Identification and classification of hubs in brain networks

Graph-based network analysis of resting-state functional MRI

In the past decade, resting-state functional MRI (R-fMRI) measures of brain activity have attracted considerable attention. Based on changes in the blood oxygen level-dependent signal, R-fMRI offers a novel way to assess the brain's spontaneous or intrinsic (i.e., task-free) activity with both high spatial and temporal resolutions. The properties of both the intra- and inter-regional connectivity of resting-state brain activity have been well documented, promoting our understanding of the brain as a complex network. Specifically, the topological organization of brain networks has been recently studied with graph theory. In this review, we will summarize the recent advances in graph-based brain network analyses of R-fMRI signals, both in typical and atypical populations. Application of these approaches to R-fMRI data has demonstrated non-trivial topological properties of functional networks in the human brain. Among these is the knowledge that the brain's intrinsic activity is organized as a small-world, highly efficient network, with significant modularity and highly connected hub regions. These network properties have also been found to change throughout normal development, aging, and in various pathological conditions. The literature reviewed here suggests that graph-based network analyses are capable of uncovering system-level changes associated with different processes in the resting brain, which could provide novel insights into the understanding of the underlying physiological mechanisms of brain function. We also highlight several potential research topics in the future.

Anterior insula activations in perceptual paradigms: often observed but barely understood

Anterior insular cortex is among the non-sensory brain regions most commonly found activated in functional brain imaging studies on visual and auditory perception. However, most of these studies do not explicitly address the functional role of this specific brain region in perception, but rather report its activation as a by-product. Here, we attempt to characterize the involvement of anterior insular cortex in various perceptual paradigms, including studies of visual awareness, perceptual decision making, cross-modal sensory processes and the role of spontaneous neural activity fluctuations in perception. We conclude that anterior insular cortex may be associated with perception in that it underpins heightened alertness of either stimulus- or task-driven origin, or both. Such a mechanism could integrate endogenous and exogenous functional demands under the joint criterion of whether they challenge an individual's homeostasis.

Optimal hierarchical modular topologies for producing limited sustained activation of neural networks

An essential requirement for the representation of functional patterns in complex neural networks, such as the mammalian cerebral cortex, is the existence of stable regimes of network activation, typically arising from a limited parameter range. In this range of limited sustained activity (LSA), the activity of neural populations in the network persists between the extremes of either quickly dying out or activating the whole network. Hierarchical modular networks were previously found to show a wider parameter range for LSA than random or small-world networks not possessing hierarchical organization or multiple modules. Here we explored how variation in the number of hierarchical levels and modules per level influenced network dynamics and occurrence of LSA. We tested hierarchical configurations of different network sizes, approximating the large-scale networks linking cortical columns in one hemisphere of the rat, cat, or macaque monkey brain. Scaling of the network size affected the number of hierarchical levels and modules in the optimal networks, also depending on whether global edge density or the numbers of connections per node were kept constant. For constant edge density, only few network configurations, possessing an intermediate number of levels and a large number of modules, led to a large range of LSA independent of brain size. For a constant number of node connections, there was a trend for optimal configurations in larger-size networks to possess a larger number of hierarchical levels or more modules. These results may help to explain the trend to greater network complexity apparent in larger brains and may indicate that this complexity is required for maintaining stable levels of neural activation.

Functional genomics and networks: new approaches in the extraction of complex gene modules

The engine that makes the cell work is made of an intricate network of molecular interactions. Nowadays, the elements and relationships of this complex network can be studied with several types of high-throughput techniques. The dream of having a global picture of the cell from different perspectives that can jointly explain cell behavior is, at least technically, feasible. However, this task can only be accomplished by filling the gap between data and information. The availability of methods capable of accurately managing, integrating and analyzing the results from these experiments is crucial for this purpose. Here, we review the new challenges raised by the availability of different genomic data, as well as the new proposals presented to cope with the increasing data complexity. Special emphasis is given to approaches that explore the transcriptome trying to describe the modules of genes that account for the traits studied.

Neural correlates of 'pessimistic' attitude in depression

BACKGROUND

Preparing for potentially threatening events in the future is essential for survival. Anticipating the future to be unpleasant is also a cognitive key feature of depression. We hypothesized that 'pessimism'-related emotion processing would characterize brain activity in major depression.MethodDuring functional magnetic resonance imaging, depressed patients and a healthy control group were cued to expect and then perceive pictures of known emotional valences--pleasant, unpleasant and neutral--and stimuli of unknown valence that could have been either pleasant or unpleasant. Brain activation associated with the 'unknown' expectation was compared with the 'known' expectation conditions.

RESULTS

While anticipating pictures of unknown valence, activation patterns in depressed patients within the medial and dorsolateral prefrontal areas, inferior frontal gyrus, insula and medial thalamus were similar to activations associated with expecting unpleasant pictures, but not with expecting positive pictures. The activity within a majority of these areas correlated with the depression scores. Differences between healthy and depressed persons were found particularly for medial and dorsolateral prefrontal and insular activations.

CONCLUSIONS

Brain activation in depression during expecting events of unknown emotional valence was comparable with activation while expecting certainly negative, but not positive events. This neurobiological finding is consistent with cognitive models supposing that depressed patients develop a 'pessimistic' attitude towards events with an unknown emotional meaning. Thereby, particularly the role of brain areas associated with the processing of cognitive and executive control and of the internal state is emphasized in contributing to major depression.

