Using Pareto optimality to explore the topology and dynamics of the human connectome

A simulated annealing algorithm for randomizing weighted networks

Scientific discovery in connectomics relies on network null models. The prominence of network features is conventionally evaluated against null distributions estimated using randomized networks. Modern imaging technologies provide an increasingly rich array of biologically meaningful edge weights. Despite the prevalence of weighted graph analysis in connectomics, randomization models that only preserve binary node degree remain most widely used. Here we propose a simulated annealing procedure for generating randomized networks that preserve weighted degree (strength) sequences. We show that the procedure outperforms other rewiring algorithms and generalizes to multiple network formats, including directed and signed networks, as well as diverse real-world networks. Throughout, we use morphospace representation to assess the sampling behavior of the algorithm and the variability of the resulting ensemble. Finally, we show that accurate strength preservation yields different inferences about brain network organization. Collectively, this work provides a simple but powerful method to analyze richly detailed next-generation connectomics datasets.

A Hybrid Routing Pattern in Human Brain Structural Network Revealed By Evolutionary Computation

The human brain functional connectivity network (FCN) is constrained and shaped by the communication processes in the structural connectivity network (SCN). The underlying communication mechanism thus becomes a critical issue for understanding the formation and organization of the FCN. A number of communication models supported by different routing strategies have been proposed, with shortest path (SP), random diffusion (DIF), and spatial navigation (NAV) as the most typical, respectively requiring network global knowledge, local knowledge, and both for path seeking. Yet these models all assumed every brain region to use one routing strategy uniformly, ignoring convergent evidence that supports the regional heterogeneity in both terms of biological substrates and functional roles. In this regard, the current study developed a hybrid communication model that allowed each brain region to choose a routing strategy from SP, DIF, and NAV independently. A genetic algorithm was designed to uncover the underlying region-wise hybrid routing strategy (namely HYB). The HYB was found to outperform the three typical routing strategies in predicting FCN and facilitating robust communication. Analyses on HYB further revealed that brain regions in lower-order functional modules inclined to route signals using global knowledge, while those in higher-order functional modules preferred DIF that requires only local knowledge. Compared to regions that used global knowledge for routing, regions using DIF had denser structural connections, participated in more functional modules, but played a less dominant role within modules. Together, our findings further evidenced that hybrid routing underpins efficient SCN communication and locally heterogeneous structure-function coupling.

Non-homogenous axonal bouton distribution in whole-brain single-cell neuronal networks

We examined the distribution of pre-synaptic contacts in axons of mouse neurons and constructed whole-brain single-cell neuronal networks using an extensive dataset of 1,891 fully reconstructed neurons. We found that bouton locations were not homogeneous throughout the axon and among brain regions. As our algorithm was able to generate whole-brain single-cell connectivity matrices from full morphology reconstruction datasets, we further found that non-homogeneous bouton locations have a significant impact on network wiring, including degree distribution, triad census, and community structure. By perturbing neuronal morphology, we further explored the link between anatomical details and network topology. In our in silico exploration, we found that dendritic and axonal tree span would have the greatest impact on network wiring, followed by synaptic contact deletion. Our results suggest that neuroanatomical details must be carefully addressed in studies of whole-brain networks at the single-cell level.

Resolving inter-regional communication capacity in the human connectome

Applications of graph theory to the connectome have inspired several models of how neural signaling unfolds atop its structure. Analytic measures derived from these communication models have mainly been used to extract global characteristics of brain networks, obscuring potentially informative inter-regional relationships. Here we develop a simple standardization method to investigate polysynaptic communication pathways between pairs of cortical regions. This procedure allows us to determine which pairs of nodes are topologically closer and which are further than expected on the basis of their degree. We find that communication pathways delineate canonical functional systems. Relating nodal communication capacity to meta-analytic probabilistic patterns of functional specialization, we also show that areas that are most closely integrated within the network are associated with higher order cognitive functions. We find that these regions' proclivity towards functional integration could naturally arise from the brain's anatomical configuration through evenly distributed connections among multiple specialized communities. Throughout, we consider two increasingly constrained null models to disentangle the effects of the network's topology from those passively endowed by spatial embedding. Altogether, the present findings uncover relationships between polysynaptic communication pathways and the brain's functional organization across multiple topological levels of analysis and demonstrate that network integration facilitates cognitive integration.

