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1.  Activity Dependent Degeneration Explains Hub Vulnerability in Alzheimer's Disease 
PLoS Computational Biology  2012;8(8):e1002582.
Brain connectivity studies have revealed that highly connected ‘hub’ regions are particularly vulnerable to Alzheimer pathology: they show marked amyloid-β deposition at an early stage. Recently, excessive local neuronal activity has been shown to increase amyloid deposition. In this study we use a computational model to test the hypothesis that hub regions possess the highest level of activity and that hub vulnerability in Alzheimer's disease is due to this feature. Cortical brain regions were modeled as neural masses, each describing the average activity (spike density and spectral power) of a large number of interconnected excitatory and inhibitory neurons. The large-scale network consisted of 78 neural masses, connected according to a human DTI-based cortical topology. Spike density and spectral power were positively correlated with structural and functional node degrees, confirming the high activity of hub regions, also offering a possible explanation for high resting state Default Mode Network activity. ‘Activity dependent degeneration’ (ADD) was simulated by lowering synaptic strength as a function of the spike density of the main excitatory neurons, and compared to random degeneration. Resulting structural and functional network changes were assessed with graph theoretical analysis. Effects of ADD included oscillatory slowing, loss of spectral power and long-range synchronization, hub vulnerability, and disrupted functional network topology. Observed transient increases in spike density and functional connectivity match reports in Mild Cognitive Impairment (MCI) patients, and may not be compensatory but pathological. In conclusion, the assumption of excessive neuronal activity leading to degeneration provides a possible explanation for hub vulnerability in Alzheimer's disease, supported by the observed relation between connectivity and activity and the reproduction of several neurophysiologic hallmarks. The insight that neuronal activity might play a causal role in Alzheimer's disease can have implications for early detection and interventional strategies.
Author Summary
An intriguing recent observation is that deposition of the amyloid-β protein, one of the hallmarks of Alzheimer's disease, mainly occurs in brain regions that are highly connected to other regions. To test the hypothesis that these ‘hub’ regions are more vulnerable due to a higher neuronal activity level, we examined the relation between brain connectivity and activity in a computational model of the human brain. Furthermore, we simulated progressive damage to brain regions based on their level of activity, and investigated its effect on the structure and dynamics of the remaining brain network. We show that brain hub regions are indeed the most active ones, and that by damaging networks according to regional activity levels, we can reproduce not only hub vulnerability but a range of phenomena encountered in actual neurophysiological data of Alzheimer patients as well: loss and slowing of brain activity in Alzheimer, loss of synchronization between areas, and similar changes in functional network organization. The results of this study suggest that excessive, connectivity dependent neuronal activity plays a role in the development of Alzheimer, and that the further investigation of factors regulating regional brain activity might help detect, elucidate and counter the disease mechanism.
PMCID: PMC3420961  PMID: 22915996
2.  The Functional Connectome of Speech Control 
PLoS Biology  2015;13(7):e1002209.
In the past few years, several studies have been directed to understanding the complexity of functional interactions between different brain regions during various human behaviors. Among these, neuroimaging research installed the notion that speech and language require an orchestration of brain regions for comprehension, planning, and integration of a heard sound with a spoken word. However, these studies have been largely limited to mapping the neural correlates of separate speech elements and examining distinct cortical or subcortical circuits involved in different aspects of speech control. As a result, the complexity of the brain network machinery controlling speech and language remained largely unknown. Using graph theoretical analysis of functional MRI (fMRI) data in healthy subjects, we quantified the large-scale speech network topology by constructing functional brain networks of increasing hierarchy from the resting state to motor output of meaningless syllables to complex production of real-life speech as well as compared to non-speech-related sequential finger tapping and pure tone discrimination networks. We identified a segregated network of highly connected local neural communities (hubs) in the primary sensorimotor and parietal regions, which formed a commonly shared core hub network across the examined conditions, with the left area 4p playing an important role in speech network organization. These sensorimotor core hubs exhibited features of flexible hubs based on their participation in several functional domains across different networks and ability to adaptively switch long-range functional connectivity depending on task content, resulting in a distinct community structure of each examined network. Specifically, compared to other tasks, speech production was characterized by the formation of six distinct neural communities with specialized recruitment of the prefrontal cortex, insula, putamen, and thalamus, which collectively forged the formation of the functional speech connectome. In addition, the observed capacity of the primary sensorimotor cortex to exhibit operational heterogeneity challenged the established concept of unimodality of this region.
This study uses graph theory to analyze functional MRI data recorded from speakers as they produce single syllables or whole sentences, revealing the complexity of the brain network machinery that controls speech and language.
Author Summary
Speech production is a complex process that requires the orchestration of multiple brain regions. However, our current understanding of the large-scale neural architecture during speaking remains scant, as research has mostly focused on examining distinct brain circuits involved in distinct aspects of speech control. Here, we performed graph theoretical analyses of functional MRI data acquired from healthy subjects in order to reveal how brain regions relate to one another while speaking. We constructed functional brain networks of increasing hierarchy from rest to simple vocal motor output to the production of real-life speech, and compared these to nonspeech control tasks such as finger tapping and pure tone discrimination. We discovered a specialized network of densely connected sensorimotor regions, which formed a common processing core across all conditions. Specifically, the primary sensorimotor cortex participated in multiple functional domains across different networks and modulated long-range connections depending on task content, which challenges the established concept of low-order unimodal function of this region. Compared to other tasks, speech production was characterized by the formation of six distinct neural communities with specialized recruitment of the prefrontal cortex, insula, putamen, and thalamus, which collectively formed the functional speech connectome.
PMCID: PMC4512708  PMID: 26204475
3.  An atlas of gene regulatory networks reveals multiple three-gene mechanisms for interpreting morphogen gradients 
Although >450 different topologies can achieve the same multicellular patterning function, they can be grouped into six main classes, which operate using different underlying dynamics.Alternative designs for the same functions can therefore split into two types: (a) topology alterations that retain the same underlying dynamics and (b) alterations that utilize a completely different underlying dynamical mechanism.This segregation of networks into distinct dynamical mechanisms can be revealed by the shape of the topology atlas itself.Cell–cell communication is not usually part of the causal mechanism underlying a band-pass response during morphogen interpretation, but it can tune the result or increase robustness.
Understanding how gene regulatory networks (GRNs) achieve particular biological functions is a central question in systems biology. Systems biology promises to go beyond a case-by-case understanding of individual networks to map out the complete design space of mechanistic possibilities that underlie biological functions. Can such maps serve as useful theoretical frameworks in which to explore the general design principles for these functions? Towards addressing these questions, we created the first design space for a morphogen interpretation function.