Eigenvector centrality mapping for analyzing connectivity patterns in fMRI data of the human brain

Functional magnetic resonance data acquired in a task-absent condition ("resting state") require new data analysis techniques that do not depend on an activation model. In this work, we introduce an alternative assumption- and parameter-free method based on a particular form of node centrality called eigenvector centrality. Eigenvector centrality attributes a value to each voxel in the brain such that a voxel receives a large value if it is strongly correlated with many other nodes that are themselves central within the network. Google's PageRank algorithm is a variant of eigenvector centrality. Thus far, other centrality measures - in particular "betweenness centrality" - have been applied to fMRI data using a pre-selected set of nodes consisting of several hundred elements. Eigenvector centrality is computationally much more efficient than betweenness centrality and does not require thresholding of similarity values so that it can be applied to thousands of voxels in a region of interest covering the entire cerebrum which would have been infeasible using betweenness centrality. Eigenvector centrality can be used on a variety of different similarity metrics. Here, we present applications based on linear correlations and on spectral coherences between fMRI times series. This latter approach allows us to draw conclusions of connectivity patterns in different spectral bands. We apply this method to fMRI data in task-absent conditions where subjects were in states of hunger or satiety. We show that eigenvector centrality is modulated by the state that the subjects were in. Our analyses demonstrate that eigenvector centrality is a computationally efficient tool for capturing intrinsic neural architecture on a voxel-wise level.

Graph theoretical analysis of structural and functional connectivity MRI in normal and pathological brain networks

Graph theoretical analysis of structural and functional connectivity MRI data (ie. diffusion tractography or cortical volume correlation and resting-state or task-related (effective) fMRI, respectively) has provided new measures of human brain organization in vivo. The most striking discovery is that the whole-brain network exhibits "small-world" properties shared with many other complex systems (social, technological, information, biological). This topology allows a high efficiency at different spatial and temporal scale with a very low wiring and energy cost. Its modular organization also allows for a high level of adaptation. In addition, degree distribution of brain networks demonstrates highly connected hubs that are crucial for the whole-network functioning. Many of these hubs have been identified in regions previously defined as belonging to the default-mode network (potentially explaining the high basal metabolism of this network) and the attentional networks. This could explain the crucial role of these hub regions in physiology (task-related fMRI data) as well as in pathophysiology. Indeed, such topological definition provides a reliable framework for predicting behavioral consequences of focal or multifocal lesions such as stroke, tumors or multiple sclerosis. It also brings new insights into a better understanding of pathophysiology of many neurological or psychiatric diseases affecting specific local or global brain networks such as epilepsy, Alzheimer's disease or schizophrenia. Graph theoretical analysis of connectivity MRI data provides an outstanding framework to merge anatomical and functional data in order to better understand brain pathologies.

Quantifying auditory event-related responses in multichannel human intracranial recordings

Multichannel intracranial recordings are used increasingly to study the functional organization of human cortex. Intracranial recordings of event-related activity, or electrocorticography (ECoG), are based on high density electrode arrays implanted directly over cortex, combining good temporal and spatial resolution. Developing appropriate statistical methods for analyzing event-related responses in these high dimensional ECoG datasets remains a major challenge for clinical and systems neuroscience. We present a novel methodological framework that combines complementary, existing methods adapted for statistical analysis of auditory event-related responses in multichannel ECoG recordings. This analytic framework integrates single-channel (time-domain, time–frequency) and multichannel analyses of event-related ECoG activity to determine statistically significant evoked responses, induced spectral responses, and effective (causal) connectivity. Implementation of this quantitative approach is illustrated using multichannel ECoG data from recent studies of auditory processing in patients with epilepsy. Methods described include a time–frequency matching pursuit algorithm adapted for modeling brief, transient cortical spectral responses to sound, and a recently developed method for estimating effective connectivity using multivariate autoregressive modeling to measure brief event-related changes in multichannel functional interactions. A semi-automated spatial normalization method for comparing intracranial electrode locations across patients is also described. The individual methods presented are published and readily accessible. We discuss the benefits of integrating multiple complementary methods in a unified and comprehensive quantitative approach. Methodological considerations in the analysis of multichannel ECoG data, including corrections for multiple comparisons are discussed, as well as remaining challenges in the development of new statistical approaches.

Cortical hubs form a module for multisensory integration on top of the hierarchy of cortical networks

Sensory stimuli entering the nervous system follow particular paths of processing, typically separated (segregated) from the paths of other modal information. However, sensory perception, awareness and cognition emerge from the combination of information (integration). The corticocortical networks of cats and macaque monkeys display three prominent characteristics: (i) modular organisation (facilitating the segregation), (ii) abundant alternative processing paths and (iii) the presence of highly connected hubs. Here, we study in detail the organisation and potential function of the cortical hubs by graph analysis and information theoretical methods. We find that the cortical hubs form a spatially delocalised, but topologically central module with the capacity to integrate multisensory information in a collaborative manner. With this, we resolve the underlying anatomical substrate that supports the simultaneous capacity of the cortex to segregate and to integrate multisensory information.

Network algorithmics and the emergence of the cortical synaptic-weight distribution

When a neuron fires and the resulting action potential travels down its axon toward other neurons' dendrites, the effect on each of those neurons is mediated by the strength of the synapse that separates it from the firing neuron. This strength, in turn, is affected by the postsynaptic neuron's response through a mechanism that is thought to underlie important processes such as learning and memory. Although of difficult quantification, cortical synaptic strengths have been found to obey a long-tailed unimodal distribution peaking near the lowest values (approximately lognormal), thus confirming some of the predictive models built previously. Most of these models are causally local, in the sense that they refer to the situation in which a number of neurons all fire directly at the same postsynaptic neuron. Consequently, they necessarily embody assumptions regarding the generation of action potentials by the presynaptic neurons that have little biological interpretability. We introduce a network model of large groups of interconnected neurons and demonstrate, making none of the assumptions that characterize the causally local models, that its long-term behavior gives rise to a distribution of synaptic weights (the mathematical surrogates of synaptic strengths) with the same properties that were experimentally observed. In our model, the action potentials that create a neuron's input are, ultimately, the product of network-wide causal chains relating what happens at a neuron to the firings of others. Our model is then of a causally global nature and predicates the emergence of the synaptic-weight distribution on network structure and function. As such, it has the potential to become instrumental also in the study of other emergent cortical phenomena.

Can structure predict function in the human brain?

