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1.  Network morphospace 
Journal of the Royal Society Interface  2015;12(103):20140881.
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.
doi:10.1098/rsif.2014.0881
PMCID: PMC4305402  PMID: 25540237
evolution; complexity; graph theory; Pareto optimality; brain connectivity
2.  A Network Convergence Zone in the Hippocampus 
PLoS Computational Biology  2014;10(12):e1003982.
The hippocampal formation is a key structure for memory function in the brain. The functional anatomy of the brain suggests that the hippocampus may be a convergence zone, as it receives polysensory input from distributed association areas throughout the neocortex. However, recent quantitative graph-theoretic analyses of the static large-scale connectome have failed to demonstrate the centrality of the hippocampus; in the context of the whole brain, the hippocampus is not among the most connected or reachable nodes. Here we show that when communication dynamics are taken into account, the hippocampus is a key hub in the connectome. Using a novel computational model, we demonstrate that large-scale brain network topology is organized to funnel and concentrate information flow in the hippocampus, supporting the long-standing hypothesis that this region acts as a critical convergence zone. Our results indicate that the functional capacity of the hippocampus is shaped by its embedding in the large-scale connectome.
Author Summary
The hippocampus is a key structure in the brain, particularly for memory function. The functional anatomy of brain networks suggests that the hippocampus should be a hub in the large-scale connectome, yet so far quantitative network analyses have failed to find any evidence of such a role. Here we show that when communication dynamics are taken into account, the hippocampus is a critical hub in the connectome. Using an anatomically realistic structural network and a novel computational model, we demonstrate that the topology of the brain is organized in a way that allows information to be funneled towards the hippocampus. Despite the fact that the hippocampus is neither among the most connected or most central nodes in the network, it receives a disproportionately high level of information flow and acts a convergence zone for the network. This convergence effect demonstrates that the architecture of the cerebral cortex facilitates integration and binding of information in the hippocampus.
doi:10.1371/journal.pcbi.1003982
PMCID: PMC4256084  PMID: 25474350
3.  Robust estimation of fractal measures for characterizing the structural complexity of the human brain: optimization and reproducibility 
NeuroImage  2013;83:646-657.
High-resolution isotropic three-dimensional reconstructions of human brain gray and white matter structures can be characterized to quantify aspects of their shape, volume and topological complexity. In particular, methods based on fractal analysis have been applied in neuroimaging studies to quantify the structural complexity of the brain in both healthy and impaired conditions. The usefulness of such measures for characterizing individual differences in brain structure critically depends on their within-subject reproducibility in order to allow the robust detection of between-subject differences. This study analyzes key analytic parameters of three fractal-based methods that rely on the box-counting algorithm with the aim to maximize within-subject reproducibility of the fractal characterizations of different brain objects, including the pial surface, the cortical ribbon volume, the white matter volume and the grey matter/white matter boundary. Two separate datasets originating from different imaging centers were analyzed, comprising, 50 subjects with three and 24 subjects with four successive scanning sessions per subject, respectively. The reproducibility of fractal measures was statistically assessed by computing their intra-class correlations. Results reveal differences between different fractal estimators and allow the identification of several parameters that are critical for high reproducibility. Highest reproducibility with intra-class correlations in the range of 0.9–0.95 is achieved with the correlation dimension. Further analyses of the fractal dimensions of parcellated cortical and subcortical gray matter regions suggest robustly estimated and region-specific patterns of individual variability. These results are valuable for defining appropriate parameter configurations when studying changes in fractal descriptors of human brain structure, for instance in studies of neurological diseases that do not allow repeated measurements or for disease-course longitudinal studies.