In order to generate a design space for such a function, we enumerated all possible wiring designs of GRNs consisting of three genes and tested their ability to perform one particular morphogen interpretation function; stripe formation, as it represents a simplified form of the much studied French flag problem and is a commonly found gene expression pattern (Figure 1A). We found that only 5% of GRNs had the ability to generate a single stripe of gene expression when simulated with a fixed morphogen input in a one-dimensional model.
We hypothesized that the core mechanisms for producing the stripe of gene expression should be represented by topologies that contain only the necessary and sufficient gene–gene interactions for that function. Hence, we utilized the notions of complexity and neighborhood to generate a complexity atlas. GRNs of such an atlas (represented by nodes) are considered neighbors if they differ by a single gene–gene interaction (neighboring GRN nodes are connected by edges). Such a metagraph (graph of graphs) can then be reorganized using complexity (number of gene–gene interactions) to determine a GRNs position in the y axis, whereas GRNs are spaced in the x axis with the aim of reducing edge crossing (Figure 5A). This reorganization reveals a striking structure, where ‘stalactites' of complexity can be seen protruding from the bottom of the atlas. Each of these stalactites converges on a single ‘core' topology that by extensive analysis we find represents a distinct mechanism.
The mechanisms employ a diverse range of distinct space–time behaviors, and the underlying core topologies display design features such as modularity and feed-forward. We mapped the mechanisms to the complexity atlas by analyzing how each particular GRN of the atlas was working. The GRNs functioning via the different mechanisms are highlighted by the different colors in Figure 5A. Mechanisms thus occupy large regions of separated topology space, suggesting them to be discrete. Analyzing transitions between mechanisms through parameter space confirms this to be the case.
We find that three of the mechanisms are employed in real patterning systems, including both blastoderm patterning in Drosophila and mesoderm specification in Xenopus (Figure 5B). The remaining three mechanisms are thus candidates for employment in other patterning systems. We explored the performance features of these mechanisms, which suggest that some have features such as robustness to parameter variation that make them highly likely to be employed in particular patterning contexts.
Only one of the six-core mechanisms absolutely requires cell–cell communication for functionality, prompting us to predict that cell–cell communication will rarely be responsible for the basic dose response of morphogen interpretation networks. However, we show how cell–cell communication has an important role in robust stripe generation in the face of a noisy morphogen input and in fine tuning the quantitative details of stripe patterning.
In summary, the complexity atlas approach is an amendable approach to any system with a clear genotype–function relationship. We demonstrate how certain functions such as morphogen interpretation may have a range of potential solutions in contrast to previous studies that analyzed more constrained functions. Furthermore, we demonstrate how such an approach can be utilized to define a ‘design space' for a given biological function that describes the different mechanistic possibilities and how they relate to one another (Figure 5). Such a design space can be used practically as a guide to discern which patterning mechanisms are likely be at work in a particular context throwing up less intuitive possibilities with powerful performance features.
The interpretation of morphogen gradients is a pivotal concept in developmental biology, and several mechanisms have been proposed to explain how gene regulatory networks (GRNs) achieve concentration-dependent responses. However, the number of different mechanisms that may exist for cells to interpret morphogens, and the importance of design features such as feedback or local cell–cell communication, is unclear. A complete understanding of such systems will require going beyond a case-by-case analysis of real morphogen interpretation mechanisms and mapping out a complete GRN ‘design space.' Here, we generate a first atlas of design space for GRNs capable of patterning a homogeneous field of cells into discrete gene expression domains by interpreting a fixed morphogen gradient. We uncover multiple very distinct mechanisms distributed discretely across the atlas, thereby expanding the repertoire of morphogen interpretation network motifs. Analyzing this diverse collection of mechanisms also allows us to predict that local cell–cell communication will rarely be responsible for the basic dose-dependent response of morphogen interpretation networks.
PMCID: PMC3010108  PMID: 21045819
design space; gene network; morphogen; patterning; systems biology
4.  Disrupted Modularity and Local Connectivity of Brain Functional Networks in Childhood-Onset Schizophrenia 
Modularity is a fundamental concept in systems neuroscience, referring to the formation of local cliques or modules of densely intra-connected nodes that are sparsely inter-connected with nodes in other modules. Topological modularity of brain functional networks can quantify theoretically anticipated abnormality of brain network community structure – so-called dysmodularity – in developmental disorders such as childhood-onset schizophrenia (COS). We used graph theory to investigate topology of networks derived from resting-state fMRI data on 13 COS patients and 19 healthy volunteers. We measured functional connectivity between each pair of 100 regional nodes, focusing on wavelet correlation in the frequency interval 0.05–0.1 Hz, then applied global and local thresholding rules to construct graphs from each individual association matrix over the full range of possible connection densities. We show how local thresholding based on the minimum spanning tree facilitates group comparisons of networks by forcing the connectedness of sparse graphs. Threshold-dependent graph theoretical results are compatible with the results of a k-means unsupervised learning algorithm and a multi-resolution (spin glass) approach to modularity, both of which also find community structure but do not require thresholding of the association matrix. In general modularity of brain functional networks was significantly reduced in COS, due to a relatively reduced density of intra-modular connections between neighboring regions. Other network measures of local organization such as clustering were also decreased, while complementary measures of global efficiency and robustness were increased, in the COS group. The group differences in complex network properties were mirrored by differences in simpler statistical properties of the data, such as the variability of the global time series and the internal homogeneity of the time series within anatomical regions of interest.
PMCID: PMC2965020  PMID: 21031030
graph theory; brain; network; modularity; schizophrenia; clustering; fMRI
5.  Functional neural network analysis in frontotemporal dementia and Alzheimer's disease using EEG and graph theory 
BMC Neuroscience  2009;10:101.
Although a large body of knowledge about both brain structure and function has been gathered over the last decades, we still have a poor understanding of their exact relationship. Graph theory provides a method to study the relation between network structure and function, and its application to neuroscientific data is an emerging research field. We investigated topological changes in large-scale functional brain networks in patients with Alzheimer's disease (AD) and frontotemporal lobar degeneration (FTLD) by means of graph theoretical analysis of resting-state EEG recordings. EEGs of 20 patients with mild to moderate AD, 15 FTLD patients, and 23 non-demented individuals were recorded in an eyes-closed resting-state. The synchronization likelihood (SL), a measure of functional connectivity, was calculated for each sensor pair in 0.5–4 Hz, 4–8 Hz, 8–10 Hz, 10–13 Hz, 13–30 Hz and 30–45 Hz frequency bands. The resulting connectivity matrices were converted to unweighted graphs, whose structure was characterized with several measures: mean clustering coefficient (local connectivity), characteristic path length (global connectivity) and degree correlation (network 'assortativity'). All results were normalized for network size and compared with random control networks.