Over the past decade, scientific interest in the properties of large-scale spontaneous neural dynamics has intensified. Concurrently, novel technologies have been developed for characterizing the connective anatomy of intra-regional circuits and inter-regional fiber pathways. It will soon be possible to build computational models that incorporate these newly detailed structural network measurements to make predictions of neural dynamics at multiple scales. Here, we review the practicality and the value of these efforts, while at the same time considering in which cases and to what extent structure does determine neural function. Studies of the healthy brain, of neural development, and of pathology all yield examples of direct correspondences between structural linkage and dynamical correlation. Theoretical arguments further support the notion that brain network topology and spatial embedding should strongly influence network dynamics. Although future models will need to be tested more quantitatively and against a wider range of empirical neurodynamic features, our present large-scale models can already predict the macroscopic pattern of dynamic correlation across the brain. We conclude that as neuroscience grapples with datasets of increasing completeness and complexity, and attempts to relate the structural and functional architectures discovered at different neural scales, the value of computational modeling will continue to grow.

Age- and gender-related differences in the cortical anatomical network

Neuroanatomical differences attributable to aging and gender have been well documented, and these differences may be associated with differences in behaviors and cognitive performance. However, little is known about the dynamic organization of anatomical connectivity within the cerebral cortex, which may underlie population differences in brain function. In this study, we investigated age and sex effects on the anatomical connectivity patterns of 95 normal subjects ranging in age from 19 to 85 years. Using the connectivity probability derived from diffusion magnetic resonance imaging tractography, we characterized the cerebral cortex as a weighted network of connected regions. This approach captures the underlying organization of anatomical connectivity for each subject at a regional level. Advanced graph theoretical analysis revealed that the resulting cortical networks exhibited "small-world" character (i.e., efficient information transfer both at local and global scale). In particular, the precuneus and posterior cingulate gyrus were consistently observed as centrally connected regions, independent of age and sex. Additional analysis revealed a reduction in overall cortical connectivity with age. There were also changes in the underlying network organization that resulted in decreased local efficiency, and also a shift of regional efficiency from the parietal and occipital to frontal and temporal neocortex in older brains. In addition, women showed greater overall cortical connectivity and the underlying organization of their cortical networks was more efficient, both locally and globally. There were also distributed regional differences in efficiency between sexes. Our results provide new insights into the substrates that underlie behavioral and cognitive differences in aging and sex.

Graph-theory based parcellation of functional subunits in the brain from resting-state fMRI data

Resting-state fMRI provides a method to examine the functional network of the brain under spontaneous fluctuations. A number of studies have proposed using resting-state BOLD data to parcellate the brain into functional subunits. In this work, we present two state-of-the-art graph-based partitioning approaches, and investigate their application to the problem of brain network segmentation using resting-state fMRI. The two approaches, the normalized cut (Ncut) and the modularity detection algorithm, are also compared to the Gaussian mixture model (GMM) approach. We show that the Ncut approach performs consistently better than the modularity detection approach, and it also outperforms the GMM approach for in vivo fMRI data. Resting-state fMRI data were acquired from 43 healthy subjects, and the Ncut algorithm was used to parcellate several different cortical regions of interest. The group-wise delineation of the functional subunits based on resting-state fMRI was highly consistent with the parcellation results from two task-based fMRI studies (one with 18 subjects and the other with 20 subjects). The findings suggest that whole-brain parcellation of the cortex using resting-state fMRI is feasible, and that the Ncut algorithm provides the appropriate technique for this task.

Intrinsic functional connectivity as a tool for human connectomics: theory, properties, and optimization

Resting state functional connectivity MRI (fcMRI) is widely used to investigate brain networks that exhibit correlated fluctuations. While fcMRI does not provide direct measurement of anatomic connectivity, accumulating evidence suggests it is sufficiently constrained by anatomy to allow the architecture of distinct brain systems to be characterized. fcMRI is particularly useful for characterizing large-scale systems that span distributed areas (e.g., polysynaptic cortical pathways, cerebro-cerebellar circuits, cortical-thalamic circuits) and has complementary strengths when contrasted with the other major tool available for human connectomics-high angular resolution diffusion imaging (HARDI). We review what is known about fcMRI and then explore fcMRI data reliability, effects of preprocessing, analysis procedures, and effects of different acquisition parameters across six studies (n = 98) to provide recommendations for optimization. Run length (2-12 min), run structure (1 12-min run or 2 6-min runs), temporal resolution (2.5 or 5.0 s), spatial resolution (2 or 3 mm), and the task (fixation, eyes closed rest, eyes open rest, continuous word-classification) were varied. Results revealed moderate to high test-retest reliability. Run structure, temporal resolution, and spatial resolution minimally influenced fcMRI results while fixation and eyes open rest yielded stronger correlations as contrasted to other task conditions. Commonly used preprocessing steps involving regression of nuisance signals minimized nonspecific (noise) correlations including those associated with respiration. The most surprising finding was that estimates of correlation strengths stabilized with acquisition times as brief as 5 min. The brevity and robustness of fcMRI positions it as a powerful tool for large-scale explorations of genetic influences on brain architecture. We conclude by discussing the strengths and limitations of fcMRI and how it can be combined with HARDI techniques to support the emerging field of human connectomics.

Whole-brain anatomical networks: does the choice of nodes matter?