Brain network communication: concepts, models and applications

Understanding communication and information processing in nervous systems is a central goal of neuroscience. Over the past two decades, advances in connectomics and network neuroscience have opened new avenues for investigating polysynaptic communication in complex brain networks. Recent work has brought into question the mainstay assumption that connectome signalling occurs exclusively via shortest paths, resulting in a sprawling constellation of alternative network communication models. This Review surveys the latest developments in models of brain network communication. We begin by drawing a conceptual link between the mathematics of graph theory and biological aspects of neural signalling such as transmission delays and metabolic cost. We organize key network communication models and measures into a taxonomy, aimed at helping researchers navigate the growing number of concepts and methods in the literature. The taxonomy highlights the pros, cons and interpretations of different conceptualizations of connectome signalling. We showcase the utility of network communication models as a flexible, interpretable and tractable framework to study brain function by reviewing prominent applications in basic, cognitive and clinical neurosciences. Finally, we provide recommendations to guide the future development, application and validation of network communication models.

Trade-offs among cost, integration, and segregation in the human connectome

The human brain structural network is thought to be shaped by the optimal trade-off between cost and efficiency. However, most studies on this problem have focused on only the trade-off between cost and global efficiency (i.e., integration) and have overlooked the efficiency of segregated processing (i.e., segregation), which is essential for specialized information processing. Direct evidence on how trade-offs among cost, integration, and segregation shape the human brain network remains lacking. Here, adopting local efficiency and modularity as segregation factors, we used a multiobjective evolutionary algorithm to investigate this problem. We defined three trade-off models, which represented trade-offs between cost and integration (Dual-factor model), and trade-offs among cost, integration, and segregation (local efficiency or modularity; Tri-factor model), respectively. Among these, synthetic networks with optimal trade-off among cost, integration, and modularity (Tri-factor model [Q]) showed the best performance. They had a high recovery rate of structural connections and optimal performance in most network features, especially in segregated processing capacity and network robustness. Morphospace of this trade-off model could further capture the variation of individual behavioral/demographic characteristics in a domain-specific manner. Overall, our results highlight the importance of modularity in the formation of the human brain structural network and provide new insights into the original cost-efficiency trade-off hypothesis.

From calcium imaging to graph topology

Systems neuroscience is facing an ever-growing mountain of data. Recent advances in protein engineering and microscopy have together led to a paradigm shift in neuroscience; using fluorescence, we can now image the activity of every neuron through the whole brain of behaving animals. Even in larger organisms, the number of neurons that we can record simultaneously is increasing exponentially with time. This increase in the dimensionality of the data is being met with an explosion of computational and mathematical methods, each using disparate terminology, distinct approaches, and diverse mathematical concepts. Here we collect, organize, and explain multiple data analysis techniques that have been, or could be, applied to whole-brain imaging, using larval zebrafish as an example model. We begin with methods such as linear regression that are designed to detect relations between two variables. Next, we progress through network science and applied topological methods, which focus on the patterns of relations among many variables. Finally, we highlight the potential of generative models that could provide testable hypotheses on wiring rules and network progression through time, or disease progression. While we use examples of imaging from larval zebrafish, these approaches are suitable for any population-scale neural network modeling, and indeed, to applications beyond systems neuroscience. Computational approaches from network science and applied topology are not limited to larval zebrafish, or even to systems neuroscience, and we therefore conclude with a discussion of how such methods can be applied to diverse problems across the biological sciences.

Pareto optimality, economy-effectiveness trade-offs and ion channel degeneracy: improving population modelling for single neurons

Neurons encounter unavoidable evolutionary trade-offs between multiple tasks. They must consume as little energy as possible while effectively fulfilling their functions. Cells displaying the best performance for such multi-task trade-offs are said to be Pareto optimal, with their ion channel configurations underpinning their functionality. Ion channel degeneracy, however, implies that multiple ion channel configurations can lead to functionally similar behaviour. Therefore, instead of a single model, neuroscientists often use populations of models with distinct combinations of ionic conductances. This approach is called population (database or ensemble) modelling. It remains unclear, which ion channel parameters in the vast population of functional models are more likely to be found in the brain. Here we argue that Pareto optimality can serve as a guiding principle for addressing this issue by helping to identify the subpopulations of conductance-based models that perform best for the trade-off between economy and functionality. In this way, the high-dimensional parameter space of neuronal models might be reduced to geometrically simple low-dimensional manifolds, potentially explaining experimentally observed ion channel correlations. Conversely, Pareto inference might also help deduce neuronal functions from high-dimensional Patch-seq data. In summary, Pareto optimality is a promising framework for improving population modelling of neurons and their circuits.

A Riemannian approach to predicting brain function from the structural connectome

Ongoing brain function is largely determined by the underlying wiring of the brain, but the specific rules governing this relationship remain unknown. Emerging literature has suggested that functional interactions between brain regions emerge from the structural connections through mono- as well as polysynaptic mechanisms. Here, we propose a novel approach based on diffusion maps and Riemannian optimization to emulate this dynamic mechanism in the form of random walks on the structural connectome and predict functional interactions as a weighted combination of these random walks. Our proposed approach was evaluated in two different cohorts of healthy adults (Human Connectome Project, HCP; Microstructure-Informed Connectomics, MICs). Our approach outperformed existing approaches and showed that performance plateaus approximately around the third random walk. At macroscale, we found that the largest number of walks was required in nodes of the default mode and frontoparietal networks, underscoring an increasing relevance of polysynaptic communication mechanisms in transmodal cortical networks compared to primary and unimodal systems.