In AD, the clustering coefficient decreased in the lower alpha and beta bands (p < 0.001), and the characteristic path length decreased in the lower alpha and gamma bands (p < 0.05) compared to controls. In FTLD no significant differences with controls were found in these measures. The degree correlation decreased in both alpha bands in AD compared to controls (p < 0.05), but increased in the FTLD lower alpha band compared with controls (p < 0.01).
With decreasing local and global connectivity parameters, the large-scale functional brain network organization in AD deviates from the optimal 'small-world' network structure towards a more 'random' type. This is associated with less efficient information exchange between brain areas, supporting the disconnection hypothesis of AD. Surprisingly, FTLD patients show changes in the opposite direction, towards a (perhaps excessively) more 'ordered' network structure, possibly reflecting a different underlying pathophysiological process.
PMCID: PMC2736175  PMID: 19698093
6.  The UCLA multimodal connectivity database: a web-based platform for brain connectivity matrix sharing and analysis 
Brain connectomics research has rapidly expanded using functional MRI (fMRI) and diffusion-weighted MRI (dwMRI). A common product of these varied analyses is a connectivity matrix (CM). A CM stores the connection strength between any two regions (“nodes”) in a brain network. This format is useful for several reasons: (1) it is highly distilled, with minimal data size and complexity, (2) graph theory can be applied to characterize the network's topology, and (3) it retains sufficient information to capture individual differences such as age, gender, intelligence quotient (IQ), or disease state. Here we introduce the UCLA Multimodal Connectivity Database (, an openly available website for brain network analysis and data sharing. The site is a repository for researchers to publicly share CMs derived from their data. The site also allows users to select any CM shared by another user, compute graph theoretical metrics on the site, visualize a report of results, or download the raw CM. To date, users have contributed over 2000 individual CMs, spanning different imaging modalities (fMRI, dwMRI) and disorders (Alzheimer's, autism, Attention Deficit Hyperactive Disorder). To demonstrate the site's functionality, whole brain functional and structural connectivity matrices are derived from 60 subjects' (ages 26–45) resting state fMRI (rs-fMRI) and dwMRI data and uploaded to the site. The site is utilized to derive graph theory global and regional measures for the rs-fMRI and dwMRI networks. Global and nodal graph theoretical measures between functional and structural networks exhibit low correspondence. This example demonstrates how this tool can enhance the comparability of brain networks from different imaging modalities and studies. The existence of this connectivity-based repository should foster broader data sharing and enable larger-scale meta-analyses comparing networks across imaging modality, age group, and disease state.
PMCID: PMC3508475  PMID: 23226127
graph theory; data sharing; functional connectivity; structural connectivity; resting-state fMRI; diffusion-weighted MRI
7.  Neuronal Functional Connection Graphs among Multiple Areas of the Rat Somatosensory System during Spontaneous and Evoked Activities 
PLoS Computational Biology  2013;9(6):e1003104.
Small-World Networks (SWNs) represent a fundamental model for the comprehension of many complex man-made and biological networks. In the central nervous system, SWN models have been shown to fit well both anatomical and functional maps at the macroscopic level. However, the functional microscopic level, where the nodes of a network are represented by single neurons, is still poorly understood. At this level, although recent evidences suggest that functional connection graphs exhibit small-world organization, it is not known whether and how these maps, potentially distributed in multiple brain regions, change across different conditions, such as spontaneous and stimulus-evoked activities. We addressed these questions by analyzing the data from simultaneous multi-array extracellular recordings in three brain regions of rats, diversely involved in somatosensory information processing: the ventropostero-lateral thalamic nuclei, the primary somatosensory cortex and the centro-median thalamic nuclei. From both spike and Local Field Potential (LFP) recordings, we estimated the functional connection graphs by using the Normalized Compression Similarity for spikes and the Phase Synchrony for LFPs. Then, by using graph-theoretical statistics, we characterized the functional topology both during spontaneous activity and sensory stimulation. Our main results show that: (i) spikes and LFPs show SWN organization during spontaneous activity; (ii) after stimulation onset, while substantial functional graph reconfigurations occur both in spike and LFPs, small-worldness is nonetheless preserved; (iii) the stimulus triggers a significant increase of inter-area LFP connections without modifying the topology of intra-area functional connections. Finally, investigating computationally the functional substrate that supports the observed phenomena, we found that (iv) the fundamental concept of cell assemblies, transient groups of activating neurons, can be described by small-world networks. Our results suggest that activity of neurons from multiple areas of the rat somatosensory system contributes to the integration of local computations arisen in distributed functional cell assemblies according to the principles of SWNs.
Author Summary
Cell assemblies (sequences of neuronal activations), seem to represent a functional unit of information processing. However, it remains unclear how groups of neurons may organize their activity during information processing, working as a sole functional unit. One prominent principle in complex network theory is covered by small-world networks, in which each node is easily reachable by each other and organized in highly dense clusters. Small-world networks have been already observed on large scales in human and primate brain areas while their presence at the neuronal level remains unclear. The aim of this work was to investigate the possibility that functional, related neural populations, encompassing multiple brain regions, could be organized in small-world networks. We investigated the coherent neuronal activity among multiple rat brain regions involved in somatosensory information processing. We found that the recorded neuronal populations represented small-world networks and that these topologies were maintained during stimulations. Furthermore, by using simulations to explore the hidden substrates supporting the observed topological features, we inferred that small-world networks represent a plausible topology for cell assemblies. This work suggests that the coherent activity of neurons from multiple brain areas promotes the integration of local computations, the functional principle of small-world networks.
PMCID: PMC3681651  PMID: 23785273
8.  Altered functional and structural brain network organization in autism☆ 
NeuroImage : Clinical  2012;2:79-94.