Whole-brain anatomical connectivity in living humans can be modeled as a network with diffusion-MRI and tractography. Network nodes are associated with distinct grey-matter regions, while white-matter fiber bundles serve as interconnecting network links. However, the lack of a gold standard for regional parcellation in brain MRI makes the definition of nodes arbitrary, meaning that network nodes are defined using templates employing either random or anatomical parcellation criteria. Consequently, the number of nodes included in networks studied by different authors has varied considerably, from less than 100 up to more than 10(4). Here, we systematically and quantitatively assess the behavior, structure and topological attributes of whole-brain anatomical networks over a wide range of nodal scales, a variety of grey-matter parcellations as well as different diffusion-MRI acquisition protocols. We show that simple binary decisions about network organization, such as whether small-worldness or scale-freeness is evident, are unaffected by spatial scale, and that the estimates of various organizational parameters (e.g. small-worldness, clustering, path length, and efficiency) are consistent across different parcellation scales at the same resolution (i.e. the same number of nodes). However, these parameters vary considerably as a function of spatial scale; for example small-worldness exhibited a difference of 95% between the widely-used automated anatomical labeling (AAL) template (approximately 100 nodes) and a 4000-node random parcellation (sigma(AAL)=1.9 vs. sigma(4000)=53.6+/-2.2). These findings indicate that any comparison of network parameters across studies must be made with reference to the spatial scale of the nodal parcellation.

Comparison of characteristics between region-and voxel-based network analyses in resting-state fMRI data

Small-world networks are a class of networks that exhibit efficient long-distance communication and tightly interconnected local neighborhoods. In recent years, functional and structural brain networks have been examined using network theory-based methods, and consistently shown to have small-world properties. Moreover, some voxel-based brain networks exhibited properties of scale-free networks, a class of networks with mega-hubs. However, there are considerable inconsistencies across studies in the methods used and the results observed, particularly between region-based and voxel-based brain networks. We constructed functional brain networks at multiple resolutions using the same resting-state fMRI data, and compared various network metrics, degree distribution, and localization of nodes of interest. It was found that the networks with higher resolutions exhibited the properties of small-world networks more prominently. It was also found that voxel-based networks were more robust against network fragmentation compared to region-based networks. Although the degree distributions of all networks followed an exponentially truncated power law rather than true power law, the higher the resolution, the closer the distribution was to a power law. The voxel-based analyses also enhanced visualization of the results in the 3D brain space. It was found that nodes with high connectivity tended have high efficiency, a co-localization of properties that was not as consistently observed in the region-based networks. Our results demonstrate benefits of constructing the brain network at the finest scale the experiment will permit.

Identifying the brain's most globally connected regions

Recent advances in brain connectivity methods have made it possible to identify hubs-the brain's most globally connected regions. Such regions are essential for coordinating brain functions due to their connectivity with numerous regions with a variety of specializations. Current structural and functional connectivity methods generally agree that default mode network (DMN) regions have among the highest global brain connectivity (GBC). We developed two novel statistical approaches using resting state functional connectivity MRI-weighted and unweighted GBC (wGBC and uGBC)-to test the hypothesis that the highest global connectivity also occurs in the cognitive control network (CCN), a network anti-correlated with the DMN across a variety of tasks. High global connectivity was found in both CCN and DMN. The newly developed wGBC approach improves upon existing methods by quantifying inter-subject consistency, quantifying the highest GBC values by percentage, and avoiding arbitrarily connection strength thresholding. The uGBC approach is based on graph theory and includes many of these improvements, but still requires an arbitrary connection threshold. We found high GBC in several subcortical regions (e.g., hippocampus, basal ganglia) only with wGBC despite the regions' extensive anatomical connectivity. These results demonstrate the complementary utility of wGBC and uGBC analyses for the characterization of the most highly connected, and thus most functionally important, regions of the brain. Additionally, the high connectivity of both the CCN and the DMN demonstrates that brain regions outside primary sensory-motor networks are highly involved in coordinating information throughout the brain.

Altered anatomical network in early blindness revealed by diffusion tensor tractography

The topological architecture of the cerebral anatomical network reflects the structural organization of the human brain. Recently, topological measures based on graph theory have provided new approaches for quantifying large-scale anatomical networks. Diffusion MRI studies have revealed the efficient small-world properties and modular structure of the anatomical network in normal subjects. However, no previous study has used diffusion MRI to reveal changes in the brain anatomical network in early blindness. Here, we utilized diffusion tensor imaging to construct binary anatomical networks for 17 early blind subjects and 17 age- and gender-matched sighted controls. We established the existence of structural connections between any pair of the 90 cortical and sub-cortical regions using deterministic tractography. Compared with controls, early blind subjects showed a decreased degree of connectivity, a reduced global efficiency, and an increased characteristic path length in their brain anatomical network, especially in the visual cortex. Moreover, we revealed some regions with motor or somatosensory function have increased connections with other brain regions in the early blind, which suggested experience-dependent compensatory plasticity. This study is the first to show alterations in the topological properties of the anatomical network in early blindness. From the results, we suggest that analyzing the brain's anatomical network obtained using diffusion MRI data provides new insights into the understanding of the brain's re-organization in the specific population with early visual deprivation.

An adaptive complex network model for brain functional networks

Brain functional networks are graph representations of activity in the brain, where the vertices represent anatomical regions and the edges their functional connectivity. These networks present a robust small world topological structure, characterized by highly integrated modules connected sparsely by long range links. Recent studies showed that other topological properties such as the degree distribution and the presence (or absence) of a hierarchical structure are not robust, and show different intriguing behaviors. In order to understand the basic ingredients necessary for the emergence of these complex network structures we present an adaptive complex network model for human brain functional networks. The microscopic units of the model are dynamical nodes that represent active regions of the brain, whose interaction gives rise to complex network structures. The links between the nodes are chosen following an adaptive algorithm that establishes connections between dynamical elements with similar internal states. We show that the model is able to describe topological characteristics of human brain networks obtained from functional magnetic resonance imaging studies. In particular, when the dynamical rules of the model allow for integrated processing over the entire network scale-free non-hierarchical networks with well defined communities emerge. On the other hand, when the dynamical rules restrict the information to a local neighborhood, communities cluster together into larger ones, giving rise to a hierarchical structure, with a truncated power law degree distribution.