doi:10.1016/j.neuroimage.2013.06.072
PMCID: PMC3897251  PMID: 23831414
4.  Using Pareto optimality to explore the topology and dynamics of the human connectome 
Graph theory has provided a key mathematical framework to analyse the architecture of human brain networks. This architecture embodies an inherently complex relationship between connection topology, the spatial arrangement of network elements, and the resulting network cost and functional performance. An exploration of these interacting factors and driving forces may reveal salient network features that are critically important for shaping and constraining the brain's topological organization and its evolvability. Several studies have pointed to an economic balance between network cost and network efficiency with networks organized in an ‘economical’ small-world favouring high communication efficiency at a low wiring cost. In this study, we define and explore a network morphospace in order to characterize different aspects of communication efficiency in human brain networks. Using a multi-objective evolutionary approach that approximates a Pareto-optimal set within the morphospace, we investigate the capacity of anatomical brain networks to evolve towards topologies that exhibit optimal information processing features while preserving network cost. This approach allows us to investigate network topologies that emerge under specific selection pressures, thus providing some insight into the selectional forces that may have shaped the network architecture of existing human brains.
doi:10.1098/rstb.2013.0530
PMCID: PMC4150305  PMID: 25180308
graph theory; network science; brain connectivity; diffusion imaging
5.  On nodes and modes in resting state fMRI 
Neuroimage  2014;99(100):533-547.
This paper examines intrinsic brain networks in light of recent developments in the characterisation of resting state fMRI timeseries — and simulations of neuronal fluctuations based upon the connectome. Its particular focus is on patterns or modes of distributed activity that underlie functional connectivity. We first demonstrate that the eigenmodes of functional connectivity – or covariance among regions or nodes – are the same as the eigenmodes of the underlying effective connectivity, provided we limit ourselves to symmetrical connections. This symmetry constraint is motivated by appealing to proximity graphs based upon multidimensional scaling. Crucially, the principal modes of functional connectivity correspond to the dynamically unstable modes of effective connectivity that decay slowly and show long term memory. Technically, these modes have small negative Lyapunov exponents that approach zero from below. Interestingly, the superposition of modes – whose exponents are sampled from a power law distribution – produces classical 1/f (scale free) spectra. We conjecture that the emergence of dynamical instability – that underlies intrinsic brain networks – is inevitable in any system that is separated from external states by a Markov blanket. This conjecture appeals to a free energy formulation of nonequilibrium steady-state dynamics. The common theme that emerges from these theoretical considerations is that endogenous fluctuations are dominated by a small number of dynamically unstable modes. We use this as the basis of a dynamic causal model (DCM) of resting state fluctuations — as measured in terms of their complex cross spectra. In this model, effective connectivity is parameterised in terms of eigenmodes and their Lyapunov exponents — that can also be interpreted as locations in a multidimensional scaling space. Model inversion provides not only estimates of edges or connectivity but also the topography and dimensionality of the underlying scaling space. Here, we focus on conceptual issues with simulated fMRI data and provide an illustrative application using an empirical multi-region timeseries.
Highlights
•This paper presents a formal characterisation of [eigen] modes in resting state fMRI.•It relates these to dynamical instabilities in neuronal (inferential) processing.•It uses the equivalence between eigenmodes of functional and effective connectivity for dynamical causal modelling.
doi:10.1016/j.neuroimage.2014.05.056
PMCID: PMC4121089  PMID: 24862075
Dynamic causal modelling; Effective connectivity; Functional connectivity; Resting state; fMRI; Proximity graph; Free energy; Criticality; Self-organisation; Lyapunov exponents; Bayesian
6.  Patterns of information flow in local cortical networks 
BMC Neuroscience  2014;15(Suppl 1):P213.
doi:10.1186/1471-2202-15-S1-P213
PMCID: PMC4126424
7.  Impulsivity and the Modular Organization of Resting-State Neural Networks 
Cerebral Cortex (New York, NY)  2012;23(6):1444-1452.
Impulsivity is a complex trait associated with a range of maladaptive behaviors, including many forms of psychopathology. Previous research has implicated multiple neural circuits and neurotransmitter systems in impulsive behavior, but the relationship between impulsivity and organization of whole-brain networks has not yet been explored. Using graph theory analyses, we characterized the relationship between impulsivity and the functional segregation (“modularity”) of the whole-brain network architecture derived from resting-state functional magnetic resonance imaging (fMRI) data. These analyses revealed remarkable differences in network organization across the impulsivity spectrum. Specifically, in highly impulsive individuals, regulatory structures including medial and lateral regions of the prefrontal cortex were isolated from subcortical structures associated with appetitive drive, whereas these brain areas clustered together within the same module in less impulsive individuals. Further exploration of the modular organization of whole-brain networks revealed novel shifts in the functional connectivity between visual, sensorimotor, cortical, and subcortical structures across the impulsivity spectrum. The current findings highlight the utility of graph theory analyses of resting-state fMRI data in furthering our understanding of the neurobiological architecture of complex behaviors.