Structural and functional underconnectivity have been reported for multiple brain regions, functional systems, and white matter tracts in individuals with autism spectrum disorders (ASD). Although recent developments in complex network analysis have established that the brain is a modular network exhibiting small-world properties, network level organization has not been carefully examined in ASD. Here we used resting-state functional MRI (n = 42 ASD, n = 37 typically developing; TD) to show that children and adolescents with ASD display reduced short and long-range connectivity within functional systems (i.e., reduced functional integration) and stronger connectivity between functional systems (i.e., reduced functional segregation), particularly in default and higher-order visual regions. Using graph theoretical methods, we show that pairwise group differences in functional connectivity are reflected in network level reductions in modularity and clustering (local efficiency), but shorter characteristic path lengths (higher global efficiency). Structural networks, generated from diffusion tensor MRI derived fiber tracts (n = 51 ASD, n = 43 TD), displayed lower levels of white matter integrity yet higher numbers of fibers. TD and ASD individuals exhibited similar levels of correlation between raw measures of structural and functional connectivity (n = 35 ASD, n = 35 TD). However, a principal component analysis combining structural and functional network properties revealed that the balance of local and global efficiency between structural and functional networks was reduced in ASD, positively correlated with age, and inversely correlated with ASD symptom severity. Overall, our findings suggest that modeling the brain as a complex network will be highly informative in unraveling the biological basis of ASD and other neuropsychiatric disorders.
► Complex network analysis of resting-state fMRI and DTI tractography in autism ► Local and long-range functional connectivity is reduced in ASD. ► Reduced local efficiency and modularity of functional networks in ASD ► Altered age-related trajectory of global efficiency for structural networks in ASD
PMCID: PMC3777708  PMID: 24179761
Resting-state functional connectivity; Diffusion tensor imaging; Graph theory; Brain networks; Autism spectrum disorders
9.  Functional Brain Networks Develop from a “Local to Distributed” Organization 
PLoS Computational Biology  2009;5(5):e1000381.
The mature human brain is organized into a collection of specialized functional networks that flexibly interact to support various cognitive functions. Studies of development often attempt to identify the organizing principles that guide the maturation of these functional networks. In this report, we combine resting state functional connectivity MRI (rs-fcMRI), graph analysis, community detection, and spring-embedding visualization techniques to analyze four separate networks defined in earlier studies. As we have previously reported, we find, across development, a trend toward ‘segregation’ (a general decrease in correlation strength) between regions close in anatomical space and ‘integration’ (an increased correlation strength) between selected regions distant in space. The generalization of these earlier trends across multiple networks suggests that this is a general developmental principle for changes in functional connectivity that would extend to large-scale graph theoretic analyses of large-scale brain networks. Communities in children are predominantly arranged by anatomical proximity, while communities in adults predominantly reflect functional relationships, as defined from adult fMRI studies. In sum, over development, the organization of multiple functional networks shifts from a local anatomical emphasis in children to a more “distributed” architecture in young adults. We argue that this “local to distributed” developmental characterization has important implications for understanding the development of neural systems underlying cognition. Further, graph metrics (e.g., clustering coefficients and average path lengths) are similar in child and adult graphs, with both showing “small-world”-like properties, while community detection by modularity optimization reveals stable communities within the graphs that are clearly different between young children and young adults. These observations suggest that early school age children and adults both have relatively efficient systems that may solve similar information processing problems in divergent ways.
Author Summary
The first two decades of life represent a period of extraordinary developmental change in sensory, motor, and cognitive abilities. One of the ultimate goals of developmental cognitive neuroscience is to link the complex behavioral milestones that occur throughout this time period with the equally intricate functional and structural changes of the underlying neural substrate. Achieving this goal would not only give us a deeper understanding of normal development but also a richer insight into the nature of developmental disorders. In this report, we use computational analyses, in combination with a recently developed MRI technique that measures spontaneous brain activity, to help us to understand the principles that guide the maturation of the human brain. We find that brain regions in children communicate with other regions more locally but that over age communication becomes more distributed. Interestingly, the efficiency of communication in children (measured as a ‘small world’ network) is comparable to that of the adult. We argue that these findings have important implications for understanding both the maturation and the function of neural systems in typical and atypical development.
PMCID: PMC2671306  PMID: 19412534
10.  Influence of Wiring Cost on the Large-Scale Architecture of Human Cortical Connectivity 
PLoS Computational Biology  2014;10(4):e1003557.
In the past two decades some fundamental properties of cortical connectivity have been discovered: small-world structure, pronounced hierarchical and modular organisation, and strong core and rich-club structures. A common assumption when interpreting results of this kind is that the observed structural properties are present to enable the brain's function. However, the brain is also embedded into the limited space of the skull and its wiring has associated developmental and metabolic costs. These basic physical and economic aspects place separate, often conflicting, constraints on the brain's connectivity, which must be characterized in order to understand the true relationship between brain structure and function. To address this challenge, here we ask which, and to what extent, aspects of the structural organisation of the brain are conserved if we preserve specific spatial and topological properties of the brain but otherwise randomise its connectivity. We perform a comparative analysis of a connectivity map of the cortical connectome both on high- and low-resolutions utilising three different types of surrogate networks: spatially unconstrained (‘random’), connection length preserving (‘spatial’), and connection length optimised (‘reduced’) surrogates. We find that unconstrained randomisation markedly diminishes all investigated architectural properties of cortical connectivity. By contrast, spatial and reduced surrogates largely preserve most properties and, interestingly, often more so in the reduced surrogates. Specifically, our results suggest that the cortical network is less tightly integrated than its spatial constraints would allow, but more strongly segregated than its spatial constraints would necessitate. We additionally find that hierarchical organisation and rich-club structure of the cortical connectivity are largely preserved in spatial and reduced surrogates and hence may be partially attributable to cortical wiring constraints. In contrast, the high modularity and strong s-core of the high-resolution cortical network are significantly stronger than in the surrogates, underlining their potential functional relevance in the brain.
Author Summary
Macroscopic regions in the grey matter of the human brain are intricately connected by white-matter pathways, forming the extremely complex network of the brain. Analysing this brain network may provide us insights on how anatomy enables brain function and, ultimately, cognition and consciousness. Various important principles of organization have indeed been consistently identified in the brain's structural connectivity, such as a small-world and modular architecture. However, it is currently unclear which of these principles are functionally relevant, and which are merely the consequence of more basic constraints of the brain, such as its three-dimensional spatial embedding into the limited volume of the skull or the high metabolic cost of long-range connections. In this paper, we model what aspects of the structural organization of the brain are affected by its wiring constraints by assessing how far these aspects are preserved in brain-like networks with varying spatial wiring constraints. We find that all investigated features of brain organization also appear in spatially constrained networks, but we also discover that several of the features are more pronounced in the brain than its wiring constraints alone would necessitate. These findings suggest the functional relevance of the ‘over-expressed’ properties of brain architecture.
