emphasized here as they have important theoretical ramifications. First, the connectome is principally about structure, about the extensive but finite set of physical links between neural elements. The physical reality of structural connectivity provides an important point of methodological convergence— different empirical methods for mapping structural connections should eventually provide a consistent anatomical description. In contrast, functional connectivity, which unfolds within the structural network, is significantly more variable across time, reflecting changes in internal state or neural responses to stimuli or task demand. Functional connectivity is generally defined as a statistical dependence between remote neural elements or regions, and it can be measured with a number of rather disparate methods that offer different perspectives on brain dynamics.15 For example, electrophysiology and magnetic resonance imaging (MRI) measure very different physiological signals and operate on different time and spatial scales. Additionally, time

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series data can be processed to extract temporal correlations, spectral coherence, or measures of causal signal flow, with each approach representing different aspects of neural dynamics. Thus, the configuration of a “functional connectome” depends critically on the choice of empirical methodology and analysis. Second, it is important to underline that the connectome is a description of brain connectivity. The term description implies compression of raw data with the goal of extracting maximal information. To use an analogy, an architectural description of a physical structure summarizes the major features of the design, but it does not offer a list of the dimensions and positions of all the building blocks. In the case of the brain, it is not necessary to demand that the connectome be an exact replica of the connectional anatomy down to the finest ramifications of neurites and individual synaptic boutons. Instead, the connectome should aim at a description of brain architecture that ranges over multiple levels of organization, reflecting the multiscale nature of brain connectivity. This program may include the microscale of cells and synapses, but it does not imply that all structural connectivity must be reduced to this level. Like any good road map, the human connectome should provide a multiscale description of the topological and spatial layout of connectional anatomy. Such a description will require the development of sophisticated neuroinformatics methods and tools to represent information about connectivity across multiple spatial scales, for example, in the form of digital brain atlases.16 Third, the main thrust behind the concept of the connectome is that it is the description of a network. The human connectome is not just a large collection of data. Instead it is a mathematical object that naturally fits within a larger theoretical framework and thus links neuroscience to modern developments in network science and complex systems. Ongoing research in these rapidly evolving areas of science will be instrumental for enabling the connectome to realize its full potential as a theoretical foundation for cognitive neuroscience. New theoretical, mathematical, and statistical approaches are urgently needed to fully reveal the organization of the human connectome and its fundamental role in cognition. Some beginnings have been made, notably in graph theory and dynamical systems, and it seems likely that connectomics will continue to benefit from

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efforts to map and model other complex biological systems. Connectome data sets are invaluable for linking structure to function, a relationship that is of pivotal concern in the biological sciences and one of the central themes of this review. In biology, structure is encountered at many scales, from molecules to the morphology of cells and entire living forms. Structure enables function, conservatively defined here as the set of possible actions and interactions of a biological system. For example, the three-dimensional structure of a macromolecule is critically important for its interactions with other chemical species, its capacity to bind substrates or catalyze chemical reactions, and its localization within specific cellular compartments. Similarly, the connectivity structure of a neural system defines its internal dynamic states and its range of responses to external perturbation. The importance of the connectome derives from the fundamental role of the network of structural connectivity in shaping the rich and dynamic set of functional interactions of neural elements that underlie human cognition and behavior. The term connectome was suggested in deliberate analogy to the genome, and there are many parallels between these two constructs. Both are fundamental structural data sets that inform us about the possible “functions” (actions and interactions) of a biological system. Both involve structural elements at different levels of scale (neurons, neuronal populations, and brain regions; base pairs, genes, and genetic regulatory networks). Both exhibit variability across organisms of the same, as well as different, species. Both give rise to complex system dynamics (firing patterns of neurons and neuronal populations; spatiotemporal expression profiles of gene products). Additional parallels have emerged due to recent discoveries in gene modification and transcription. Structural units of the connectome as well as the genome can be modified by environmental interactions. It is well known that brain connectivity is altered through various forms of neuronal plasticity, and recent work has shown that individual genomes can be subject to epigenetic biochemical modification.17 Finally, both connectomes and genomes are embedded in three-dimensional space, and the spatial proximity of their elements influences their functional dynamics. The location of neural elements is crucially important for the development of connectivity and the resulting functional