doi:10.1093/cercor/bhs126
PMCID: PMC3643719  PMID: 22645253
Barratt Impulsiveness Scale; functional connectivity; graph theory; neuroimaging; resting-state fMRI
8.  Mechanisms of Zero-Lag Synchronization in Cortical Motifs 
PLoS Computational Biology  2014;10(4):e1003548.
Zero-lag synchronization between distant cortical areas has been observed in a diversity of experimental data sets and between many different regions of the brain. Several computational mechanisms have been proposed to account for such isochronous synchronization in the presence of long conduction delays: Of these, the phenomenon of “dynamical relaying” – a mechanism that relies on a specific network motif – has proven to be the most robust with respect to parameter mismatch and system noise. Surprisingly, despite a contrary belief in the community, the common driving motif is an unreliable means of establishing zero-lag synchrony. Although dynamical relaying has been validated in empirical and computational studies, the deeper dynamical mechanisms and comparison to dynamics on other motifs is lacking. By systematically comparing synchronization on a variety of small motifs, we establish that the presence of a single reciprocally connected pair – a “resonance pair” – plays a crucial role in disambiguating those motifs that foster zero-lag synchrony in the presence of conduction delays (such as dynamical relaying) from those that do not (such as the common driving triad). Remarkably, minor structural changes to the common driving motif that incorporate a reciprocal pair recover robust zero-lag synchrony. The findings are observed in computational models of spiking neurons, populations of spiking neurons and neural mass models, and arise whether the oscillatory systems are periodic, chaotic, noise-free or driven by stochastic inputs. The influence of the resonance pair is also robust to parameter mismatch and asymmetrical time delays amongst the elements of the motif. We call this manner of facilitating zero-lag synchrony resonance-induced synchronization, outline the conditions for its occurrence, and propose that it may be a general mechanism to promote zero-lag synchrony in the brain.
Author Summary
Understanding large-scale neuronal dynamics – and how they relate to the cortical anatomy – is one of the key areas of neuroscience research. Despite a wealth of recent research, the key principles of this relationship have yet to be established. Here we employ computational modeling to study neuronal dynamics on small subgraphs – or motifs – across a hierarchy of spatial scales. We establish a novel organizing principle that we term a “resonance pair” (two mutually coupled nodes), which promotes stable, zero-lag synchrony amongst motif nodes. The bidirectional coupling between a resonance pair acts to mutually adjust their dynamics onto a common and relatively stable synchronized regime, which then propagates and stabilizes the synchronization of other nodes within the motif. Remarkably, we find that this effect can propagate along chains of coupled nodes and hence holds the potential to promote stable zero-lag synchrony in larger sub-networks of cortical systems. Our findings hence suggest a potential unifying account of the existence of zero-lag synchrony, an important phenomenon that may underlie crucial cognitive processes in the brain. Moreover, such pairs of mutually coupled oscillators are found in a wide variety of physical and biological systems suggesting a new, broadly relevant and unifying principle.
doi:10.1371/journal.pcbi.1003548
PMCID: PMC3998884  PMID: 24763382
9.  Communication Efficiency and Congestion of Signal Traffic in Large-Scale Brain Networks 
PLoS Computational Biology  2014;10(1):e1003427.