PMCID: PMC3974635  PMID: 24699277
11.  A Comparative Study of Theoretical Graph Models for Characterizing Structural Networks of Human Brain 
Previous studies have investigated both structural and functional brain networks via graph-theoretical methods. However, there is an important issue that has not been adequately discussed before: what is the optimal theoretical graph model for describing the structural networks of human brain? In this paper, we perform a comparative study to address this problem. Firstly, large-scale cortical regions of interest (ROIs) are localized by recently developed and validated brain reference system named Dense Individualized Common Connectivity-based Cortical Landmarks (DICCCOL) to address the limitations in the identification of the brain network ROIs in previous studies. Then, we construct structural brain networks based on diffusion tensor imaging (DTI) data. Afterwards, the global and local graph properties of the constructed structural brain networks are measured using the state-of-the-art graph analysis algorithms and tools and are further compared with seven popular theoretical graph models. In addition, we compare the topological properties between two graph models, namely, stickiness-index-based model (STICKY) and scale-free gene duplication model (SF-GD), that have higher similarity with the real structural brain networks in terms of global and local graph properties. Our experimental results suggest that among the seven theoretical graph models compared in this study, STICKY and SF-GD models have better performances in characterizing the structural human brain network.
PMCID: PMC3863486  PMID: 24369454
12.  Hierarchical Alteration of Brain Structural and Functional Networks in Female Migraine Sufferers 
PLoS ONE  2012;7(12):e51250.
Little is known about the changes of brain structural and functional connectivity networks underlying the pathophysiology in migraine. We aimed to investigate how the cortical network reorganization is altered by frequent cortical overstimulation associated with migraine.
Methodology/Principal Findings
Gray matter volumes and resting-state functional magnetic resonance imaging signal correlations were employed to construct structural and functional networks between brain regions in 43 female patients with migraine (PM) and 43 gender-matched healthy controls (HC) by using graph theory-based approaches. Compared with the HC group, the patients showed abnormal global topology in both structural and functional networks, characterized by higher mean clustering coefficients without significant change in the shortest absolute path length, which indicated that the PM lost optimal topological organization in their cortical networks. Brain hubs related to pain-processing revealed abnormal nodal centrality in both structural and functional networks, including the precentral gyrus, orbital part of the inferior frontal gyrus, parahippocampal gyrus, anterior cingulate gyrus, thalamus, temporal pole of the middle temporal gyrus and the inferior parietal gyrus. Negative correlations were found between migraine duration and regions with abnormal centrality. Furthermore, the dysfunctional connections in patients' cortical networks formed into a connected component and three dysregulated modules were identified involving pain-related information processing and motion-processing visual networks.
Our results may reflect brain alteration dynamics resulting from migraine and suggest that long-term and high-frequency headache attacks may cause both structural and functional connectivity network reorganization. The disrupted information exchange between brain areas in migraine may be reshaped into a hierarchical modular structure progressively.
PMCID: PMC3515541  PMID: 23227257
13.  Driving and Driven Architectures of Directed Small-World Human Brain Functional Networks 
PLoS ONE  2011;6(8):e23460.
Recently, increasing attention has been focused on the investigation of the human brain connectome that describes the patterns of structural and functional connectivity networks of the human brain. Many studies of the human connectome have demonstrated that the brain network follows a small-world topology with an intrinsically cohesive modular structure and includes several network hubs in the medial parietal regions. However, most of these studies have only focused on undirected connections between regions in which the directions of information flow are not taken into account. How the brain regions causally influence each other and how the directed network of human brain is topologically organized remain largely unknown. Here, we applied linear multivariate Granger causality analysis (GCA) and graph theoretical approaches to a resting-state functional MRI dataset with a large cohort of young healthy participants (n = 86) to explore connectivity patterns of the population-based whole-brain functional directed network. This directed brain network exhibited prominent small-world properties, which obviously improved previous results of functional MRI studies showing weak small-world properties in the directed brain networks in terms of a kernel-based GCA and individual analysis. This brain network also showed significant modular structures associated with 5 well known subsystems: fronto-parietal, visual, paralimbic/limbic, subcortical and primary systems. Importantly, we identified several driving hubs predominantly located in the components of the attentional network (e.g., the inferior frontal gyrus, supplementary motor area, insula and fusiform gyrus) and several driven hubs predominantly located in the components of the default mode network (e.g., the precuneus, posterior cingulate gyrus, medial prefrontal cortex and inferior parietal lobule). Further split-half analyses indicated that our results were highly reproducible between two independent subgroups. The current study demonstrated the directions of spontaneous information flow and causal influences in the directed brain networks, thus providing new insights into our understanding of human brain functional connectome.
PMCID: PMC3155571  PMID: 21858129
14.  Anomalous Gray Matter Structural Networks in Patients with Hepatitis B Virus-Related Cirrhosis without Overt Hepatic Encephalopathy 
PLoS ONE  2015;10(3):e0119339.
Background and Purpose
Increasing evidence suggests that cirrhosis may affect the connectivity among different brain regions in patients before overt hepatic encephalopathy (OHE) occurs. However, there has been no study investigating the structural reorganization of these altered connections at the network level. The primary focus of this study was to investigate the abnormal topological organization of the structural network in patients with hepatitis B virus-related cirrhosis (HBV-RC) without OHE using structural MRI.
Using graph theoretical analysis, we compared the global and regional topological properties of gray matter structural networks between 28 patients with HBV-RC without OHE and 30 age-, sex- and education-matched healthy controls. The structural correlation networks were constructed for the two groups based on measures of gray matter volume.
The brain network of the HBV-RC group exhibited a significant decrease in the clustering coefficient and reduced small-worldness at the global level across a range of network densities. Regionally, brain areas with altered nodal degree/betweenness centrality were observed predominantly in association cortices (frontal and temporal regions) (p < 0.05, uncorrected), including a significantly decreased nodal degree in the inferior temporal gyrus (p < 0.001, uncorrected). Furthermore, the HBV-RC group exhibited a loss of association hubs and the emergence of an increased number of non-association hubs compared with the healthy controls.
The results of this large-scale gray matter structural network study suggest reduced topological organization efficiency in patients with HBV-RC without OHE. Our findings provide new insight concerning the mechanisms of neurobiological reorganization in the HBV-RC brain from a network perspective.