Neural optimization: Understanding trade-offs with Pareto theory

Nervous systems, like any organismal structure, have been shaped by evolutionary processes to increase fitness. The resulting neural 'bauplan' has to account for multiple objectives simultaneously, including computational function, as well as additional factors such as robustness to environmental changes and energetic limitations. Oftentimes these objectives compete, and quantification of the relative impact of individual optimization targets is non-trivial. Pareto optimality offers a theoretical framework to decipher objectives and trade-offs between them. We, therefore, highlight Pareto theory as a useful tool for the analysis of neurobiological systems from biophysically detailed cells to large-scale network structures and behavior. The Pareto approach can help to assess optimality, identify relevant objectives and their respective impact, and formulate testable hypotheses.

Inter-individual body mass variations relate to fractionated functional brain hierarchies

Variations in body mass index (BMI) have been suggested to relate to atypical brain organization, yet connectome-level substrates of BMI and their neurobiological underpinnings remain unclear. Studying 325 healthy young adults, we examined associations between functional connectivity and inter-individual BMI variations. We utilized non-linear connectome manifold learning techniques to represent macroscale functional organization along continuous hierarchical axes that dissociate low level and higher order brain systems. We observed an increased differentiation between unimodal and heteromodal association networks in individuals with higher BMI, indicative of a disrupted modular architecture and hierarchy of the brain. Transcriptomic decoding and gene enrichment analyses identified genes previously implicated in genome-wide associations to BMI and specific cortical, striatal, and cerebellar cell types. These findings illustrate functional connectome substrates of BMI variations in healthy young adults and point to potential molecular associations.

Cost-efficiency trade-offs of the human brain network revealed by a multiobjective evolutionary algorithm

It is widely believed that the formation of brain network architecture is under the pressure of optimal trade-off between reducing wiring cost and promoting communication efficiency. However, the questions of whether this trade-off exists in empirical human brain structural networks and, if so, how it takes effect are still not well understood. Here, we employed a multiobjective evolutionary algorithm to directly and quantitatively explore the cost-efficiency trade-off in human brain structural networks. Using this algorithm, we generated a population of synthetic networks with optimal but diverse cost-efficiency trade-offs. It was found that these synthetic networks could not only reproduce a large portion of connections in the empirical brain structural networks but also embed a resembling small-world organization. Moreover, the synthetic and empirical brain networks were found similar in terms of the spatial arrangement of hub regions and the modular structure, which are two important topological features widely assumed to be outcomes of cost-efficiency trade-offs. The synthetic networks had high robustness against random attacks as the empirical brain networks did. Additionally, we also revealed some differences between the synthetic networks and the empirical brain networks, including lower segregated processing capacity and weaker robustness against targeted attacks in the synthetic networks. These findings provide direct and quantitative evidence that the structure of human brain networks is indeed largely influenced by optimal cost-efficiency trade-offs. We also suggest that some additional factors (e.g., segregated processing capacity) might jointly determine the network organization with cost and efficiency.

Models of communication and control for brain networks: distinctions, convergence, and future outlook

Recent advances in computational models of signal propagation and routing in the human brain have underscored the critical role of white-matter structure. A complementary approach has utilized the framework of network control theory to better understand how white matter constrains the manner in which a region or set of regions can direct or control the activity of other regions. Despite the potential for both of these approaches to enhance our understanding of the role of network structure in brain function, little work has sought to understand the relations between them. Here, we seek to explicitly bridge computational models of communication and principles of network control in a conceptual review of the current literature. By drawing comparisons between communication and control models in terms of the level of abstraction, the dynamical complexity, the dependence on network attributes, and the interplay of multiple spatiotemporal scales, we highlight the convergence of and distinctions between the two frameworks. Based on the understanding of the intertwined nature of communication and control in human brain networks, this work provides an integrative perspective for the field and outlines exciting directions for future work.