Spontaneous brain activity in the newborn brain during natural sleep--an fMRI study in infants born at full term

Recent progress in functional neuroimaging research has provided the opportunity to probe at the brain's intrinsic functional architecture. Synchronized spontaneous neuronal activity is present in the form of resting-state networks in the brain even in the absence of external stimuli. The objective of this study was to investigate the presence of resting-state networks in the unsedated infant brain born at full term. Using functional MRI, we investigated spontaneous low-frequency signal fluctuations in 19 healthy full-term infants. Resting-state functional MRI data acquired during natural sleep was analyzed using independent component analysis. We found five resting-state networks in the unsedated infant brain born at full term, encompassing sensory cortices, parietal and temporal areas, and the prefrontal cortex. In addition, we found evidence for a resting-state network that enclosed the bilateral basal ganglia.

Signal propagation in cortical networks: a digital signal processing approach

This work reports a digital signal processing approach to representing and modeling transmission and combination of signals in cortical networks. The signal dynamics is modeled in terms of diffusion, which allows the information processing undergone between any pair of nodes to be fully characterized in terms of a finite impulse response (FIR) filter. Diffusion without and with time decay are investigated. All filters underlying the cat and macaque cortical organization are found to be of low-pass nature, allowing the cortical signal processing to be summarized in terms of the respective cutoff frequencies (a high cutoff frequency meaning little alteration of signals through their intermixing). Several findings are reported and discussed, including the fact that the incorporation of temporal activity decay tends to provide more diversified cutoff frequencies. Different filtering intensity is observed for each community in those networks. In addition, the brain regions involved in object recognition tend to present the highest cutoff frequencies for both the cat and macaque networks.

Hierarchical functional modularity in the resting-state human brain

Functional magnetic resonance imaging (fMRI) studies have shown that anatomically distinct brain regions are functionally connected during the resting state. Basic topological properties in the brain functional connectivity (BFC) map have highlighted the BFC's small-world topology. Modularity, a more advanced topological property, has been hypothesized to be evolutionary advantageous, contributing to adaptive aspects of anatomical and functional brain connectivity. However, current definitions of modularity for complex networks focus on nonoverlapping clusters, and are seriously limited by disregarding inclusive relationships. Therefore, BFC's modularity has been mainly qualitatively investigated. Here, we introduce a new definition of modularity, based on a recently improved clustering measurement, which overcomes limitations of previous definitions, and apply it to the study of BFC in resting state fMRI of 53 healthy subjects. Results show hierarchical functional modularity in the brain.

Deciphering the connectivity structure of biological networks using MixNet

Background

As biological networks often show complex topological features, mathematical methods are required to extract meaningful information. Clustering methods are useful in this setting, as they allow the summary of the network's topology into a small number of relevant classes. Different strategies are possible for clustering, and in this article we focus on a model-based strategy that aims at clustering nodes based on their connectivity profiles.

Results

We present MixNet, the first publicly available computer software that analyzes biological networks using mixture models. We apply this method to various networks such as the E. coli transcriptional regulatory network, the macaque cortex network, a foodweb network and the Buchnera aphidicola metabolic network. This method is also compared with other approaches such as module identification or hierarchical clustering.

Conclusion

We show how MixNet can be used to extract meaningful biological information, and to give a summary of the networks topology that highlights important biological features. This approach is powerful as MixNet is adaptive to the network under study, and finds structural information without any a priori on the structure that is investigated. This makes MixNet a very powerful tool to summarize and decipher the connectivity structure of biological networks.

Modeling the impact of lesions in the human brain

Lesions of anatomical brain networks result in functional disturbances of brain systems and behavior which depend sensitively, often unpredictably, on the lesion site. The availability of whole-brain maps of structural connections within the human cerebrum and our increased understanding of the physiology and large-scale dynamics of cortical networks allow us to investigate the functional consequences of focal brain lesions in a computational model. We simulate the dynamic effects of lesions placed in different regions of the cerebral cortex by recording changes in the pattern of endogenous ("resting-state") neural activity. We find that lesions produce specific patterns of altered functional connectivity among distant regions of cortex, often affecting both cortical hemispheres. The magnitude of these dynamic effects depends on the lesion location and is partly predicted by structural network properties of the lesion site. In the model, lesions along the cortical midline and in the vicinity of the temporo-parietal junction result in large and widely distributed changes in functional connectivity, while lesions of primary sensory or motor regions remain more localized. The model suggests that dynamic lesion effects can be predicted on the basis of specific network measures of structural brain networks and that these effects may be related to known behavioral and cognitive consequences of brain lesions.

Symbiotic relationship between brain structure and dynamics

BACKGROUND

Brain structure and dynamics are interdependent through processes such as activity-dependent neuroplasticity. In this study, we aim to theoretically examine this interdependence in a model of spontaneous cortical activity. To this end, we simulate spontaneous brain dynamics on structural connectivity networks, using coupled nonlinear maps. On slow time scales structural connectivity is gradually adjusted towards the resulting functional patterns via an unsupervised, activity-dependent rewiring rule. The present model has been previously shown to generate cortical-like, modular small-world structural topology from initially random connectivity. We provide further biophysical justification for this model and quantitatively characterize the relationship between structure, function and dynamics that accompanies the ensuing self-organization.

RESULTS

We show that coupled chaotic dynamics generate ordered and modular functional patterns, even on a random underlying structural connectivity. Consequently, structural connectivity becomes more modular as it rewires towards these functional patterns. Functional networks reflect the underlying structural networks on slow time scales, but significantly less so on faster time scales. In spite of ordered functional topology, structural networks remain robustly interconnected--and therefore small-world--due to the presence of central, inter-modular hub nodes. The noisy dynamics of these hubs enable them to persist despite ongoing rewiring and despite their comparative absence in functional networks.

CONCLUSION

Our results outline a theoretical mechanism by which brain dynamics may facilitate neuroanatomical self-organization. We find time scale dependent differences between structural and functional networks. These differences are likely to arise from the distinct dynamics of central structural nodes.