PMCID: PMC4364769  PMID: 25786256
15.  Mapping the Alzheimer’s Brain with Connectomics 
Alzheimer’s disease (AD) is the most common form of dementia. As an incurable, progressive, and neurodegenerative disease, it causes cognitive and memory deficits. However, the biological mechanisms underlying the disease are not thoroughly understood. In recent years, non-invasive neuroimaging and neurophysiological techniques [e.g., structural magnetic resonance imaging (MRI), diffusion MRI, functional MRI, and EEG/MEG] and graph theory based network analysis have provided a new perspective on structural and functional connectivity patterns of the human brain (i.e., the human connectome) in health and disease. Using these powerful approaches, several recent studies of patients with AD exhibited abnormal topological organization in both global and regional properties of neuronal networks, indicating that AD not only affects specific brain regions, but also alters the structural and functional associations between distinct brain regions. Specifically, disruptive organization in the whole-brain networks in AD is involved in the loss of small-world characters and the re-organization of hub distributions. These aberrant neuronal connectivity patterns were associated with cognitive deficits in patients with AD, even with genetic factors in healthy aging. These studies provide empirical evidence to support the existence of an aberrant connectome of AD. In this review we will summarize recent advances discovered in large-scale brain network studies of AD, mainly focusing on graph theoretical analysis of brain connectivity abnormalities. These studies provide novel insights into the pathophysiological mechanisms of AD and could be helpful in developing imaging biomarkers for disease diagnosis and monitoring.
PMCID: PMC3251821  PMID: 22291664
connectome; graph theory; small-world; cortical thickness; genetics; DTI; fMRI; EEG/MEG
16.  Graph theoretical analysis of developmental patterns of the white matter network 
Understanding the development of human brain organization is critical for gaining insight into how the enhancement of cognitive processes is related to the fine-tuning of the brain network. However, the developmental trajectory of the large-scale white matter (WM) network is not fully understood. Here, using graph theory, we examine developmental changes in the organization of WM networks in 180 typically-developing participants. WM networks were constructed using whole brain tractography and 78 cortical regions of interest were extracted from each participant. The subjects were first divided into 5 equal sample size (n = 36) groups (early childhood: 6.0–9.7 years; late childhood: 9.8–12.7 years; adolescence: 12.9–17.5 years; young adult: 17.6–21.8 years; adult: 21.9–29.6 years). Most prominent changes in the topological properties of developing brain networks occur at late childhood and adolescence. During late childhood period, the structural brain network showed significant increase in the global efficiency but decrease in modularity, suggesting a shift of topological organization toward a more randomized configuration. However, while preserving most topological features, there was a significant increase in the local efficiency at adolescence, suggesting the dynamic process of rewiring and rebalancing brain connections at different growth stages. In addition, several pivotal hubs were identified that are vital for the global coordination of information flow over the whole brain network across all age groups. Significant increases of nodal efficiency were present in several regions such as precuneus at late childhood. Finally, a stable and functionally/anatomically related modular organization was identified throughout the development of the WM network. This study used network analysis to elucidate the topological changes in brain maturation, paving the way for developing novel methods for analyzing disrupted brain connectivity in neurodevelopmental disorders.
PMCID: PMC3814848  PMID: 24198774
graph theory; neurodevelopment; anatomical connectivity; modular networks; small world network
17.  The Anatomical Distance of Functional Connections Predicts Brain Network Topology in Health and Schizophrenia 
Cerebral Cortex (New York, NY)  2012;23(1):127-138.
The human brain is a topologically complex network embedded in anatomical space. Here, we systematically explored relationships between functional connectivity, complex network topology, and anatomical (Euclidean) distance between connected brain regions, in the resting-state functional magnetic resonance imaging brain networks of 20 healthy volunteers and 19 patients with childhood-onset schizophrenia (COS). Normal between-subject differences in average distance of connected edges in brain graphs were strongly associated with variation in topological properties of functional networks. In addition, a club or subset of connector hubs was identified, in lateral temporal, parietal, dorsal prefrontal, and medial prefrontal/cingulate cortical regions. In COS, there was reduced strength of functional connectivity over short distances especially, and therefore, global mean connection distance of thresholded graphs was significantly greater than normal. As predicted from relationships between spatial and topological properties of normal networks, this disorder-related proportional increase in connection distance was associated with reduced clustering and modularity and increased global efficiency of COS networks. Between-group differences in connection distance were localized specifically to connector hubs of multimodal association cortex. In relation to the neurodevelopmental pathogenesis of schizophrenia, we argue that the data are consistent with the interpretation that spatial and topological disturbances of functional network organization could arise from excessive “pruning” of short-distance functional connections in schizophrenia.
PMCID: PMC3513955  PMID: 22275481
brain network; functional connectivity; graph theory; normal development; schizophrenia
18.  Network Analysis of Intrinsic Functional Brain Connectivity in Alzheimer's Disease 
PLoS Computational Biology  2008;4(6):e1000100.
Functional brain networks detected in task-free (“resting-state”) functional magnetic resonance imaging (fMRI) have a small-world architecture that reflects a robust functional organization of the brain. Here, we examined whether this functional organization is disrupted in Alzheimer's disease (AD). Task-free fMRI data from 21 AD subjects and 18 age-matched controls were obtained. Wavelet analysis was applied to the fMRI data to compute frequency-dependent correlation matrices. Correlation matrices were thresholded to create 90-node undirected-graphs of functional brain networks. Small-world metrics (characteristic path length and clustering coefficient) were computed using graph analytical methods. In the low frequency interval 0.01 to 0.05 Hz, functional brain networks in controls showed small-world organization of brain activity, characterized by a high clustering coefficient and a low characteristic path length. In contrast, functional brain networks in AD showed loss of small-world properties, characterized by a significantly lower clustering coefficient (p<0.01), indicative of disrupted local connectivity. Clustering coefficients for the left and right hippocampus were significantly lower (p<0.01) in the AD group compared to the control group. Furthermore, the clustering coefficient distinguished AD participants from the controls with a sensitivity of 72% and specificity of 78%. Our study provides new evidence that there is disrupted organization of functional brain networks in AD. Small-world metrics can characterize the functional organization of the brain in AD, and our findings further suggest that these network measures may be useful as an imaging-based biomarker to distinguish AD from healthy aging.
