Using graph theoretical analysis of multi channel EEG to evaluate the neural efficiency hypothesis

Development of large-scale functional brain networks in children

Large-scale rewiring of brain circuits in children leads to emergence of hierarchical organization in the mature adult brain.

The ontogeny of large-scale functional organization of the human brain is not well understood. Here we use network analysis of intrinsic functional connectivity to characterize the organization of brain networks in 23 children (ages 7–9 y) and 22 young-adults (ages 19–22 y). Comparison of network properties, including path-length, clustering-coefficient, hierarchy, and regional connectivity, revealed that although children and young-adults' brains have similar “small-world” organization at the global level, they differ significantly in hierarchical organization and interregional connectivity. We found that subcortical areas were more strongly connected with primary sensory, association, and paralimbic areas in children, whereas young-adults showed stronger cortico-cortical connectivity between paralimbic, limbic, and association areas. Further, combined analysis of functional connectivity with wiring distance measures derived from white-matter fiber tracking revealed that the development of large-scale brain networks is characterized by weakening of short-range functional connectivity and strengthening of long-range functional connectivity. Importantly, our findings show that the dynamic process of over-connectivity followed by pruning, which rewires connectivity at the neuronal level, also operates at the systems level, helping to reconfigure and rebalance subcortical and paralimbic connectivity in the developing brain. Our study demonstrates the usefulness of network analysis of brain connectivity to elucidate key principles underlying functional brain maturation, paving the way for novel studies of disrupted brain connectivity in neurodevelopmental disorders such as autism.

The disruption of normal brain organization in humans is believed to underlie a number of behavioral conditions, such as autism spectrum disorders (ASD) and attention-deficit/hyperactivity disorder (ADHD). To gain insight into how normal brain organization develops, we mapped functional brain connectivity in children and young adults, and used a network analysis to characterize and compare the organization of brain networks. Comparison of network properties revealed that while children and young adults' brains have similar organization at the global level, there were several key differences in connectivity. For example, children's brains had less of a hierarchical organization than young-adults. Most importantly, we show that the dynamic process of over-connectivity followed by pruning, which rewires connectivity at the neuronal level, also operates at the systems level, reconfiguring and rebalancing subcortical and paralimbic connectivity in the developing brain. Our findings demonstrate the utility of using network analyses of multimodal brain connectivity to study maturation of brain circuits, and suggest new avenues for future research on neurodevelopmental disorders such as ASD and ADHD.

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.

Network inference with confidence from multivariate time series

Networks--collections of interacting elements or nodes--abound in the natural and manmade worlds. For many networks, complex spatiotemporal dynamics stem from patterns of physical interactions unknown to us. To infer these interactions, it is common to include edges between those nodes whose time series exhibit sufficient functional connectivity, typically defined as a measure of coupling exceeding a predetermined threshold. However, when uncertainty exists in the original network measurements, uncertainty in the inferred network is likely, and hence a statistical propagation of error is needed. In this manuscript, we describe a principled and systematic procedure for the inference of functional connectivity networks from multivariate time series data. Our procedure yields as output both the inferred network and a quantification of uncertainty of the most fundamental interest: uncertainty in the number of edges. To illustrate this approach, we apply a measure of linear coupling to simulated data and electrocorticogram data recorded from a human subject during an epileptic seizure. We demonstrate that the procedure is accurate and robust in both the determination of edges and the reporting of uncertainty associated with that determination.

Brain anatomical network and intelligence

Intuitively, higher intelligence might be assumed to correspond to more efficient information transfer in the brain, but no direct evidence has been reported from the perspective of brain networks. In this study, we performed extensive analyses to test the hypothesis that individual differences in intelligence are associated with brain structural organization, and in particular that higher scores on intelligence tests are related to greater global efficiency of the brain anatomical network. We constructed binary and weighted brain anatomical networks in each of 79 healthy young adults utilizing diffusion tensor tractography and calculated topological properties of the networks using a graph theoretical method. Based on their IQ test scores, all subjects were divided into general and high intelligence groups and significantly higher global efficiencies were found in the networks of the latter group. Moreover, we showed significant correlations between IQ scores and network properties across all subjects while controlling for age and gender. Specifically, higher intelligence scores corresponded to a shorter characteristic path length and a higher global efficiency of the networks, indicating a more efficient parallel information transfer in the brain. The results were consistently observed not only in the binary but also in the weighted networks, which together provide convergent evidence for our hypothesis. Our findings suggest that the efficiency of brain structural organization may be an important biological basis for intelligence.