Signal diffusion along connectome gradients and inter-hub routing differentially contribute to dynamic human brain function

Human cognition is dynamic, alternating over time between externally-focused states and more abstract, often self-generated, patterns of thought. Although cognitive neuroscience has documented how networks anchor particular modes of brain function, mechanisms that describe transitions between distinct functional states remain poorly understood. Here, we examined how time-varying changes in brain function emerge within the constraints imposed by macroscale structural network organization. Studying a large cohort of healthy adults (n = 326), we capitalized on manifold learning techniques that identify low dimensional representations of structural connectome organization and we decomposed neurophysiological activity into distinct functional states and their transition patterns using Hidden Markov Models. Structural connectome organization predicted dynamic transitions anchored in sensorimotor systems and those between sensorimotor and transmodal states. Connectome topology analyses revealed that transitions involving sensorimotor states traversed short and intermediary distances and adhered strongly to communication mechanisms of network diffusion. Conversely, transitions between transmodal states involved spatially distributed hubs and increasingly engaged long-range routing. These findings establish that the structure of the cortex is optimized to allow neural states the freedom to vary between distinct modes of processing, and so provides a key insight into the neural mechanisms that give rise to the flexibility of human cognition.

Navigable maps of structural brain networks across species

Connectomes are spatially embedded networks whose architecture has been shaped by physical constraints and communication needs throughout evolution. Using a decentralized navigation protocol, we investigate the relationship between the structure of the connectomes of different species and their spatial layout. As a navigation strategy, we use greedy routing where nearest neighbors, in terms of geometric distance, are visited. We measure the fraction of successful greedy paths and their length as compared to shortest paths in the topology of connectomes. In Euclidean space, we find a striking difference between the navigability properties of mammalian and non-mammalian species, which implies the inability of Euclidean distances to fully explain the structural organization of their connectomes. In contrast, we find that hyperbolic space, the effective geometry of complex networks, provides almost perfectly navigable maps of connectomes for all species, meaning that hyperbolic distances are exceptionally congruent with the structure of connectomes. Hyperbolic maps therefore offer a quantitative meaningful representation of connectomes that suggests a new cartography of the brain based on the combination of its connectivity with its effective geometry rather than on its anatomy only. Hyperbolic maps also provide a universal framework to study decentralized communication processes in connectomes of different species and at different scales on an equal footing.

Editorial: Linking experimental and computational connectomics

Large-scale in silico experimentation depends on the generation of connectomes beyond available anatomical structure. We suggest that linking research across the fields of experimental connectomics, theoretical neuroscience, and high-performance computing can enable a new generation of models bridging the gap between biophysical detail and global function. This Focus Feature on "Linking Experimental and Computational Connectomics" aims to bring together some examples from these domains as a step toward the development of more comprehensive generative models of multiscale connectomes.

The heterogeneity in link weights may decrease the robustness of real-world complex weighted networks

Here we report a comprehensive analysis of the robustness of seven high-quality real-world complex weighted networks to errors and attacks toward nodes and links. We use measures of the network damage conceived for a binary (e.g. largest connected cluster LCC, and binary efficiency Eff_bin) or a weighted network structure (e.g. the efficiency Eff, and the total flow TF). We find that removing a very small fraction of nodes and links with respectively higher strength and weight triggers an abrupt collapse of the weighted functioning measures while measures that evaluate the binary-topological connectedness are almost unaffected. These findings unveil a problematic response-state where the attack toward a small fraction of nodes-links returns the real-world complex networks in a connected but inefficient state. Our findings unveil how the robustness may be overestimated when focusing on the connectedness of the components only. Last, to understand how the networks robustness is affected by link weights heterogeneity, we randomly assign link weights over the topological structure of the real-world networks and we find that highly heterogeneous networks show a faster efficiency decrease under nodes-links removal: i.e. the robustness of the real-world complex networks against nodes-links removal is negatively correlated with link weights heterogeneity.

REBUS and the Anarchic Brain: Toward a Unified Model of the Brain Action of Psychedelics

This paper formulates the action of psychedelics by integrating the free-energy principle and entropic brain hypothesis. We call this formulation relaxed beliefs under psychedelics (REBUS) and the anarchic brain, founded on the principle that-via their entropic effect on spontaneous cortical activity-psychedelics work to relax the precision of high-level priors or beliefs, thereby liberating bottom-up information flow, particularly via intrinsic sources such as the limbic system. We assemble evidence for this model and show how it can explain a broad range of phenomena associated with the psychedelic experience. With regard to their potential therapeutic use, we propose that psychedelics work to relax the precision weighting of pathologically overweighted priors underpinning various expressions of mental illness. We propose that this process entails an increased sensitization of high-level priors to bottom-up signaling (stemming from intrinsic sources), and that this heightened sensitivity enables the potential revision and deweighting of overweighted priors. We end by discussing further implications of the model, such as that psychedelics can bring about the revision of other heavily weighted high-level priors, not directly related to mental health, such as those underlying partisan and/or overly-confident political, religious, and/or philosophical perspectives. SIGNIFICANCE STATEMENT: Psychedelics are capturing interest, with efforts underway to bring psilocybin therapy to marketing authorisation and legal access within a decade, spearheaded by the findings of a series of phase 2 trials. In this climate, a compelling unified model of how psychedelics alter brain function to alter consciousness would have appeal. Towards this end, we have sought to integrate a leading model of global brain function, hierarchical predictive coding, with an often-cited model of the acute action of psychedelics, the entropic brain hypothesis. The resulting synthesis states that psychedelics work to relax high-level priors, sensitising them to liberated bottom-up information flow, which, with the right intention, care provision and context, can help guide and cultivate the revision of entrenched pathological priors.