The complex connectivity of the cerebral cortex suggests that inter-regional communication is a primary function. Using computational modeling, we show that anatomical connectivity may be a major determinant for global information flow in brain networks. A macaque brain network was implemented as a communication network in which signal units flowed between grey matter nodes along white matter paths. Compared to degree-matched surrogate networks, information flow on the macaque brain network was characterized by higher loss rates, faster transit times and lower throughput, suggesting that neural connectivity may be optimized for speed rather than fidelity. Much of global communication was mediated by a “rich club” of hub regions: a sub-graph comprised of high-degree nodes that are more densely interconnected with each other than predicted by chance. First, macaque communication patterns most closely resembled those observed for a synthetic rich club network, but were less similar to those seen in a synthetic small world network, suggesting that the former is a more fundamental feature of brain network topology. Second, rich club regions attracted the most signal traffic and likewise, connections between rich club regions carried more traffic than connections between non-rich club regions. Third, a number of rich club regions were significantly under-congested, suggesting that macaque connectivity actively shapes information flow, funneling traffic towards some nodes and away from others. Together, our results indicate a critical role of the rich club of hub nodes in dynamic aspects of global brain communication.
Author Summary
A fundamental question in systems neuroscience is how the structural connectivity of the cerebral cortex shapes global communication. Here, using computational modeling in conjunction with an anatomically realistic structural network, we show that cortico-cortical communication is constrained by high-level features of brain network topology. We find that neural network topology is configured in a way that prioritizes speed of information flow over reliability and total throughput. The defining characteristic of the information processing architecture of the network is a densely interconnected rich club of hub nodes. Namely, rich club nodes and connections between rich club nodes absorb the greatest proportion of total signal traffic. In addition, rich club connectivity appears to actively shape information flow, whereby signal traffic is biased towards some nodes and away from others. Finally, synthetic networks containing a rich club could almost perfectly reproduce the information flow patterns of the real anatomical network. Altogether, our data demonstrate that a central collective of highly interconnected hubs serves to facilitate cortico-cortical communication. By simulating communication on a static structural network we have revealed a dynamic aspect of the global information processing architecture and the critical role played by the rich club of hub nodes.
doi:10.1371/journal.pcbi.1003427
PMCID: PMC3886893  PMID: 24415931
10.  Multi-scale integration and predictability in resting state brain activity 
The human brain displays heterogeneous organization in both structure and function. Here we develop a method to characterize brain regions and networks in terms of information-theoretic measures. We look at how these measures scale when larger spatial regions as well as larger connectome sub-networks are considered. This framework is applied to human brain fMRI recordings of resting-state activity and DSI-inferred structural connectivity. We find that strong functional coupling across large spatial distances distinguishes functional hubs from unimodal low-level areas, and that this long-range functional coupling correlates with structural long-range efficiency on the connectome. We also find a set of connectome regions that are both internally integrated and coupled to the rest of the brain, and which resemble previously reported resting-state networks. Finally, we argue that information-theoretic measures are useful for characterizing the functional organization of the brain at multiple scales.
doi:10.3389/fninf.2014.00066
PMCID: PMC4109611  PMID: 25104933
human connectome; resting-state; integrative regions; information theory; multivariate mutual information; complexity measures
11.  Dynamic functional connectivity: Promise, issues, and interpretations 
NeuroImage  2013;80:10.1016/j.neuroimage.2013.05.079.
The brain must dynamically integrate, coordinate, and respond to internal and external stimuli across multiple time scales. Non-invasive measurements of brain activity with fMRI have greatly advanced our understanding of the large-scale functional organization supporting these fundamental features of brain function. Conclusions from previous resting-state fMRI investigations were based upon static descriptions of functional connectivity (FC), and only recently studies have begun to capitalize on the wealth of information contained within the temporal features of spontaneous BOLD FC. Emerging evidence suggests that dynamic FC metrics may index changes in macroscopic neural activity patterns underlying critical aspects of cognition and behavior, though limitations with regard to analysis and interpretation remain. Here, we review recent findings, methodological considerations, neural and behavioral correlates, and future directions in the emerging field of dynamic FC investigations.
doi:10.1016/j.neuroimage.2013.05.079
PMCID: PMC3807588  PMID: 23707587
Functional connectivity; Resting state; Dynamics; Spontaneous activity; Functional MRI (fMRI); Fluctuations
12.  Structure and function of complex brain networks 
An increasing number of theoretical and empirical studies approach the function of the human brain from a network perspective. The analysis of brain networks is made feasible by the development of new imaging acquisition methods as well as new tools from graph theory and dynamical systems. This review surveys some of these methodological advances and summarizes recent findings on the architecture of structural and functional brain networks. Studies of the structural connectome reveal several modules or network communities that are interlinked by hub regions mediating communication processes between modules. Recent network analyses have shown that network hubs form a densely linked collective called a “rich club,” centrally positioned for attracting and dispersing signal traffic. In parallel, recordings of resting and task-evoked neural activity have revealed distinct resting-state networks that contribute to functions in distinct cognitive domains. Network methods are increasingly applied in a clinical context, and their promise for elucidating neural substrates of brain and mental disorders is discussed.