Uncovering intrinsic modular organization of spontaneous brain activity in humans

The characterization of topological architecture of complex brain networks is one of the most challenging issues in neuroscience. Slow (<0.1 Hz), spontaneous fluctuations of the blood oxygen level dependent (BOLD) signal in functional magnetic resonance imaging are thought to be potentially important for the reflection of spontaneous neuronal activity. Many studies have shown that these fluctuations are highly coherent within anatomically or functionally linked areas of the brain. However, the underlying topological mechanisms responsible for these coherent intrinsic or spontaneous fluctuations are still poorly understood. Here, we apply modern network analysis techniques to investigate how spontaneous neuronal activities in the human brain derived from the resting-state BOLD signals are topologically organized at both the temporal and spatial scales. We first show that the spontaneous brain functional networks have an intrinsically cohesive modular structure in which the connections between regions are much denser within modules than between them. These identified modules are found to be closely associated with several well known functionally interconnected subsystems such as the somatosensory/motor, auditory, attention, visual, subcortical, and the "default" system. Specifically, we demonstrate that the module-specific topological features can not be captured by means of computing the corresponding global network parameters, suggesting a unique organization within each module. Finally, we identify several pivotal network connectors and paths (predominantly associated with the association and limbic/paralimbic cortex regions) that are vital for the global coordination of information flow over the whole network, and we find that their lesions (deletions) critically affect the stability and robustness of the brain functional system. Together, our results demonstrate the highly organized modular architecture and associated topological properties in the temporal and spatial brain functional networks of the human brain that underlie spontaneous neuronal dynamics, which provides important implications for our understanding of how intrinsically coherent spontaneous brain activity has evolved into an optimal neuronal architecture to support global computation and information integration in the absence of specific stimuli or behaviors.

Cortical hubs revealed by intrinsic functional connectivity: mapping, assessment of stability, and relation to Alzheimer's disease

Recent evidence suggests that some brain areas act as hubs interconnecting distinct, functionally specialized systems. These nexuses are intriguing because of their potential role in integration and also because they may augment metabolic cascades relevant to brain disease. To identify regions of high connectivity in the human cerebral cortex, we applied a computationally efficient approach to map the degree of intrinsic functional connectivity across the brain. Analysis of two separate functional magnetic resonance imaging datasets (each n = 24) demonstrated hubs throughout heteromodal areas of association cortex. Prominent hubs were located within posterior cingulate, lateral temporal, lateral parietal, and medial/lateral prefrontal cortices. Network analysis revealed that many, but not all, hubs were located within regions previously implicated as components of the default network. A third dataset (n = 12) demonstrated that the locations of hubs were present across passive and active task states, suggesting that they reflect a stable property of cortical network architecture. To obtain an accurate reference map, data were combined across 127 participants to yield a consensus estimate of cortical hubs. Using this consensus estimate, we explored whether the topography of hubs could explain the pattern of vulnerability in Alzheimer's disease (AD) because some models suggest that regions of high activity and metabolism accelerate pathology. Positron emission tomography amyloid imaging in AD (n = 10) compared with older controls (n = 29) showed high amyloid-beta deposition in the locations of cortical hubs consistent with the possibility that hubs, while acting as critical way stations for information processing, may also augment the underlying pathological cascade in AD.

Graph analysis of cortical networks reveals complex anatomical communication substrate

Sensory information entering the nervous system follows independent paths of processing such that specific features are individually detected. However, sensory perception, awareness, and cognition emerge from the combination of information. Here we have analyzed the corticocortical network of the cat, looking for the anatomical substrate which permits the simultaneous segregation and integration of information in the brain. We find that cortical communications are mainly governed by three topological factors of the underlying network: (i) a large density of connections, (ii) segregation of cortical areas into clusters, and (iii) the presence of highly connected hubs aiding the multisensory processing and integration. Statistical analysis of the shortest paths reveals that, while information is highly accessible to all cortical areas, the complexity of cortical information processing may arise from the rich and intricate alternative paths in which areas can influence each other.

Complex brain networks: graph theoretical analysis of structural and functional systems

Recent developments in the quantitative analysis of complex networks, based largely on graph theory, have been rapidly translated to studies of brain network organization. The brain's structural and functional systems have features of complex networks--such as small-world topology, highly connected hubs and modularity--both at the whole-brain scale of human neuroimaging and at a cellular scale in non-human animals. In this article, we review studies investigating complex brain networks in diverse experimental modalities (including structural and functional MRI, diffusion tensor imaging, magnetoencephalography and electroencephalography in humans) and provide an accessible introduction to the basic principles of graph theory. We also highlight some of the technical challenges and key questions to be addressed by future developments in this rapidly moving field.

Efficient network reconstruction from dynamical cascades identifies small-world topology of neuronal avalanches

Cascading activity is commonly found in complex systems with directed interactions such as metabolic networks, neuronal networks, or disease spreading in social networks. Substantial insight into a system's organization can be obtained by reconstructing the underlying functional network architecture from the observed activity cascades. Here we focus on Bayesian approaches and reduce their computational demands by introducing the Iterative Bayesian (IB) and Posterior Weighted Averaging (PWA) methods. We introduce a special case of PWA, cast in nonparametric form, which we call the normalized count (NC) algorithm. NC efficiently reconstructs random and small-world functional network topologies and architectures from subcritical, critical, and supercritical cascading dynamics and yields significant improvements over commonly used correlation methods. With experimental data, NC identified a functional and structural small-world topology and its corresponding traffic in cortical networks with neuronal avalanche dynamics.

In many complex systems found across disciplines, such as biological cells and organisms, social networks, economic systems, and the Internet, individual elements interact with each other, thereby forming large networks whose structure is often not known. In these complex networks, local events can easily propagate, resulting in diverse spatio-temporal activity cascades, or avalanches. Examples of such cascading activity are the propagation of diseases in social networks, cascades of chemical reactions inside a cell, the propagation of neuronal activity in the brain, and e-mail forwarding on the Internet. Although the observation of a single cascade provides limited insight into the organization of a complex network, the observation of many cascades allows for the reconstruction of very robust features of network organization, providing valuable insight into network function as well as network failure. The current work develops new algorithms for an efficient reconstruction of relatively large networks in the context of cascading activity. When applied to the brain, these algorithms uncover the structural and functional features of gray matter networks that display activity cascades in the form of neuronal avalanches.