Networks of interconnected brain regions coordinate brain activities. Information is processed in the grey matter (cortex and subcortical structures) and passed along the network via whitish, fatty-coated fiber bundles, the white matter. Using maps of these white matter tracks, we provided evidence that higher intelligence may result from more efficient information transfer. Specifically, we hypothesized that higher IQ derives from higher global efficiency of the brain anatomical network. Seventy-nine healthy young adults were divided into general and high IQ groups. We used diffusion tensor tractography, which maps brain white matter fibers, to construct anatomical brain networks for each subject and calculated the network properties using both binary and weighted networks. We consistently found that the high intelligence group's brain network was significantly more efficient than was the general intelligence group's. Moreover, IQ scores were significantly correlated with network properties, such as shorter path lengths and higher overall efficiency, indicating that the information transfer in the brain was more efficient. These converging evidences support the hypothesis that the efficiency of the organization of the brain structure may be an important biological basis for intelligence.

Neuronal networks in Alzheimer's disease

Alzheimer's disease (AD) is a progressive, neurodegenerative disease that can be clinically characterized by impaired memory and many other cognitive functions. Previous studies have demonstrated that the impairment is accompanied by not only regional brain abnormalities but also changes in neuronal connectivity between anatomically distinct brain regions. Specifically, using neurophysiological and neuroimaging techniques as well as advanced graph theory-based computational approaches, several recent studies have suggested that AD patients have disruptive neuronal integrity in large-scale structural and functional brain systems underlying high-level cognition, as demonstrated by a loss of small-world network characteristics. Small world is an attractive model for the description of complex brain networks because it can support both segregated and integrated information processing. The altered small-world organization thus reflects aberrant neuronal connectivity in the AD brain that is most likely to explain cognitive deficits caused by this disease. In this review, we will summarize recent advances in the brain network research on AD, focusing mainly on the large-scale structural and functional descriptions. The literature reviewed here suggests that AD patients are associated with integrative abnormalities in the distributed neuronal networks, which could provide new insights into the disease mechanism in AD and help us to uncover an imaging-based biomarker for the diagnosis and monitoring of the disease.

Parcellation-dependent small-world brain functional networks: a resting-state fMRI study

Recent studies have demonstrated small-world properties in both functional and structural brain networks that are constructed based on different parcellation approaches. However, one fundamental but vital issue of the impact of different brain parcellation schemes on the network topological architecture remains unclear. Here, we used resting-state functional MRI (fMRI) to investigate the influences of different brain parcellation atlases on the topological organization of brain functional networks. Whole-brain fMRI data were divided into ninety and seventy regions of interest according to two predefined anatomical atlases, respectively. Brain functional networks were constructed by thresholding the correlation matrices among the parcellated regions and further analyzed using graph theoretical approaches. Both atlas-based brain functional networks were found to show robust small-world properties and truncated power-law connectivity degree distributions, which are consistent with previous brain functional and structural networks studies. However, more importantly, we found that there were significant differences in multiple topological parameters (e.g., small-worldness and degree distribution) between the two groups of brain functional networks derived from the two atlases. This study provides quantitative evidence on how the topological organization of brain networks is affected by the different parcellation strategies applied.

Altered small-world brain functional networks in children with attention-deficit/hyperactivity disorder

In this study, we investigated the changes in topological architectures of brain functional networks in attention-deficit/hyperactivity disorder (ADHD). Functional magnetic resonance images (fMRI) were obtained from 19 children with ADHD and 20 healthy controls during resting state. Brain functional networks were constructed by thresholding the correlation matrix between 90 cortical and subcortical regions and further analyzed by applying graph theoretical approaches. Experimental results showed that, although brain networks of both groups exhibited economical small-world topology, altered functional networks were demonstrated in the brain of ADHD when compared with the normal controls. In particular, increased local efficiencies combined with a decreasing tendency in global efficiencies found in ADHD suggested a disorder-related shift of the topology toward regular networks. Additionally, significant alterations in nodal efficiency were also found in ADHD, involving prefrontal, temporal, and occipital cortex regions, which were compatible with previous ADHD studies. The present study provided the first evidence for brain dysfunction in ADHD from the viewpoint of global organization of brain functional networks by using resting-state fMRI.