PMCID: PMC3811098  PMID: 24174898
neuroimaging; connectome; graph theory; resting state; diffusion imaging; tractography; functional MRI
13.  Zero-lag synchronization in cortical motifs 
BMC Neuroscience  2013;14(Suppl 1):P37.
doi:10.1186/1471-2202-14-S1-P37
PMCID: PMC3704765
15.  Altered Functional and Structural Connectivity Networks in Psychogenic Non-Epileptic Seizures 
PLoS ONE  2013;8(5):e63850.
Psychogenic non-epileptic seizures (PNES) are paroxysmal behaviors that resemble epileptic seizures but lack abnormal electrical activity. Recent studies suggest aberrant functional connectivity involving specific brain regions in PNES. Little is known, however, about alterations of topological organization of whole-brain functional and structural connectivity networks in PNES. We constructed functional connectivity networks from resting-state functional MRI signal correlations and structural connectivity networks from diffusion tensor imaging tractography in 17 PNES patients and 20 healthy controls. Graph theoretical analysis was employed to compute network properties. Moreover, we investigated the relationship between functional and structural connectivity networks. We found that PNES patients exhibited altered small-worldness in both functional and structural networks and shifted towards a more regular (lattice-like) organization, which could serve as a potential imaging biomarker for PNES. In addition, many regional characteristics were altered in structural connectivity network, involving attention, sensorimotor, subcortical and default-mode networks. These regions with altered nodal characteristics likely reflect disease-specific pathophysiology in PNES. Importantly, the coupling strength of functional-structural connectivity was decreased and exhibited high sensitivity and specificity to differentiate PNES patients from healthy controls, suggesting that the decoupling strength of functional-structural connectivity might be an important characteristic reflecting the mechanisms of PNES. This is the first study to explore the altered topological organization in PNES combining functional and structural connectivity networks, providing a new way to understand the pathophysiological mechanisms of PNES.
doi:10.1371/journal.pone.0063850
PMCID: PMC3661726  PMID: 23717498
16.  Using Confirmatory Factor Analysis to Measure Contemporaneous Activation of Defined Neuronal Networks in Functional Magnetic Resonance Imaging 
Neuroimage  2012;60(4):1982-1991.
Functional neuroimaging often generates large amounts of data on regions of interest. Such data can be addressed effectively with a widely-used statistical technique based on measurement theory that has not yet been applied to neuroimaging. Confirmatory factor analysis is a convenient hypothesis-driven modeling environment that can be used to conduct formal statistical tests comparing alternative hypotheses regarding the elements of putative neuronal networks. In such models, measures of each activated region of interest are treated as indicators of an underlying latent construct that represents the contemporaneous activation of the elements in the network. As such, confirmatory factor analysis focuses analyses on the activation of hypothesized networks as a whole, improves statistical power by modeling measurement error, and provides a theory-based approach to data reduction with a robust statistical basis. This approach is illustrated using data on seven regions of interest in a hypothesized mesocorticostriatal reward system in a sample of 262 adult volunteers assessed during a card-guessing reward task. A latent construct reflecting contemporaneous activation of the reward system was found to be significantly associated with a latent construct measuring impulsivity, particularly in males.
doi:10.1016/j.neuroimage.2012.02.002
PMCID: PMC3321108  PMID: 22348884
confirmatory factor analysis; functional neuroimaging; measurement theory; mesocorticostriatal reward system; impulsivity; sex differences
17.  Structural and Functional Aspects Relating to Cost and Benefit of Rich Club Organization in the Human Cerebral Cortex 
Cerebral Cortex (New York, NY)  2013;24(9):2258-2267.