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interactions. Interestingly, the spatial arrangement of chromatin within cell nuclei has been shown to be associated with organized patterns of gene expression.18 This comparison suggests that neither genomes nor connectomes should be viewed as rigid programs or “blueprints” for biological function. Their functions result not so much from the reading-out of encoded instructions but rather involves the collective and coordinated actions of genetic and neural elements in complex networks. The organization of these networks enables the robustness and flexibility of biological systems.19,20 Structural variations can alter the dynamic expression of genomes and connectomes and thus impact their functionality. The sensitivity of functional dynamics to structural change is an important rationale for pursuing efforts to catalogue and map genome and connectome data sets. Empirical approaches to mapping the human connectome The structural connectivity of the human brain is organized on multiple nested spatial scales. Roughly, three scales of organization may be distinguished:13 the microscale of single neurons and synapses, the macroscale of anatomically distinct brain regions and pathways, and the mesoscale of neuronal populations and their interconnecting circuitry. Methods for mapping the connectome exist at each of these levels, and all of them are needed to achieve a complete representation of the human brain’s connections. At the microscale, there is significant progress on mapping cellular connectivity and reconstructing neuronal processes and synapses with electron or light microscopy. At the mesoscale, tracing of axonal projections with neuroanatomical markers and histological sectioning are poised to deliver whole-brain connectional data in several nonhuman species. At the macroscale, the development of noninvasive neuroimaging techniques allows the observation of connectional anatomy across the whole brain in live human participants. To this day, the mapping of the cellular connections in Caenorhabditis elegans, accomplished 25 years ago,21 stands as the only complete connectome of any organism. This unique data set was the result of painstaking reconstruction of three-dimensional neuronal morphology and connectivity from thousands of ultrathin serial sections scanned by

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electron microscopy (EM). Extensive analysis of the C. elegans nervous system has since provided important insights into its wiring economy22 and connectional topology.23 In recent years, the advent of new imaging, serial sectioning, and computational reconstruction techniques has fueled renewed efforts to map cellular anatomy of neural tissue at extremely high, subcellular resolution. New computational methods for automated three-dimensional reconstruction of sectioned EM volumes are under development,24 an indispensable step given the size and complexity of serial EM data sets. EM reconstruction is currently the only approach with sufficient spatial resolution to trace individual axonal processes as well as dendritic spines, and it can in principle be used to map non-neuronal cell types and reveal their relation to neurons. Other approaches in microscopy involve the combination of optical fluorescence and scanning EM of thin sections of neural tissue, or “array tomography,”25 as well as the labeling of individual neurons with immunofluorescent markers followed by three-dimensional optical imaging.26,27 The latter technique has been successfully applied to a set of neuromuscular connections in the mouse28 delivering important insights into the spatial layout of axonal processes and their high degree of structural variability, both within and across individuals. Imaging of neurons expressing fluorescent markers in a column of rat primary somatosensory cortex has provided detailed information on the number and spatial distribution of neuronal cell bodies and the pattern of thalamic afferents.29 The development of these and other fluorescence imaging techniques, particularly in combination with genetic approaches,30 will undoubtedly yield unprecedented quantitative data on the composition and connectivity of neural circuits in various animal species. Histological sectioning and imaging of the whole human brain is made difficult by its sheer size and volume. Modern refinements of histological techniques allow the collection of complete sets of thin histological sections from individual postmortem brains. Quantitative analysis of these sections can visualize variations in cellular microarchitecture as well as axonal connectivity. In a new technological development, histological sections of postmortem human brains are examined with polarized light imaging, a technique that provides information