Functional optimality of the sulcus pattern of the human brain

We develop a mathematical model of information transmission across the biological neural network of the human brain. The overall function of the brain consists of the emergent processes resulting from the spread of information through the neural network. The capacity of the brain is therefore related to the rate at which it can transmit information through the neural network. The particular transmission model under consideration allows for information to be transmitted along multiple paths between points of the cortex. The resulting transmission rates are governed by potential theory. According to this theory, the brain has preferred and quantized transmission modes that correspond to eigenfunctions of the classical Steklov eigenvalue problem, with the reciprocal eigenvalues quantifying the corresponding transmission rates. We take the model as a basis for testing the hypothesis that the sulcus pattern of the human brain has evolved to maximize the rate of transmission of information between points in the cerebral cortex. We show that the introduction of sulci, or cuts, in an otherwise smooth domain indeed increases the overall transmission rate. We demonstrate this result by means of numerical experiments concerned with a spherical domain with a varying number of slits on its surface.

A spectrum of routing strategies for brain networks

Communication of signals among nodes in a complex network poses fundamental problems of efficiency and cost. Routing of messages along shortest paths requires global information about the topology, while spreading by diffusion, which operates according to local topological features, is informationally “cheap” but inefficient. We introduce a stochastic model for network communication that combines local and global information about the network topology to generate biased random walks on the network. The model generates a continuous spectrum of dynamics that converge onto shortest-path and random-walk (diffusion) communication processes at the limiting extremes. We implement the model on two cohorts of human connectome networks and investigate the effects of varying the global information bias on the network’s communication cost. We identify routing strategies that approach a (highly efficient) shortest-path communication process with a relatively small global information bias on the system’s dynamics. Moreover, we show that the cost of routing messages from and to hub nodes varies as a function of the global information bias driving the system’s dynamics. Finally, we implement the model to identify individual subject differences from a communication dynamics point of view. The present framework departs from the classical shortest paths vs. diffusion dichotomy, unifying both models under a single family of dynamical processes that differ by the extent to which global information about the network topology influences the routing patterns of neural signals traversing the network.

Brain network communication is typically approached from the perspective of the length of inferred paths and the cost of building and maintaining network connections. However, these analyses often disregard the dynamical processes taking place on the network and the additional costs that these processes incur. Here, we introduce a framework to study communication-cost trade-offs on a broad range of communication processes modeled as biased random walks. We control the system’s dynamics that dictates the flow of messages traversing a network by biasing node’s routing strategies with different degrees of “knowledge” about the topology of the network. On the human connectome, this framework uncovers a spectrum of dynamic communication processes, some of which can achieve efficient routing strategies at low informational cost.

Pareto Optimality in Multilayer Network Growth

We model the formation of multilayer transportation networks as a multiobjective optimization process, where service providers compete for passengers, and the creation of routes is determined by a multiobjective cost function encoding a trade-off between efficiency and competition. The resulting model reproduces well real-world systems as diverse as airplane, train, and bus networks, thus suggesting that such systems are indeed compatible with the proposed local optimization mechanisms. In the specific case of airline transportation systems, we show that the networks of routes operated by each company are placed very close to the theoretical Pareto front in the efficiency-competition plane, and that most of the largest carriers of a continent belong to the corresponding Pareto front. Our results shed light on the fundamental role played by multiobjective optimization principles in shaping the structure of large-scale multilayer transportation systems, and provide novel insights to service providers on the strategies for the smart selection of novel routes.

Mapping higher-order relations between brain structure and function with embedded vector representations of connectomes

Connectomics generates comprehensive maps of brain networks, represented as nodes and their pairwise connections. The functional roles of nodes are defined by their direct and indirect connectivity with the rest of the network. However, the network context is not directly accessible at the level of individual nodes. Similar problems in language processing have been addressed with algorithms such as word2vec that create embeddings of words and their relations in a meaningful low-dimensional vector space. Here we apply this approach to create embedded vector representations of brain networks or connectome embeddings (CE). CE can characterize correspondence relations among brain regions, and can be used to infer links that are lacking from the original structural diffusion imaging, e.g., inter-hemispheric homotopic connections. Moreover, we construct predictive deep models of functional and structural connectivity, and simulate network-wide lesion effects using the face processing system as our application domain. We suggest that CE offers a novel approach to revealing relations between connectome structure and function.