Recent findings have demonstrated that a small set of highly connected brain regions may play a central role in enabling efficient communication between cortical regions, together forming a densely interconnected “rich club.” However, the density and spatial layout of the rich club also suggest that it constitutes a costly feature of brain architecture. Here, combining anatomical T1, diffusion tensor imaging, magnetic transfer imaging, and functional MRI, several aspects of structural and functional connectivity of the brain's rich club were examined. Our findings suggest that rich club regions and rich club connections exhibit high levels of wiring volume, high levels of white matter organization, high levels of metabolic energy usage, long maturational trajectories, more variable regional time series, and more inter-regional functional couplings. Taken together, these structural and functional measures extend the notion that rich club organization represents a high-cost feature of brain architecture that puts a significant strain on brain resources. The high cost of the rich club may, however, be offset by significant functional benefits that the rich club confers to the brain network as a whole.
doi:10.1093/cercor/bht064
PMCID: PMC4128699  PMID: 23551922
connectivity; connectome; diffusion tensor imaging; fMRI; rich club
18.  Disturbed resting state EEG synchronization in bipolar disorder: A graph-theoretic analysis☆ 
NeuroImage : Clinical  2013;2:414-423.
Disruption of functional connectivity may be a key feature of bipolar disorder (BD) which reflects disturbances of synchronization and oscillations within brain networks. We investigated whether the resting electroencephalogram (EEG) in patients with BD showed altered synchronization or network properties. Resting-state EEG was recorded in 57 BD type-I patients and 87 healthy control subjects. Functional connectivity between pairs of EEG channels was measured using synchronization likelihood (SL) for 5 frequency bands (δ, θ, α, β, and γ). Graph-theoretic analysis was applied to SL over the electrode array to assess network properties. BD patients showed a decrease of mean synchronization in the alpha band, and the decreases were greatest in fronto-central and centro-parietal connections. In addition, the clustering coefficient and global efficiency were decreased in BD patients, whereas the characteristic path length increased. We also found that the normalized characteristic path length and small-worldness were significantly correlated with depression scores in BD patients. These results suggest that BD patients show impaired neural synchronization at rest and a disruption of resting-state functional connectivity.
Highlights
•Global synchronization of BD patients was reduced in the alpha-band at resting.•De-synchronized connectivity was localized in fronto-centro-parietal connections.•Global topology of BD had decreased network clustering and increased path length.•BD showed the less efficient network processing.•Network characteristics of BD patients were associated with depression severity.
doi:10.1016/j.nicl.2013.03.007
PMCID: PMC3777715  PMID: 24179795
b, node betweenness centrality; BD, bipolar disorder; C, clustering coefficients; DSM-IV, diagnostic and statistical manual of mental disorders, the 4th-edition; DTI, diffusion tensor imaging (image); EEG, electroencephalogram; EOG, electrooculogram; Eg, global efficiency; El, local efficiency; FA, fractional anisotropy; FDR, false discovery rate; fMRI, functional magnetic resonance imaging; GABA, gamma-amino butyric acid; L, characteristic path length; MADRS, Montgomery–Asberg Depression Rating Scale; MEG, magnetoencephalogram; MRI, magnetic resonance imaging; NBS, network-based statistics; NC, normal healthy control; PLI, phase lag index; s, node strength; SCID, Structured Clinical Interview for DSM Disorders; SL, synchronization likelihood; WASI, Wechsler Abbreviated Scale of Intelligence; WM, white matter; YMRS, Young Mania Rating Scale; γ, normalized clustering coefficients; λ, normalized characteristic path length; σ, small-worldness; Bipolar disorder; Electroencephalogram; Synchronization likelihood; Graph theory; Functional connectivity; Resting state
19.  Exploring the Morphospace of Communication Efficiency in Complex Networks 
PLoS ONE  2013;8(3):e58070.