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on the local direction and inclination of axonal fibers.31,32 The resulting distributions of vectors of fiber orientation can then be used to reconstruct axonal pathways across the section, and possibly the entire brain. The technique allows imaging of axons at high spatial resolution, and it may become an important method for the verification and validation of fiber anatomy revealed by in vivo diffusion imaging (see below). The labeling of axonal projections with the aid of neuroanatomical tracers has revealed the connectional anatomy of several mammalian brains in extraordinary detail. The accuracy and sensitivity of several of these tracers are well established, and the approach is particularly well suited for the mapping of long-distance pathways. Collations of hundreds of individual anatomical reports on inter-regional connections in the brains of several mammalian species33,34 have provided a wealth of insight into the anatomical organization of functionally segregated brain regions and their hierarchical and clustered arrangement.35–38 Bohland et al. have proposed a connectome map for the mouse brain be constructed through the systematic application of neuroanatomical tracers.39 While invasive anatomical tracing techniques cannot be carried out in human brains, injection of lipophilic dyes in postmortem tissue have provided data on patterns of connectional anatomy in at least some cortical regions.40,41 Significant challenges remain before microscopy methods can be deployed to fully map the human connectome at the microscale. Tracing individual human axons with EM or light microscopy will require error-free tracing and reconstruction over distances of many millimeters, from stacks of thousands of serial sections that individually cover an area of only a few square microns. Acquisition, storage, and manipulation of raw data from even a single human-sized brain demands computational resources far in excess of anything available today—it has been estimated that a single microscale human connectome would consist of one trillion gigabytes, or one exabyte.42 While these challenges may seem daunting at present, they may well be overcome as high-throughput automated microscopy and reconstruction become a reality. Other limitations, shared by all ex vivo approaches to connectomics, concern the difficulty of establishing links between anatomy and function. For example, any connection diagram derived at the cellular scale represents a snapshot in

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time of a highly dynamic and variable architecture, since changes in the morphology and connectivity of individual neurons are not captured. And microscopy can only be carried out on sections from postmortem tissue, which, in the case of humans, is difficult to obtain and preserve, and the use of which eliminates the possibility of examining neural structure in relation to neural dynamics or behavior. MRI currently represents the best option for noninvasive mapping of large-scale structural connections in the human brain.43 Some connectome maps have been assembled on the basis of structural MRI data. Correlations in the thickness or volume of gray matter between two cortical areas, usually measured across multiple participants, are partially predictive of the presence of anatomical connections. The mechanism that accounts for these correlations is currently unknown but possibly involves correlated metabolic or trophic processes, or genetic or experience-related factors. Some of the very first whole-brain connection matrices were assembled on the basis of cross-subject correlations in cortical thickness,44,45 and the approach continues to be of use in clinical comparisons.46 Diffusion MRI provides information about the spatial orientation of myelinated fiber tracts in cerebral white matter.47 An important challenge is the resolution of complex fiber architecture in parts of the brain where multiple fiber tracts intersect. Various imaging techniques that use multiple diffusion directions48,49 can resolve heterogeneous fiber orientations within single voxels. The subsequent application of computational algorithms allows the inference of anatomical tracts from the observed distribution of diffusion anisotropy across the imaged brain volume. Deterministic approaches to tractography rely on finding optimal streamlines within the diffusion tensor field, while probabilistic approaches aim to provide statistical estimates for the existence of fiber pathways. Various combinations of diffusion imaging and tractography techniques have been applied to the human brain and the brains of other mammalian species, with continual improvements in resolution and sensitivity. For example, high-resolution diffusion spectrum imaging (DSI) has recently mapped the complex fiber anatomy of the human cerebellum,50 as well as the developing circuitry of the cat brain.51 Cross-validation of connectivity obtained from diffusion imaging against histological tract tracing is difficult to carry out in

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humans, due to the lack of tract tracing data, but is feasible in other species, including the macaque monkey. Initial studies suggest that there is considerable agreement between diffusion imaging and tract tracing, at least with respect to a number of long-range cerebral pathways.52,53 However, fundamental limitations of diffusion imaging, for example, its inability to resolve the direction of axonal pathways and detect axonal branching, remain to be addressed. The construction and analysis of structural and functional brain networks from empirical data involve a series of steps (Fig. 1).11 First, network nodes must be defined, which, in the case of noninvasive neuroimaging studies, generally involves a parcellation into coherent regions on the basis of histological, connectional, or functional imaging data. Once nodes are defined their mutual association is measured, expressing their structural linkage or functional coupling. These pair-wise measurements are then aggregated into a connection matrix, which represents a graph or network. Finally, network metrics can be computed to extract relevant characteristics of the graph’s topology, which can be compared between individuals, subject groups, recording modalities, or to random graphs constructed according to specific null hypotheses. While each step presents significant methodological challenges, perhaps the most important decision involves node definition. Noninvasive imaging in humans does not allow the observation of individual neurons and instead delivers measures of neural population activity averaged over millimetersized volume elements. The voxel partition is arbitrary and does not reflect any underlying biological structure. Voxels need to be grouped into coherent regions to yield a biologically meaningful partition of the brain volume. An ideal partition should maximize the connectional information contained in the resulting graph achieving high specificity and low redundancy of connection patterns among discrete parcels or regions of the brain. A partition that is too coarse will result in mixing of connection patterns and hence loss of information about specific connections. An extremely finegrained partition will tend to contain a lot of redundant information, as many nodes are simply copies of each other. Random partitioning into roughly equal-sized brain nodes53,54 offers a trade-off between coarse and fine parcellation since it allows the