Convergence and divergence are mostly reciprocated properties of the connections in the network of cortical areas

Cognition is based on the integrated functioning of hierarchically organized cortical processing streams in a manner yet to be clarified. Because integration fundamentally depends on convergence and the complementary notion of divergence of the neuronal connections, we analysed integration by measuring the degree of convergence/divergence through the connections in the network of cortical areas. By introducing a new index, we explored the complementary convergent and divergent nature of connectional reciprocity and delineated the backward and forward cortical sub-networks for the first time. Integrative properties of the areas defined by the degree of convergence/divergence through their afferents and efferents exhibited distinctive characteristics at different levels of the cortical hierarchy. Areas previously identified as hubs exhibit information bottleneck properties. Cortical networks largely deviate from random graphs where convergence and divergence are balanced at low reciprocity level. In the cortex, which is dominated by reciprocal connections, balance appears only by further increasing the number of reciprocal connections. The results point to the decisive role of the optimal number and placement of reciprocal connections in large-scale cortical integration. Our findings also facilitate understanding of the functional interactions between the cortical areas and the information flow or its equivalents in highly recurrent natural and artificial networks.

Emergence of functional hierarchy in a multiple timescale neural network model: a humanoid robot experiment

It is generally thought that skilled behavior in human beings results from a functional hierarchy of the motor control system, within which reusable motor primitives are flexibly integrated into various sensori-motor sequence patterns. The underlying neural mechanisms governing the way in which continuous sensori-motor flows are segmented into primitives and the way in which series of primitives are integrated into various behavior sequences have, however, not yet been clarified. In earlier studies, this functional hierarchy has been realized through the use of explicit hierarchical structure, with local modules representing motor primitives in the lower level and a higher module representing sequences of primitives switched via additional mechanisms such as gate-selecting. When sequences contain similarities and overlap, however, a conflict arises in such earlier models between generalization and segmentation, induced by this separated modular structure. To address this issue, we propose a different type of neural network model. The current model neither makes use of separate local modules to represent primitives nor introduces explicit hierarchical structure. Rather than forcing architectural hierarchy onto the system, functional hierarchy emerges through a form of self-organization that is based on two distinct types of neurons, each with different time properties ("multiple timescales"). Through the introduction of multiple timescales, continuous sequences of behavior are segmented into reusable primitives, and the primitives, in turn, are flexibly integrated into novel sequences. In experiments, the proposed network model, coordinating the physical body of a humanoid robot through high-dimensional sensori-motor control, also successfully situated itself within a physical environment. Our results suggest that it is not only the spatial connections between neurons but also the timescales of neural activity that act as important mechanisms leading to functional hierarchy in neural systems.

Hierarchical organization of human cortical networks in health and schizophrenia

The complex organization of connectivity in the human brain is incompletely understood. Recently, topological measures based on graph theory have provided a new approach to quantify large-scale cortical networks. These methods have been applied to anatomical connectivity data on nonhuman species, and cortical networks have been shown to have small-world topology, associated with high local and global efficiency of information transfer. Anatomical networks derived from cortical thickness measurements have shown the same organizational properties of the healthy human brain, consistent with similar results reported in functional networks derived from resting state functional magnetic resonance imaging (MRI) and magnetoencephalographic data. Here we show, using anatomical networks derived from analysis of inter-regional covariation of gray matter volume in MRI data on 259 healthy volunteers, that classical divisions of cortex (multimodal, unimodal, and transmodal) have some distinct topological attributes. Although all cortical divisions shared nonrandom properties of small-worldness and efficient wiring (short mean Euclidean distance between connected regions), the multimodal network had a hierarchical organization, dominated by frontal hubs with low clustering, whereas the transmodal network was assortative. Moreover, in a sample of 203 people with schizophrenia, multimodal network organization was abnormal, as indicated by reduced hierarchy, the loss of frontal and the emergence of nonfrontal hubs, and increased connection distance. We propose that the topological differences between divisions of normal cortex may represent the outcome of different growth processes for multimodal and transmodal networks and that neurodevelopmental abnormalities in schizophrenia specifically impact multimodal cortical organization.

Dynamical relaying can yield zero time lag neuronal synchrony despite long conduction delays

Multielectrode recordings have revealed zero time lag synchronization among remote cerebral cortical areas. However, the axonal conduction delays among such distant regions can amount to several tens of milliseconds. It is still unclear which mechanism is giving rise to isochronous discharge of widely distributed neurons, despite such latencies. Here, we investigate the synchronization properties of a simple network motif and found that, even in the presence of large axonal conduction delays, distant neuronal populations self-organize into lag-free oscillations. According to our results, cortico-cortical association fibers and certain cortico-thalamo-cortical loops represent ideal circuits to circumvent the phase shifts and time lags associated with conduction delays.

Organization of excitable dynamics in hierarchical biological networks

This study investigates the contributions of network topology features to the dynamic behavior of hierarchically organized excitable networks. Representatives of different types of hierarchical networks as well as two biological neural networks are explored with a three-state model of node activation for systematically varying levels of random background network stimulation. The results demonstrate that two principal topological aspects of hierarchical networks, node centrality and network modularity, correlate with the network activity patterns at different levels of spontaneous network activation. The approach also shows that the dynamic behavior of the cerebral cortical systems network in the cat is dominated by the network's modular organization, while the activation behavior of the cellular neuronal network of Caenorhabditis elegans is strongly influenced by hub nodes. These findings indicate the interaction of multiple topological features and dynamic states in the function of complex biological networks.