Graph theoretical analysis has played a key role in characterizing global features of the topology of complex networks, describing diverse systems such as protein interactions, food webs, social relations and brain connectivity. How system elements communicate with each other depends not only on the structure of the network, but also on the nature of the system's dynamics which are constrained by the amount of knowledge and resources available for communication processes. Complementing widely used measures that capture efficiency under the assumption that communication preferentially follows shortest paths across the network (“routing”), we define analytic measures directed at characterizing network communication when signals flow in a random walk process (“diffusion”). The two dimensions of routing and diffusion efficiency define a morphospace for complex networks, with different network topologies characterized by different combinations of efficiency measures and thus occupying different regions of this space. We explore the relation of network topologies and efficiency measures by examining canonical network models, by evolving networks using a multi-objective optimization strategy, and by investigating real-world network data sets. Within the efficiency morphospace, specific aspects of network topology that differentially favor efficient communication for routing and diffusion processes are identified. Charting regions of the morphospace that are occupied by canonical, evolved or real networks allows inferences about the limits of communication efficiency imposed by connectivity and dynamics, as well as the underlying selection pressures that have shaped network topology.
doi:10.1371/journal.pone.0058070
PMCID: PMC3591454  PMID: 23505455
20.  Human Connectomics 
Current Opinion in Neurobiology  2011;22(1):144-153.
Recent advances in non-invasive neuroimaging have enabled the measurement of connections between distant regions in the living human brain, thus opening up a new field of research: Human connectomics. Different imaging modalities allow the mapping of structural connections (axonal fiber tracts) as well as functional connections (correlations in time series), and individual variations in these connections may be related to individual variations in behaviour and cognition. Connectivity analysis has already led to several important advances. Segregated brain regions may be identified by their unique patterns of connectivity, structural and functional connectivity may be compared to elucidate how dynamic interactions arise from the anatomical substrate, and the architecture of large-scale networks connecting sets of brain regions may be analyzed in detail. The combination of structural and functional connectivity has begun to reveal key patterns of human brain organization, such as the existence of distinct modules or sub-networks that become engaged in different cognitive tasks. Collectively, advances in human connectomics open up the possibility of studying how brain connections mediate regional brain function and thence behaviour.
doi:10.1016/j.conb.2011.08.005
PMCID: PMC3294015  PMID: 21908183
Neuroimaging; Network; Neuroanatomy; Connectome; Diffusion Imaging; fMRI; Resting-State
21.  Optimization of seed density in DTI tractography for structural networks 
Journal of neuroscience methods  2011;203(1):264-272.
Diffusion tensor imaging (DTI) has been used for mapping the structural network of the human brain. The network can be constructed by choosing various brain regions as nodes and fiber tracts connecting those regions as links. The structural network generated from DTI data can be affected by noise in the scans and the choice of tractography algorithm. This study aimed to examine the effect of the number of seeds in tractography on the variance of structural networks. The variance of the network was characterized using an approach similar to the National Electrical Manufacturers Association (NEMA) standards for measurement of image noise. It was shown that the variance of the network is inversely related to the square root of seed density. Consequently, the number of seeds has a large impact on local characteristics and metrics of the network architecture. As the number of seeds increased, increased stability of structural network metrics was observed. However, more seeds can also lead to more spurious fibers and thus affect nodal degrees and edge weights, and proper thresholding may be necessary to create an appropriate weighted network. Because the variance of the network is also influenced by other imaging factors, further increase in the number of seeds has little effect in reducing the network variance. The selection of the seed number should be a balance between the network variance and computational effort.
doi:10.1016/j.jneumeth.2011.09.021
PMCID: PMC3500612  PMID: 21978486
Diffusion tensor imaging; Structural brain network; Tractography; Seeds; Variance
22.  Structural Network Topology Revealed by White Matter Tractography in Cannabis Users: A Graph Theoretical Analysis 
Brain Connectivity  2011;1(6):473-483.