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Figure 1. Extraction of brain networks from empirical data follows along four steps: (1) parcellation of the brain volume into coherent regions on the basis of structural or connectional features (MRI), or node assignment by placement of sensors and/or recording sites (EEG, MEG); (2) structural or diffusion imaging to derive estimates of structural connectivity (left) or recording of time series data to estimate functional coupling (right); (3) construction of a connection matrix representing a structural (left) or functional network (right); shown here are data from the right cerebral hemisphere, as reported previously;53,124 and (4) network analysis.

exploration of network properties over a range of node scales. Simple parcellation schemes based on anatomical landmarks are imprecise and insufficient to fully represent the true anatomical and functional diversity of the cortical architecture. More sophisticated approaches use information about structural and/or functional connectivity to define regions with a coherent connectivity profile, obtained from individual brains. These approaches build on the idea that different brain regions have

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different connection profiles (“connectional fingerprints”),55 that is, different patterns of afferent and efferent anatomical connections, or different sets of functional linkages. Measurement of the connection profiles of neural elements across the brain volume (or cortical surface) delivers gradients of connectional change with peaks corresponding to putative regional boundaries. Alternatively, homogeneous groupings that correspond to segregated brain regions can be extracted using clustering algorithms. Clustering of connectivity

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profiles was used for segmentation of thalamic nuclei56 and definition of coherent cortical regions in medial frontal cortex57 and inferior frontal cortex.58 Correlations in spontaneous neural dynamics, for example, recorded in resting-state fMRI, exhibit local differences and transitions across the cortical surface. Sharp boundaries between regions of coherent resting-state functional connectivity can be detected with image analysis tools applied to the cortical surface,59 thus allowing the noninvasive parcellation of coherent functional regions in individual subjects. Extending this approach, a detailed parcellation of lateral parietal cortex was recently carried out on the basis of functional connectivity obtained with resting-state fMRI, as well as taskevoked response profiles.60 Using a combination of gradient detection and network analysis techniques, the study identified a number of distinct parietal regions that are arranged in network communities, each of which maintains specific functional associations. As these studies demonstrate, the combination of multiple criteria for region definition including structural connectivity, resting-state functional connectivity, task-evoked activations, and possibly cytoarchitectonic data greatly improves the robustness and sensitivity for defining network nodes, the building blocks of complex brain networks at the large scale. Any effort to map the human connectome must go beyond the mapping of invariant structural patterns and also consider individual structural variability and plasticity. Structural changes occur in the course of development as well as in relation to experience. While the extent of experience-dependent structural plasticity remains unknown in most neural systems, mounting evidence suggests that many circuit elements (neurons and synapses) can undergo substantial structural modification on relatively short time scales.61 Even at the large scale of interregional pathways, recent studies have demonstrated changes in white matter architecture related to behavior.62,63 The interest in structural plasticity derives from its functional implications. Structural plasticity among neurons and brain regions alters their dynamic interactions and functional profiles. The importance of these structure– function relationships (discussed in further detail below) implies that dynamic changes of anatomical patterns should become an integral part of ef-

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forts to map the human connectome, regardless of scale.64 Network architecture of the human connectome Parcellation allows the brain to be divided into a set of discrete elements (nodes) and their mutual connections (edges) that jointly form an object called a graph (Fig. 2). Commonly used in many scientific disciplines, graphs are models of real systems that offer a comprehensive map of how the system’s elements are linked or associated with each other. As models, graphs provide a description that is highly compact and embodies numerous assumptions about the nature of the real system that is being represented.65 These assumptions must be taken into account in the analysis and functional interpretation of brain networks.66 The structure of graphs or networks can be analyzed with the help of a broad range of graph theoretical measures.67,68 A number of these measures have already been applied to the analysis of structural and functional brain networks.11,69,70 Based on the information they provide about the biological organization of brain networks, relevant graph measures roughly fall into three classes: measures of segregation, integration, and influence.66 Measures of segregation identify the degree to which the graph can be decomposed into local communities or clusters of nodes that are highly interconnected. Such network communities are often referred to as modules, a term that is used here only in its graph-theoretical connotation. Measures of integration estimate the global efficiency of communication among all nodes in the network. Measures of influence yield indices for the potential participation of individual nodes and edges in dynamic processes unfolding within the network, as expressed, for example, in the node/edge centrality. Highly influential brain nodes are also referred to as hubs and can be further classified according to their network embedding and connection profiles.71 Graph metrics have proven to be useful analytic tools for the study of brain networks. The generality of the mathematical framework of graph theory allows its application in networks derived from a wide array of recording methods, from brain anatomy to functional connectivity, and in virtually all neural systems. While most applications so far have been at the large scale of whole-brain networks derived