Author Summary
Alzheimer's disease (AD) is a brain disorder characterized by progressive impairment of episodic memory and other cognitive domains resulting in dementia and, ultimately, death. Functional neuroimaging studies have identified brain regions that show abnormal brain function in AD. Although there is converging evidence about the identity of these regions, it is not clear how this abnormality affects the functional organization of the whole brain. In order to characterize the functional organization of the brain, our approach uses small-world measures, which have also been used to study systems such as social networks and the internet. We use graph analytical methods to compute these measures of functional connectivity brain networks, which are derived from fMRI data obtained from healthy elderly controls and AD patients. The AD patients had significantly lower regional connectivity, and showed disrupted global functional organization, when compared to healthy controls. Moreover, our results indicate that cognitive decline in Alzheimer's disease patients is associated with disrupted functional connectivity in the entire brain. Our findings further suggest that small-world measures may be useful as an imaging-based biomarker to distinguish AD from healthy aging.
PMCID: PMC2435273  PMID: 18584043
19.  Resolving Anatomical and Functional Structure in Human Brain Organization: Identifying Mesoscale Organization in Weighted Network Representations 
PLoS Computational Biology  2014;10(10):e1003712.
Human brain anatomy and function display a combination of modular and hierarchical organization, suggesting the importance of both cohesive structures and variable resolutions in the facilitation of healthy cognitive processes. However, tools to simultaneously probe these features of brain architecture require further development. We propose and apply a set of methods to extract cohesive structures in network representations of brain connectivity using multi-resolution techniques. We employ a combination of soft thresholding, windowed thresholding, and resolution in community detection, that enable us to identify and isolate structures associated with different weights. One such mesoscale structure is bipartivity, which quantifies the extent to which the brain is divided into two partitions with high connectivity between partitions and low connectivity within partitions. A second, complementary mesoscale structure is modularity, which quantifies the extent to which the brain is divided into multiple communities with strong connectivity within each community and weak connectivity between communities. Our methods lead to multi-resolution curves of these network diagnostics over a range of spatial, geometric, and structural scales. For statistical comparison, we contrast our results with those obtained for several benchmark null models. Our work demonstrates that multi-resolution diagnostic curves capture complex organizational profiles in weighted graphs. We apply these methods to the identification of resolution-specific characteristics of healthy weighted graph architecture and altered connectivity profiles in psychiatric disease.
Author Summary
The human brain is a fascinating organ full of exquisite anatomical and functional detail. A striking feature of this detail lies in the presence of small modules nested within one another across hierarchical levels of organization. Here we develop and apply computational analysis tools to probe these features of brain architecture by examining network representations in which brain areas are treated as network nodes and links between areas are treated as network edges. The class of methods that we describe are referred to as “multi-resolution techniques” and enable us to identify and isolate neural structures associated with different edge properties. Our methods lead to multi-resolution curves of these network diagnostics over a range of spatial, geometric, and structural scales. For statistical comparison, we contrast our results with those obtained for several benchmark null models. Our work demonstrates that multi-resolution diagnostic curves capture complex organizational profiles in weighted graphs. We apply these methods to the identification of resolution-specific characteristics of healthy weighted graph architecture and altered connectivity profiles in psychiatric disease.
PMCID: PMC4183375  PMID: 25275860
20.  GRETNA: a graph theoretical network analysis toolbox for imaging connectomics 
Recent studies have suggested that the brain’s structural and functional networks (i.e., connectomics) can be constructed by various imaging technologies (e.g., EEG/MEG; structural, diffusion and functional MRI) and further characterized by graph theory. Given the huge complexity of network construction, analysis and statistics, toolboxes incorporating these functions are largely lacking. Here, we developed the GRaph thEoreTical Network Analysis (GRETNA) toolbox for imaging connectomics. The GRETNA contains several key features as follows: (i) an open-source, Matlab-based, cross-platform (Windows and UNIX OS) package with a graphical user interface (GUI); (ii) allowing topological analyses of global and local network properties with parallel computing ability, independent of imaging modality and species; (iii) providing flexible manipulations in several key steps during network construction and analysis, which include network node definition, network connectivity processing, network type selection and choice of thresholding procedure; (iv) allowing statistical comparisons of global, nodal and connectional network metrics and assessments of relationship between these network metrics and clinical or behavioral variables of interest; and (v) including functionality in image preprocessing and network construction based on resting-state functional MRI (R-fMRI) data. After applying the GRETNA to a publicly released R-fMRI dataset of 54 healthy young adults, we demonstrated that human brain functional networks exhibit efficient small-world, assortative, hierarchical and modular organizations and possess highly connected hubs and that these findings are robust against different analytical strategies. With these efforts, we anticipate that GRETNA will accelerate imaging connectomics in an easy, quick and flexible manner. GRETNA is freely available on the NITRC website.1
PMCID: PMC4485071  PMID: 26175682
network; graph theory; connectome; resting fMRI; small-world; hub
21.  A graph-theoretical approach in brain functional networks. Possible implications in EEG studies 
Nonlinear Biomedical Physics  2010;4(Suppl 1):S8.
Recently, it was realized that the functional connectivity networks estimated from actual brain-imaging technologies (MEG, fMRI and EEG) can be analyzed by means of the graph theory, that is a mathematical representation of a network, which is essentially reduced to nodes and connections between them.
We used high-resolution EEG technology to enhance the poor spatial information of the EEG activity on the scalp and it gives a measure of the electrical activity on the cortical surface. Afterwards, we used the Directed Transfer Function (DTF) that is a multivariate spectral measure for the estimation of the directional influences between any given pair of channels in a multivariate dataset. Finally, a graph theoretical approach was used to model the brain networks as graphs. These methods were used to analyze the structure of cortical connectivity during the attempt to move a paralyzed limb in a group (N=5) of spinal cord injured patients and during the movement execution in a group (N=5) of healthy subjects.
Analysis performed on the cortical networks estimated from the group of normal and SCI patients revealed that both groups present few nodes with a high out-degree value (i.e. outgoing links). This property is valid in the networks estimated for all the frequency bands investigated. In particular, cingulate motor areas (CMAs) ROIs act as ‘‘hubs’’ for the outflow of information in both groups, SCI and healthy. Results also suggest that spinal cord injuries affect the functional architecture of the cortical network sub-serving the volition of motor acts mainly in its local feature property.
In particular, a higher local efficiency El can be observed in the SCI patients for three frequency bands, theta (3-6 Hz), alpha (7-12 Hz) and beta (13-29 Hz).
By taking into account all the possible pathways between different ROI couples, we were able to separate clearly the network properties of the SCI group from the CTRL group. In particular, we report a sort of compensatory mechanism in the SCI patients for the Theta (3-6 Hz) frequency band, indicating a higher level of “activation” Ω within the cortical network during the motor task. The activation index is directly related to diffusion, a type of dynamics that underlies several biological systems including possible spreading of neuronal activation across several cortical regions.