Abstract
Endocannabinoid receptors modulate synaptic plasticity in the brain and may therefore impact cortical connectivity not only during development but also in response to substance abuse in later life. Such alterations may not be evident in volumetric measures utilized in brain imaging, but could affect the local and global organization of brain networks. To test this hypothesis, we used a novel computational approach to estimate network measures of structural brain connectivity derived from diffusion tensor imaging (DTI) and white matter tractography. Twelve adult cannabis (CB) users and 13 healthy subjects were evaluated using a graph theoretic analysis of both global and local brain network properties. Structural brain networks in both CB subjects and controls exhibited robust small-world network attributes in both groups. However, CB subjects showed significantly decreased global network efficiency and significantly increased clustering coefficients (degree to which nodes tend to cluster around individual nodes). CB subjects also exhibited altered patterns of local network organization in the cingulate region. Among all subjects, schizotypal and impulsive personality characteristics correlated with global efficiency but not with the clustering coefficient. Our data indicate that structural brain networks in CB subjects are less efficiently integrated and exhibit altered regional connectivity. These differences in network properties may reflect physiological processes secondary to substance abuse-induced synaptic plasticity, or differences in brain organization that increase vulnerability to substance use.
doi:10.1089/brain.2011.0053
PMCID: PMC3604771  PMID: 22432904
cannabis; delta-9-tetrahydrocannabinol; deterministic tractography; diffusion tensor imaging; graph theory; network analysis
23.  Network analysis of corticocortical connections reveals ventral and dorsal processing streams in mouse visual cortex 
The Journal of Neuroscience  2012;32(13):4386-4399.
Much of the information used for visual perception and visually guided actions is processed in complex networks of connections within the cortex. To understand how this works in the normal brain and to determine the impact of disease, mice are promising models. In primate visual cortex, information is processed in a dorsal stream specialized for visuospatial processing and guided action and a ventral stream for object recognition. Here, we traced the outputs of 10 visual areas and used quantitative graph analytic tools of modern network science to determine, from the projection strengths in 39 cortical targets, the community structure of the network. We found a high density of the cortical graph that exceeded that previously shown in monkey. Each source area showed a unique distribution of projection weights across its targets (i.e. connectivity profile) that was well-fit by a lognormal function. Importantly, the community structure was strongly dependent on the location of the source area: outputs from medial/anterior extrastriate areas were more strongly linked to parietal, motor and limbic cortex, whereas lateral extrastriate areas were preferentially connected to temporal and parahippocampal cortex. These two subnetworks resemble dorsal and ventral cortical streams in primates, demonstrating that the basic layout of cortical networks is conserved across species.
doi:10.1523/JNEUROSCI.6063-11.2012
PMCID: PMC3328193  PMID: 22457489
24.  Rich Club Organization of Macaque Cerebral Cortex and Its Role in Network Communication 
PLoS ONE  2012;7(9):e46497.
Graph-theoretical analysis of brain connectivity data has revealed significant features of brain network organization across a range of species. Consistently, large-scale anatomical networks exhibit highly nonrandom attributes including an efficient small world modular architecture, with distinct network communities that are interlinked by hub regions. The functional importance of hubs motivates a closer examination of their mutual interconnections, specifically to examine the hypothesis that hub regions are more densely linked than expected based on their degree alone, i.e. forming a central rich club. Extending recent findings of rich club topology in the cat and human brain, this report presents evidence for the existence of rich club organization in the cerebral cortex of a non-human primate, the macaque monkey, based on a connectivity data set representing a collation of numerous tract tracing studies. Rich club regions comprise portions of prefrontal, parietal, temporal and insular cortex and are widely distributed across network communities. An analysis of network motifs reveals that rich club regions tend to form star-like configurations, indicative of their central embedding within sets of nodes. In addition, rich club nodes and edges participate in a large number of short paths across the network, and thus contribute disproportionately to global communication. As rich club regions tend to attract and disperse communication paths, many of the paths follow a characteristic pattern of first increasing and then decreasing node degree. Finally, the existence of non-reciprocal projections imposes a net directional flow of paths into and out of the rich club, with some regions preferentially attracting and others dispersing signals. Overall, the demonstration of rich club organization in a non-human primate contributes to our understanding of the network principles underlying neural connectivity in the mammalian brain, and further supports the hypothesis that rich club regions and connections have a central role in global brain communication.
doi:10.1371/journal.pone.0046497
PMCID: PMC3460908  PMID: 23029538

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