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Figure 2. Schematic representation of the basic concepts of graph theory. The diagram at the left shows an undirected weighted brain network such as might result from diffusion imaging and tractography. Thresholding removes weak connections, and further analysis reveals two network communities or modules that are interlinked via a highly connected and highly central hub node (a so-called connector hub, “C”). Each module also contains highly connected nodes with mainly intramodular connections, forming provincial hubs (“P”). The diagrams at the bottom show module 2 after the connections within it have been converted to a binary format, illustrating the concepts of a network path (between nodes 1 and 2) and clustering (around node 3).

with diffusion or functional neuroimaging, cellular multielectrode recordings and optical imaging data sets are increasingly modeled as graphs and complex networks.72–74 Graph measures not only have a wide range of applications but also appear to be robust and reliable. In large-scale systems, graph metrics have been shown to be robust across multiple imaging runs,75 given a suitable partition of the brain into nodes and edges. Importantly, all graph measures are sensitive to the choice of partition and parcellation,76,77 thus underscoring the importance of the initial node and edge definition. More methodological work is needed in the area of statistical comparison of graphs across imaging modalities, subjects, or subject groups. Such comparisons can be challenging, especially if the graphs contain an unequal number of network elements, or if the correspondence of nodes and/or edges across different graphs cannot be established. Graph analyses of structural and functional data from the human brain have demonstrated an abundance of characteristic, nonrandom attributes. Large-scale structural networks derived from noninvasive diffusion imaging exhibit a high propensity for clustering of nodes into structural communi-

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ties, or modules. This tendency toward modularity is coupled with a high capacity for global information flow, as indicated by high efficiency and short path length. The combination of high clustering and short path length is one of the major characteristics of small-world networks.78,79 Robust small-world attributes were identified in several independent studies of whole-brain structural networks of human cerebral cortex.53,54,80–82 The functional significance of the “small-world-ness” of the human brain derives from the natural propensity of this architecture to promote functional segregation (local clustering) and functional integration (global efficiency), a hallmark of complex neural dynamics.83 Variations or disturbances of local and/or global anatomical or dynamic features may be sensitively reflected in corresponding small-world measures, and these measures may thus be of analytic importance. The small world of the human cerebral cortex is composed of structural modules, or network communities. A modularity analysis45 performed on a connection matrix of human cortex previously derived from intersubject correlations in cortical thickness44 identified several internally densely

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Figure 3. Graphical layout and modularity of a human connectome. (A) The structural backbone of a structural connection matrix averaged over five participants was extracted, as previously described,53 resulting in a matrix containing roughly 25% of all pathways. This connection matrix was then embedded in two-dimensional space using the Kamada–Kawai force-based energy minimization algorithm.140 This algorithm assigns spring-like forces to the network’s edges and attempts to place all nodes such that an energy term is minimized, achieving an optimally balanced layout. Labels refer to the centers of mass of several brain regions. Note that the connection topology, when graphically arranged such that “tension” along edges is minimized, roughly reproduces the arrangement of these regions in anatomical space. Right and left hemispheres are clearly separated, and the major anterior–posterior and lateral-medial cortical axes can be recognized. Regions are abbreviated as follows: (r = right hemisphere, l = left hemisphere): BSTS = bank of the superior temporal sulcus; CAC = caudal anterior cingulate cortex; IP = inferior parietal cortex; LOCC = lateral occipital cortex; LOF = lateral orbitofrontal cortex; LING = lingual gyrus; MOF = medial orbitofrontal cortex; MT = middle temporal cortex; POPE = pars opercularis; PTRI = pars triangularis; PSTC = postcentral gyrus; PC = posterior cingulate cortex; PREC = precentral gyrus; PCUN = precuneus; RAC = rostral anterior cingulate cortex; SF = superior frontal cortex; SP = superior parietal cortex. (B) Nodes are color coded by their assignment to one of six major structural modules, as previously reported.53 Note that modules comprise spatially coherent groupings of nodes. Plots at the right show anatomical locations of these modules in the right hemisphere.