Navigation of brain networks

We show that the combination of topology and geometry in mammalian cortical networks allows for near-optimal decentralized communication under navigation routing. Following a simple propagation rule based on local knowledge of the distance between cortical regions, we demonstrate that brain networks can be successfully navigated with efficiency that is comparable to shortest paths routing. This finding helps to conciliate the major progress achieved over more than a decade of connectomics research, under the assumption of communication via shortest paths, with recent questions raised by the biologically unrealistic requirements involved in the computation of optimal routes. Our results reiterate the importance of the brain’s spatial embedding, suggesting a three-way relationship between connectome geometry, topology, and communication.

Understanding the mechanisms of neural communication in large-scale brain networks remains a major goal in neuroscience. We investigated whether navigation is a parsimonious routing model for connectomics. Navigating a network involves progressing to the next node that is closest in distance to a desired destination. We developed a measure to quantify navigation efficiency and found that connectomes in a range of mammalian species (human, mouse, and macaque) can be successfully navigated with near-optimal efficiency (>80% of optimal efficiency for typical connection densities). Rewiring network topology or repositioning network nodes resulted in 45–60% reductions in navigation performance. We found that the human connectome cannot be progressively randomized or clusterized to result in topologies with substantially improved navigation performance (>5%), suggesting a topological balance between regularity and randomness that is conducive to efficient navigation. Navigation was also found to (i) promote a resource-efficient distribution of the information traffic load, potentially relieving communication bottlenecks, and (ii) explain significant variation in functional connectivity. Unlike commonly studied communication strategies in connectomics, navigation does not mandate assumptions about global knowledge of network topology. We conclude that the topology and geometry of brain networks are conducive to efficient decentralized communication.

Developmental increases in white matter network controllability support a growing diversity of brain dynamics

As the human brain develops, it increasingly supports coordinated control of neural activity. The mechanism by which white matter evolves to support this coordination is not well understood. Here we use a network representation of diffusion imaging data from 882 youth ages 8-22 to show that white matter connectivity becomes increasingly optimized for a diverse range of predicted dynamics in development. Notably, stable controllers in subcortical areas are negatively related to cognitive performance. Investigating structural mechanisms supporting these changes, we simulate network evolution with a set of growth rules. We find that all brain networks are structured in a manner highly optimized for network control, with distinct control mechanisms predicted in child vs. older youth. We demonstrate that our results cannot be explained by changes in network modularity. This work reveals a possible mechanism of human brain development that preferentially optimizes dynamic network control over static network architecture.

Generative models for network neuroscience: prospects and promise

Network neuroscience is the emerging discipline concerned with investigating the complex patterns of interconnections found in neural systems, and identifying principles with which to understand them. Within this discipline, one particularly powerful approach is network generative modelling, in which wiring rules are algorithmically implemented to produce synthetic network architectures with the same properties as observed in empirical network data. Successful models can highlight the principles by which a network is organized and potentially uncover the mechanisms by which it grows and develops. Here, we review the prospects and promise of generative models for network neuroscience. We begin with a primer on network generative models, with a discussion of compressibility and predictability, and utility in intuiting mechanisms, followed by a short history on their use in network science, broadly. We then discuss generative models in practice and application, paying particular attention to the critical need for cross-validation. Next, we review generative models of biological neural networks, both at the cellular and large-scale level, and across a variety of species including Caenorhabditis elegans, Drosophila, mouse, rat, cat, macaque and human. We offer a careful treatment of a few relevant distinctions, including differences between generative models and null models, sufficiency and redundancy, inferring and claiming mechanism, and functional and structural connectivity. We close with a discussion of future directions, outlining exciting frontiers both in empirical data collection efforts as well as in method and theory development that, together, further the utility of the generative network modelling approach for network neuroscience.

Emerging Frontiers of Neuroengineering: A Network Science of Brain Connectivity

Neuroengineering is faced with unique challenges in repairing or replacing complex neural systems that are composed of many interacting parts. These interactions form intricate patterns over large spatiotemporal scales and produce emergent behaviors that are difficult to predict from individual elements. Network science provides a particularly appropriate framework in which to study and intervene in such systems by treating neural elements (cells, volumes) as nodes in a graph and neural interactions (synapses, white matter tracts) as edges in that graph. Here, we review the emerging discipline of network neuroscience, which uses and develops tools from graph theory to better understand and manipulate neural systems from micro- to macroscales. We present examples of how human brain imaging data are being modeled with network analysis and underscore potential pitfalls. We then highlight current computational and theoretical frontiers and emphasize their utility in informing diagnosis and monitoring, brain-machine interfaces, and brain stimulation. A flexible and rapidly evolving enterprise, network neuroscience provides a set of powerful approaches and fundamental insights that are critical for the neuroengineer's tool kit.