Evolving functional network properties and synchronizability during human epileptic seizures

We assess electrical brain dynamics before, during, and after 100 human epileptic seizures with different anatomical onset locations by statistical and spectral properties of functionally defined networks. We observe a concave-like temporal evolution of characteristic path length and cluster coefficient indicative of a movement from a more random toward a more regular and then back toward a more random functional topology. Surprisingly, synchronizability was significantly decreased during the seizure state but increased already prior to seizure end. Our findings underline the high relevance of studying complex systems from the viewpoint of complex networks, which may help to gain deeper insights into the complicated dynamics underlying epileptic seizures.

Dynamical consequences of lesions in cortical networks

To understand the effects of a cortical lesion it is necessary to consider not only the loss of local neural function, but also the lesion-induced changes in the larger network of endogenous oscillatory interactions in the brain. To investigate how network embedding influences a region's functional role, and the consequences of its being damaged, we implement two models of oscillatory cortical interactions, both of which inherit their coupling architecture from the available anatomical connection data for macaque cerebral cortex. In the first model, node dynamics are governed by Kuramoto phase oscillator equations, and we investigate the sequence in which areas entrain one another in the transition to global synchrony. In the second model, node dynamics are governed by a more realistic neural mass model, and we assess long-run inter-regional interactions using a measure of directed information flow. Highly connected parietal and frontal areas are found to synchronize most rapidly, more so than equally highly connected visual and somatosensory areas, and this difference can be explained in terms of the network's clustered architecture. For both models, lesion effects extend beyond the immediate neighbors of the lesioned site, and the amplitude and dispersal of nonlocal effects are again influenced by cluster patterns in the network. Although the consequences of in vivo lesions will always depend on circuitry local to the damaged site, we conclude that lesions of parietal regions (especially areas 5 and 7a) and frontal regions (especially areas 46 and FEF) have the greatest potential to disrupt the integrative aspects of neocortical function.

Mapping the structural core of human cerebral cortex

Structurally segregated and functionally specialized regions of the human cerebral cortex are interconnected by a dense network of cortico-cortical axonal pathways. By using diffusion spectrum imaging, we noninvasively mapped these pathways within and across cortical hemispheres in individual human participants. An analysis of the resulting large-scale structural brain networks reveals a structural core within posterior medial and parietal cerebral cortex, as well as several distinct temporal and frontal modules. Brain regions within the structural core share high degree, strength, and betweenness centrality, and they constitute connector hubs that link all major structural modules. The structural core contains brain regions that form the posterior components of the human default network. Looking both within and outside of core regions, we observed a substantial correspondence between structural connectivity and resting-state functional connectivity measured in the same participants. The spatial and topological centrality of the core within cortex suggests an important role in functional integration.

Mapping anatomical connectivity patterns of human cerebral cortex using in vivo diffusion tensor imaging tractography

The characterization of the topological architecture of complex networks underlying the structural and functional organization of the brain is a basic challenge in neuroscience. However, direct evidence for anatomical connectivity networks in the human brain remains scarce. Here, we utilized diffusion tensor imaging deterministic tractography to construct a macroscale anatomical network capturing the underlying common connectivity pattern of human cerebral cortex in a large sample of subjects (80 young adults) and further quantitatively analyzed its topological properties with graph theoretical approaches. The cerebral cortex was divided into 78 cortical regions, each representing a network node, and 2 cortical regions were considered connected if the probability of fiber connections exceeded a statistical criterion. The topological parameters of the established cortical network (binarized) resemble that of a "small-world" architecture characterized by an exponentially truncated power-law distribution. These characteristics imply high resilience to localized damage. Furthermore, this cortical network was characterized by major hub regions in association cortices that were connected by bridge connections following long-range white matter pathways. Our results are compatible with previous structural and functional brain networks studies and provide insight into the organizational principles of human brain anatomical networks that underlie functional states.

Contribution of exploratory methods to the investigation of extended large-scale brain networks in functional MRI: methodologies, results, and challenges

A large-scale brain network can be defined as a set of segregated and integrated regions, that is, distant regions that share strong anatomical connections and functional interactions. Data-driven investigation of such networks has recently received a great deal of attention in blood-oxygen-level-dependent (BOLD) functional magnetic resonance imaging (fMRI). We here review the rationale for such an investigation, the methods used, the results obtained, and also discuss some issues that have to be faced for an efficient exploration.

The frontal eye field as a prediction map

Predictive processes are widespread in the motor and sensory areas of the primate brain. They enable rapid computations despite processing delays and assist in resolving noisy, ambiguous input. Here we propose that the frontal eye field, a cortical area devoted to sensorimotor aspects of eye movement control, implements a prediction map of the postsaccadic visual scene for the purpose of constructing a stable percept despite saccadic eye movements.

Shapley ratings in brain networks

Recent applications of network theory to brain networks as well as the expanding empirical databases of brain architecture spawn an interest in novel techniques for analyzing connectivity patterns in the brain. Treating individual brain structures as nodes in a directed graph model permits the application of graph theoretical concepts to the analysis of these structures within their large-scale connectivity networks. In this paper, we explore the application of concepts from graph and game theory toward this end. Specifically, we utilize the Shapley value principle, which assigns a rank to players in a coalition based upon their individual contributions to the collective profit of that coalition, to assess the contributions of individual brain structures to the graph derived from the global connectivity network. We report Shapley values for variations of a prefrontal network, as well as for a visual cortical network, which had both been extensively investigated previously. This analysis highlights particular nodes as strong or weak contributors to global connectivity. To understand the nature of their contribution, we compare the Shapley values obtained from these networks and appropriate controls to other previously described nodal measures of structural connectivity. We find a strong correlation between Shapley values and both betweenness centrality and connection density. Moreover, a stepwise multiple linear regression analysis indicates that approximately 79% of the variance in Shapley values obtained from random networks can be explained by betweenness centrality alone. Finally, we investigate the effects of local lesions on the Shapley ratings, showing that the present networks have an immense structural resistance to degradation. We discuss our results highlighting the use of such measures for characterizing the organization and functional role of brain networks.