connected modules largely corresponding to functionally distinct groups of areas related to vision, movement or language, as well as a set of interlinking hub regions including several areas of multimodal or association cortex. A study based on DSI data sets53

identified six structurally distinct modules in the frontal, temporoparietal, and medial cortex (Fig. 3). These modules mostly consisted of regions that were spatially close together, linked by a large number of short connection pathways found between

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adjacent areas. Particularly notable was a set of hubs located along the anterior–posterior medial axis of the brain that included highly connected regions such as the rostral and caudal anterior cingulate cortex, the paracentral lobule, and the precuneus. These regions were not only strongly interconnected among each other but also maintained connections with regions in virtually all other major communities in both right and left cerebral hemispheres. A topologically central core of interconnected hub regions was also described in a recent graph analysis of large-scale structural connectivity of cat and macaque monkey.84 In parallel studies, functional modules have been derived from physiological brain recordings, including resting-state fMRI. There are several common themes in the organization of structural and functional networks. For example, in agreement with structural network analysis, functional networks exhibit a strong tendency toward modular small-world architecture.85,86 Another common feature is the existence of a nested hierarchy of structural87 and functional modules.88 The full extent of this hierarchy is only partially mapped and ranges from relatively coarse groupings—for example, the hemispheres or major lobes of the cerebrum—to more fine-grained groupings, such as functional brain systems (e.g., visual, auditory, somatomotor cortices), individual segregated regions, gray matter nuclei, and even columnar arrangements of cells. An important question for future network studies concerns the relationship between structural and functional modules and their hierarchical arrangement. Answering this question will require multimodal imaging studies and network comparison across modalities. Communities or modules are coupled via hub nodes, which represent highly connected and highly central regions of the brain. Several studies of structural networks have pinpointed these regions within the parietal and frontal lobes of the cerebral cortex.81,82 At least some hubs are mutually interconnected or aggregated to form a prominent structural core, located in posteromedial cortex and comprising several cortical regions including the precuneus, posterior cingulate cortex, and superior parietal cortex.53 Large-scale functional networks exhibit a similar organization, with several studies demonstrating a set of functional clusters or modules,86 highly connected hub nodes,89 and high global efficiency.90

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Community detection is of special relevance for studies of human brain networks because structural and functional modules may coincide with sets of brain regions and pathways that jointly participate in specific cognitive tasks. For example, the application of modularity to resting-state fMRI data has revealed several distinct networks engaged in different cognitive task domains.60,91 With the arrival of robust parcellation strategies it is expected that studies of communities in structural and functional brain networks will yield increasingly refined maps of neurocognitive networks. The prominent place of the posterior medial cortex within the cortical network deserves closer scrutiny. Several graph-theoretic analyses of independently acquired diffusion imaging data sets have reported high centrality for the precuneus and posterior cingulate cortex, and neighboring regions.53,81,82,92 These regions form an aggregation of structural hubs (a structural core), combining a high degree of connectedness as expressed in the graph measures of node degree and strength, as well as high betweenness centrality. Interestingly, the same set of regions occupies an equally central position in functional networks, in particular those engaged during cognitive rest,89,93–96 and also corresponds to an area of extremely high baseline metabolic activity.97 The precuneus, a central component of the structural core, is activated during a wide variety of cognitive operations including self-referential processing, imagery, and episodic memory,98 its level of baseline activation is associated with the level of consciousness,99 and its functional connectivity changes in the transition from waking to sleep.100 Activation studies and network analyses indicate a high degree of functional specialization within subregions of the precuneus. The region exhibits differential patterns of activity during the processing of different kinds of social emotions.101 In addition, resting-state fMRI suggests that different subregions of the precuneus maintain distinct sets of functional connections with other parts of the brain.102 Taken together, the combination of physiological observations and network analysis has significantly increased our understanding of the structure and function of this highly central brain region. Linking structure and function One of the main rationales for assembling the human connectome is the importance of anatomical