The modular organization of human anatomical brain networks: Accounting for the cost of wiring

Brain networks are expected to be modular. However, existing techniques for estimating a network's modules make it difficult to assess the influence of organizational principles such as wiring cost reduction on the detected modules. Here we present a modification of an existing module detection algorithm that allowed us to focus on connections that are unexpected under a cost-reduction wiring rule and to identify modules from among these connections. We applied this technique to anatomical brain networks and showed that the modules we detected differ from those detected using the standard technique. We demonstrated that these novel modules are spatially distributed, exhibit unique functional fingerprints, and overlap considerably with rich clubs, giving rise to an alternative and complementary interpretation of the functional roles of specific brain regions. Finally, we demonstrated that, using the modified module detection approach, we can detect modules in a developmental dataset that track normative patterns of maturation. Collectively, these findings support the hypothesis that brain networks are composed of modules and provide additional insight into the function of those modules.

The geometric nature of weights in real complex networks

The topology of many real complex networks has been conjectured to be embedded in hidden metric spaces, where distances between nodes encode their likelihood of being connected. Besides of providing a natural geometrical interpretation of their complex topologies, this hypothesis yields the recipe for sustainable Internet's routing protocols, sheds light on the hierarchical organization of biochemical pathways in cells, and allows for a rich characterization of the evolution of international trade. Here we present empirical evidence that this geometric interpretation also applies to the weighted organization of real complex networks. We introduce a very general and versatile model and use it to quantify the level of coupling between their topology, their weights and an underlying metric space. Our model accurately reproduces both their topology and their weights, and our results suggest that the formation of connections and the assignment of their magnitude are ruled by different processes.

Complex networks have been conjectured to be hidden in metric spaces, which offer geometric interpretation of networks' topologies. Here the authors extend this concept to weighted networks, providing empirical evidence on the metric natures of weights, which are shown to be reproducible by a gravity model.

Functional complexity emerging from anatomical constraints in the brain: the significance of network modularity and rich-clubs

The large-scale structural ingredients of the brain and neural connectomes have been identified in recent years. These are, similar to the features found in many other real networks: the arrangement of brain regions into modules and the presence of highly connected regions (hubs) forming rich-clubs. Here, we examine how modules and hubs shape the collective dynamics on networks and we find that both ingredients lead to the emergence of complex dynamics. Comparing the connectomes of C. elegans, cats, macaques and humans to surrogate networks in which either modules or hubs are destroyed, we find that functional complexity always decreases in the perturbed networks. A comparison between simulated and empirically obtained resting-state functional connectivity indicates that the human brain, at rest, lies in a dynamical state that reflects the largest complexity its anatomical connectome can host. Last, we generalise the topology of neural connectomes into a new hierarchical network model that successfully combines modular organisation with rich-club forming hubs. This is achieved by centralising the cross-modular connections through a preferential attachment rule. Our network model hosts more complex dynamics than other hierarchical models widely used as benchmarks.

Building an EEG-fMRI Multi-Modal Brain Graph: A Concurrent EEG-fMRI Study

The topological architecture of brain connectivity has been well-characterized by graph theory based analysis. However, previous studies have primarily built brain graphs based on a single modality of brain imaging data. Here we develop a framework to construct multi-modal brain graphs using concurrent EEG-fMRI data which are simultaneously collected during eyes open (EO) and eyes closed (EC) resting states. FMRI data are decomposed into independent components with associated time courses by group independent component analysis (ICA). EEG time series are segmented, and then spectral power time courses are computed and averaged within 5 frequency bands (delta; theta; alpha; beta; low gamma). EEG-fMRI brain graphs, with EEG electrodes and fMRI brain components serving as nodes, are built by computing correlations within and between fMRI ICA time courses and EEG spectral power time courses. Dynamic EEG-fMRI graphs are built using a sliding window method, versus static ones treating the entire time course as stationary. In global level, static graph measures and properties of dynamic graph measures are different across frequency bands and are mainly showing higher values in eyes closed than eyes open. Nodal level graph measures of a few brain components are also showing higher values during eyes closed in specific frequency bands. Overall, these findings incorporate fMRI spatial localization and EEG frequency information which could not be obtained by examining only one modality. This work provides a new approach to examine EEG-fMRI associations within a graph theoretic framework with potential application to many topics.

Small-World Brain Networks Revisited

It is nearly 20 years since the concept of a small-world network was first quantitatively defined, by a combination of high clustering and short path length; and about 10 years since this metric of complex network topology began to be widely applied to analysis of neuroimaging and other neuroscience data as part of the rapid growth of the new field of connectomics. Here, we review briefly the foundational concepts of graph theoretical estimation and generation of small-world networks. We take stock of some of the key developments in the field in the past decade and we consider in some detail the implications of recent studies using high-resolution tract-tracing methods to map the anatomical networks of the macaque and the mouse. In doing so, we draw attention to the important methodological distinction between topological analysis of binary or unweighted graphs, which have provided a popular but simple approach to brain network analysis in the past, and the topology of weighted graphs, which retain more biologically relevant information and are more appropriate to the increasingly sophisticated data on brain connectivity emerging from contemporary tract-tracing and other imaging studies. We conclude by highlighting some possible future trends in the further development of weighted small-worldness as part of a deeper and broader understanding of the topology and the functional value of the strong and weak links between areas of mammalian cortex.