The present study aims at demonstrating the possible applications of graph theoretical approaches in the analyses of brain functional connectivity from EEG signals. In particular, the methodological aspects of the i) cortical activity from scalp EEG signals, ii) functional connectivity estimations iii) graph theoretical indexes are emphasized in the present paper to show their impact in a real application.
PMCID: PMC2880805  PMID: 20522269
22.  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.
PMCID: PMC2893007  PMID: 20589099
resting-state; functional connectivity; human connectome; small-world; functional MRI; graph theory; brain; network
23.  Topological Organization of Functional Brain Networks in Healthy Children: Differences in Relation to Age, Sex, and Intelligence 
PLoS ONE  2013;8(2):e55347.
Recent studies have demonstrated developmental changes of functional brain networks derived from functional connectivity using graph theoretical analysis, which has been rapidly translated to studies of brain network organization. However, little is known about sex- and IQ-related differences in the topological organization of functional brain networks during development. In this study, resting-state fMRI (rs-fMRI) was used to map the functional brain networks in 51 healthy children. We then investigated the effects of age, sex, and IQ on economic small-world properties and regional nodal properties of the functional brain networks. At a global level of whole networks, we found significant age-related increases in the small-worldness and local efficiency, significant higher values of the global efficiency in boys compared with girls, and no significant IQ-related difference. Age-related increases in the regional nodal properties were found predominately in the frontal brain regions, whereas the parietal, temporal, and occipital brain regions showed age-related decreases. Significant sex-related differences in the regional nodal properties were found in various brain regions, primarily related to the default mode, language, and vision systems. Positive correlations between IQ and the regional nodal properties were found in several brain regions related to the attention system, whereas negative correlations were found in various brain regions primarily involved in the default mode, emotion, and language systems. Together, our findings of the network topology of the functional brain networks in healthy children and its relationship with age, sex, and IQ bring new insights into the understanding of brain maturation and cognitive development during childhood and adolescence.
PMCID: PMC3563524  PMID: 23390528
24.  Combining the Finite Element Method with Structural Connectome-based Analysis for Modeling Neurotrauma: Connectome Neurotrauma Mechanics 
PLoS Computational Biology  2012;8(8):e1002619.
This article presents the integration of brain injury biomechanics and graph theoretical analysis of neuronal connections, or connectomics, to form a neurocomputational model that captures spatiotemporal characteristics of trauma. We relate localized mechanical brain damage predicted from biofidelic finite element simulations of the human head subjected to impact with degradation in the structural connectome for a single individual. The finite element model incorporates various length scales into the full head simulations by including anisotropic constitutive laws informed by diffusion tensor imaging. Coupling between the finite element analysis and network-based tools is established through experimentally-based cellular injury thresholds for white matter regions. Once edges are degraded, graph theoretical measures are computed on the “damaged” network. For a frontal impact, the simulations predict that the temporal and occipital regions undergo the most axonal strain and strain rate at short times (less than 24 hrs), which leads to cellular death initiation, which results in damage that shows dependence on angle of impact and underlying microstructure of brain tissue. The monotonic cellular death relationships predict a spatiotemporal change of structural damage. Interestingly, at 96 hrs post-impact, computations predict no network nodes were completely disconnected from the network, despite significant damage to network edges. At early times () network measures of global and local efficiency were degraded little; however, as time increased to 96 hrs the network properties were significantly reduced. In the future, this computational framework could help inform functional networks from physics-based structural brain biomechanics to obtain not only a biomechanics-based understanding of injury, but also neurophysiological insight.
Author Summary
According to the Centers for Disease Control and Prevention in the United States, approximately 1.7 million people, on average, sustain a traumatic brain injury annually. During the last few decades, brain neurotrauma biomechanics has been an active area of research involving medical clinicians and a broad range of scientists and engineers. In addition, advances and fast growth of human connectomics continues to reveal new insights into the damaged brain. With recent advances in computational methods and high performance computing, we see the need and the exciting possibility to merge brain neurotrauma biomechanics and human connectomics science to form a new area of investigation - connectome neurotrauma mechanics. For neurotrauma, the idea is simple - inform human structural connectome analysis using physics-based predictions of biomechanical brain injury. If successful, this technique may be further used to inform human functional connectome analysis, thus providing a new tool to help understand the pathophysiology of mild traumatic brain injury.
PMCID: PMC3420926  PMID: 22915997
25.  Emergence of a Small-World Functional Network in Cultured Neurons 
PLoS Computational Biology  2012;8(5):e1002522.
The functional networks of cultured neurons exhibit complex network properties similar to those found in vivo. Starting from random seeding, cultures undergo significant reorganization during the initial period in vitro, yet despite providing an ideal platform for observing developmental changes in neuronal connectivity, little is known about how a complex functional network evolves from isolated neurons. In the present study, evolution of functional connectivity was estimated from correlations of spontaneous activity. Network properties were quantified using complex measures from graph theory and used to compare cultures at different stages of development during the first 5 weeks in vitro. Networks obtained from young cultures (14 days in vitro) exhibited a random topology, which evolved to a small-world topology during maturation. The topology change was accompanied by an increased presence of highly connected areas (hubs) and network efficiency increased with age. The small-world topology balances integration of network areas with segregation of specialized processing units. The emergence of such network structure in cultured neurons, despite a lack of external input, points to complex intrinsic biological mechanisms. Moreover, the functional network of cultures at mature ages is efficient and highly suited to complex processing tasks.
Author Summary
Many social, technological and biological networks exhibit properties that are neither completely random, nor fully regular. They are known as complex networks and statistics exist to characterize their structure. Until recently, such networks have primarily been analyzed as fixed structures, which enable interaction between their components (nodes). The present work is one of the first empirical studies investigating the adaptation of complex networks [1]. Network evolution is particularly important for applying complex network analysis to biological systems, where the evolution of the network reflects the biological processes that drive it. Here, we characterize the functional networks obtained from neurons grown in vitro. Network properties are described at seven day intervals during the neurons' maturation period. Initially, neurons formed random networks, which spontaneously reorganized to a ‘small-world’ architecture. The ‘small-world’ concept derives from the study of social networks, where it is referred to as ‘six-degrees of separation’: the connection of any two individuals by as few as six acquaintances. In brain networks, this translates to rapid interaction between neurons, mediated by a few links between locally connected clusters (cliques) of neurons. This architecture is considered optimal for efficient information processing and its spontaneous emergence in cultured neurons is remarkable.
PMCID: PMC3355061  PMID: 22615555

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