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structure for our understanding of neural dynamics103 and, ultimately, cognition and behavior. As will become clear in this section, structural brain connectivity does not rigidly determine neural interactions but acts as a set of constraints, a skeleton, or scaffold that sharply reduces the dimensionality of the neural state space. Within this reduced lowdimensional space, neural dynamics remain fluid, variable and sensitive to dynamic perturbations. While this section of the chapter places emphasis on how structure shapes function, it must be noted that functional interactions and their expression in behavior at the level of the whole organisms can profoundly influence structural patterns through a variety of mechanisms of plasticity—thus, structural and functional connectivity are reciprocally linked.104 Recent empirical studies and computational models of the brain’s resting state have revealed the variable, yet robust, nature of spontaneous neural dynamics. Electrophysiological recordings of spontaneous neural activity have demonstrated the presence of rapid transitions in global patterns of functional correlations, including periods of intermittent phase locking and phase synchrony at multiple temporal and spatial scales,105 and broadly distributed synchronization times across multiple frequencies.106 Variable periods of phase locking have also been observed in magnetoencephalographic and fMRI recordings of resting-state brain activity,107 and further evidence suggests that scalefree neural dynamics involving multiple nested frequencies may have behavioral and cognitive relevance.108 The scale-free nature of spontaneous neural dynamics is potentially indicative of the presence of a “critical state,”109 a dynamic regime characterized by a highly diverse set of intrinsic neural states and a maximal dynamic range of responses to extrinsic perturbations. Consistent patterns of activation and deactivation among brain regions during transitions from task evoked to resting activity led to the discovery of a coherent “default mode”110 whose participating regions were linked into a default mode network.93 Neurophysiological correlates of default mode brain activity as observed in resting-state fMRI have been identified through combined imaging and electrophysiological recordings.111–113 Refined analysis and network modeling techniques have revealed the fine structure of resting-state brain

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dynamics in ever greater detail. Independent component analysis has demonstrated several distinct resting-state networks,114,115 some of which correspond to sets of brain regions that cooperate in specific cognitive domains such as vision, motor planning, or episodic memory. Their activation is modulated by task demands and may have diagnostic value in at least some brain disorders.116 Distinct functional networks extracted from spontaneous (default mode) brain activity also resemble networks of coactive brain regions identified during task-evoked activation studies.117 This suggests that resting brain activity partly reflects or rehearses dynamic patterns that become differentially activated in response to extrinsic task- and stimulusrelated perturbations. The systematic mapping of resting-state functional connectivity has become an integral part of the connectomics research program.118,119 The characteristic patterns of neural dynamics observed during rest appear to have an anatomical basis. Variations in the strength and integrity of specific white matter tracts are correlated with the strength of the corresponding functional connection observed in the resting state.120 Major components of the default mode network121 and other resting-state networks122 are anatomically connected. Perhaps the most compelling evidence for structure–function relationships in brain connectivity comes from the direct comparison of structural and functional networks within the same cohort of subjects.53,123,124 The comparison reveals significant positive correlations between the strengths of structural connections identified with diffusion MRI and the strengths of functional connections obtained with resting-state functional MRI. This relationship is robust relative to effects of spatial distance or variations in postprocessing of functional correlations. Another observation reinforces an important point that was made in earlier studies of functional connectivity:125 a large number of strong functional connections exist between regions that are not directly linked by an anatomical pathway. Because of the high base rate of structurally unconnected node pairs, it is impractical (indeed erroneous) to infer structural connections from functional connections by simple means such as thresholding (Fig. 4). The network aspect of functional connectivity is further underscored by the observation that many functional connections between

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Figure 4. Inference of structural connectivity (SC) from resting-state functional connectivity (rsFC) is impractical. (A) Probability distribution of rsFC strengths between connected and unconnected node pairs. Note that node pairs that are directly connected, on average, maintain stronger rsFC than unconnected node pairs. (B) The corresponding receiver operating characteristic (ROC) curve shows the relation of false positives to true positives at thresholds ranging from −1 to +1. (C) Thresholding the rsFC matrix such that 90% of all structural connections are correctly detected (≈ 16,100 out of 17,600) also falsely detects approximately 242,000 connections. For simplicity, the plot shows true and false positives in the right cerebral hemisphere only. Data as previously reported.124