Topological and geometric measurements of force-chain structure

Developing quantitative methods for characterizing structural properties of force chains in densely packed granular media is an important step toward understanding or predicting large-scale physical properties of a packing. A promising framework in which to develop such methods is network science, which can be used to translate particle locations and force contacts into a graph in which particles are represented by nodes and forces between particles are represented by weighted edges. Recent work applying network-based community-detection techniques to extract force chains opens the door to developing statistics of force-chain structure, with the goal of identifying geometric and topological differences across packings, and providing a foundation on which to build predictions of bulk material properties from mesoscale network features. Here we discuss a trio of related but fundamentally distinct measurements of the mesoscale structure of force chains in two-dimensional (2D) packings, including a statistic derived using tools from algebraic topology, which together provide a tool set for the analysis of force chain architecture. We demonstrate the utility of this tool set by detecting variations in force-chain architecture with pressure. Collectively, these techniques can be generalized to 3D packings, and to the assessment of continuous deformations of packings under stress or strain.

Generative models of the human connectome

The human connectome represents a network map of the brain's wiring diagram and the pattern into which its connections are organized is thought to play an important role in cognitive function. The generative rules that shape the topology of the human connectome remain incompletely understood. Earlier work in model organisms has suggested that wiring rules based on geometric relationships (distance) can account for many but likely not all topological features. Here we systematically explore a family of generative models of the human connectome that yield synthetic networks designed according to different wiring rules combining geometric and a broad range of topological factors. We find that a combination of geometric constraints with a homophilic attachment mechanism can create synthetic networks that closely match many topological characteristics of individual human connectomes, including features that were not included in the optimization of the generative model itself. We use these models to investigate a lifespan dataset and show that, with age, the model parameters undergo progressive changes, suggesting a rebalancing of the generative factors underlying the connectome across the lifespan.

The Structural and Functional Connectome and Prediction of Risk for Cognitive Impairment in Older Adults

The human connectome refers to a comprehensive description of the brain's structural and functional connections in terms of brain networks. As the field of brain connectomics has developed, data acquisition, subsequent processing and modeling, and ultimately the representation of the connectome have become better defined and integrated with network science approaches. In this way, the human connectome has provided a way to elucidate key features of not only the healthy brain but also diseased brains. The field has quickly evolved, offering insights into network disruptions that are characteristic for specific neurodegenerative disorders. In this paper, we provide a brief review of the field of brain connectomics, as well as a more in-depth survey of recent studies that have provided new insights into brain network pathologies, including those found in Alzheimer's disease (AD), patients with mild cognitive impairment (MCI), and finally in people classified as being "at risk". Until the emergence of brain connectomics, most previous studies had assessed neurodegenerative diseases mainly by focusing on specific and dispersed locales in the brain. Connectomics-based approaches allow us to model the brain as a network, which allows for inferences about how dynamic changes in brain function would be affected in relation to structural changes. In fact, looking at diseases using network theory gives rise to new hypotheses on mechanisms of pathophysiology and clinical symptoms. Finally, we discuss the future of this field and how understanding both the functional and structural connectome can aid in gaining sharper insight into changes in biological brain networks associated with cognitive impairment and dementia.

Network morphospace

The structure of complex networks has attracted much attention in recent years. It has been noted that many real-world examples of networked systems share a set of common architectural features. This raises important questions about their origin, for example whether such network attributes reflect common design principles or constraints imposed by selectional forces that have shaped the evolution of network topology. Is it possible to place the many patterns and forms of complex networks into a common space that reveals their relations, and what are the main rules and driving forces that determine which positions in such a space are occupied by systems that have actually evolved? We suggest that these questions can be addressed by combining concepts from two currently relatively unconnected fields. One is theoretical morphology, which has conceptualized the relations between morphological traits defined by mathematical models of biological form. The second is network science, which provides numerous quantitative tools to measure and classify different patterns of local and global network architecture across disparate types of systems. Here, we explore a new theoretical concept that lies at the intersection between both fields, the 'network morphospace'. Defined by axes that represent specific network traits, each point within such a space represents a location occupied by networks that share a set of common 'morphological' characteristics related to aspects of their connectivity. Mapping a network morphospace reveals the extent to which the space is filled by existing networks, thus allowing a distinction between actual and impossible designs and highlighting the generative potential of rules and constraints that pervade the evolution of complex systems.