unconnected region pairs can be partially predicted by the existence of indirect structural connections, or indirect network paths (Fig. 5). This point crystallizes one of the most fundamental insights about the importance of a network perspective on human brain function. All pair-wise dynamic interactions between two brain regions reflect a combination of direct effects (if a direct structural link exists) and a blend of indirect effects that involve the flow of information through numerous network paths. A corollary of this observation is that the local perturbation (activation, deletion) of nodes or edges within the network can manifest itself as disturbances of dynamic interactions in distant and seemingly unrelated brain systems.126 Specific mechanisms of how the structure of anatomical networks shapes functional connectivity can be productively addressed by pursuing computational modeling studies. Such models are useful because they allow the precise specification of structural coupling and the computation of complete

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neural time series data, without interference from physiological or scanner noise. Putative dynamic mechanisms can be fully explored by introducing perturbations in structural coupling or physiology, leading to predictions that can be empirically tested. For example, a computational neural mass model of the macaque cortex predicted that correlated fluctuations in BOLD responses were shaped by structural brain networks.127 Further, the model suggested that variations in BOLD amplitude were partially driven by the transient synchronization of neural signals between sets of brain regions at fast time scales, a prediction for which there is some supporting evidence.113 An important feature of this as well as several related dynamic models128,129 is the highly variable nature of ongoing brain dynamics. While anatomical connections play a key role in creating large-scale functional networks, they also allow for significant and ongoing fluctuations within these networks, thus giving rise to a repertoire of variable network states. Jointly, these models predict that a

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Figure 5. Strengths of direct and indirect structural connections partially predict functional connections. (A) Correlation of direct SC and rsFC strengths at node pairs that are structurally connected. (B) Direct SC and indirect SC between node pairs are strongly correlated, that is, the presence of a strong, direct connection also predicts additional strong, indirect connections. This effect is due to the high degree of clustering present in the network. (C) Indirect SC weakly predicts rsFC between structurally unconnected node pairs. All correlations have P < 0.001. Indirect SC was computed as the sum of the product of SC strengths over all indirect paths of length 2. Direct and indirect SC was resampled to a Gaussian distribution. All data and analyses as previously reported.124

dynamic functional repertoire is essential for efficient neural processing.130 The availability of connectome data sets allows the construction of neurocomputational models that can re-create key features and patterns of human brain dynamics.124,131 Comparison of modeled and empirically recorded patterns of functional connectivity suggest that dynamical models of neuronal populations coupled by a structural connection matrix can capture a large proportion of the observed variance in large-scale resting-state neural activity. More refined models that include better coverage of subcortical regions, more accurate or more highly resolved connectome data, and more sophisticated neuronal physiology, are on the horizon. Such “virtual brains”130 will offer increasingly detailed models that combine anatomy and physiology in silico, thus contributing to our understanding of the network dynamics of the human connectome. Outlook Networks have become pervasive and adaptable models to map and manage complex systems as diverse as societies, ecosystems, cellular metabolism, and, finally, brains. Their considerable appeal and power as models of neural systems derives in no small part from the insights they can deliver about the origin of dynamic brain activity unfolding within anatomical circuits, across multiple levels of scale. This brief review cannot do justice to the full range of future applications of the human connectome. In addition to the aspects covered here,

significant progress has already been made in discovering the network basis of common disorders of the brain,132 response to and recovery from brain injury,126,133,134 individual differences,90,135,136 heritability,137 normal development and aging,138,139 as well as design principles of neural circuits that derive from their spatial embedding and wiring economy.87 The arrival of genomics brought significant change to the biological sciences and generated breathtaking technological and intellectual advances, many of which are still unfolding. It will be some time before the impact of connectomics on neuroscience has fully materialized. Going beyond the collection of large data sets or the development of new empirical recording and mapping technologies, connectomics has the potential to bring about a new conceptual understanding of human brain function. The human connectome refocuses an urgent need for a solid theoretical foundation to guide future empirical research in brain connectivity and dynamics. Network theory offers a fruitful beginning and raises the tantalizing possibility that at least some of the principles underlying the functioning of the human brain are held in common among a wide range of complex systems, biological and otherwise. Acknowledgments The author gratefully acknowledges support by a grant from the James S. McDonnell Foundation.

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Conflicts of interest The author declares no conflicts of interest.

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