Heritability of "small-world" networks in the brain: a graph theoretical analysis of resting-state EEG functional connectivity

Recent studies have shown that resting-state functional networks as studied with fMRI, EEG, and MEG may be so-called small-world networks. We investigated to what extent the characteristic features of small-world networks are genetically determined. To represent functional connectivity between brain areas, we measured resting EEG in 574 twins and their siblings and calculated the synchronization likelihood between each pair of electrodes. We applied a threshold to obtain a binary graph from which we calculated the clustering coefficient C (describing local interconnectedness) and average path length L (describing global interconnectedness) for each individual. Modeling of MZ and DZ twin and sibling resemblance indicated that across various frequency bands 46-89% of the individual differences in C and 37-62% of the individual differences in L are heritable. It is asserted that C, L, and a small-world organization are viable markers of genetic differences in brain organization.

Brain network analysis from high-resolution EEG recordings by the application of theoretical graph indexes

The extraction of the salient characteristics from brain connectivity patterns is an open challenging topic since often the estimated cerebral networks have a relative large size and complex structure. Since a graph is a mathematical representation of a network, which is essentially reduced to nodes and connections between them, the use of a theoretical graph approach would extract significant information from the functional brain networks estimated through different neuroimaging techniques. The present work intends to support the development of the "brain network analysis:" a mathematical tool consisting in a body of indexes based on the graph theory able to improve the comprehension of the complex interactions within the brain. In the present work, we applied for demonstrative purpose some graph indexes to the time-varying networks estimated from a set of high-resolution EEG data in a group of healthy subjects during the performance of a motor task. The comparison with a random benchmark allowed extracting the significant properties of the estimated networks in the representative Alpha (7-12 Hz) band. Altogether, our findings aim at proving how the brain network analysis could reveal important information about the time-frequency dynamics of the functional cortical networks.

Partial directed coherence-based information flow in Parkinson's disease patients performing a visually-guided motor task

We propose a partial directed coherence (PCD) method based on a sparse multivariate autoregressive (mAR) model to investigate patterns of information flow in electroencephalography (EEG) recordings in Parkinson's disease (PD) patients performing a visually-guided motor task. The use of a sparsity constraint on the mAR matrix addresses issues such as sample size, model order selection and number of parameters to be estimated, particularly when the number of EEG channels used is large and the window size is small in order to capture dynamic changes. The proposed PDC-based information flow analysis demonstrated distinctly altered patterns of connectivity between PD patients off medication and healthy subjects, particularly with respect to net information outflow from the left sensorimotor (L Sm) region, which might indicate excessive spreading of activity in the diseased state. Disrupted patterns of connectivity in PD were partially restored by levodopa medication. In addition, PDC-based analysis proved to be more sensitive to temporally-dynamic connectivity changes as compared to traditional spectral analysis, which might be influenced primarily by large-scale changes. We suggest that the proposed sparse-PDC method is a suitable technique to investigate altered connectivity in Parkinson's disease.

Characterizing dynamic functional connectivity across sleep stages from EEG

Following a nonlinear dynamics approach, we investigated the emergence of functional clusters which are related with spontaneous brain activity during sleep. Based on multichannel EEG traces from 10 healthy subjects, we compared the functional connectivity across different sleep stages. Our exploration commences with the conjecture of a small-world patterning, present in the scalp topography of the measured electrical activity. The existence of such a communication pattern is first confirmed for our data and then precisely determined by means of two distinct measures of non-linear interdependence between time-series. A graph encapsulating the small-world network structure along with the relative interdependence strength is formed for each sleep stage and subsequently fed to a suitable clustering procedure. Finally the delineated graph components are comparatively presented for all stages revealing novel attributes of sleep architecture. Our results suggest a pivotal role for the functional coupling during the different stages and indicate interesting dynamic characteristics like its variable hemispheric asymmetry and the isolation between anterior and posterior cortical areas during REM.

Working memory in schizophrenia: an EEG study using power spectrum and coherence analysis to estimate cortical activation and network behavior

This study examined regional cortical activations and cortico-cortical connectivity in a group of 20 high-functioning patients with schizophrenia and 20 healthy controls matched for age and sex during a 0- and a 2-back working memory (WM) task. An earlier study comparing schizophrenia patients with education level-matched healthy controls revealed less "optimally" organized network during the 2-back task, whereas a second study with healthy volunteers had suggested that the degree of cortical organization may be inversely proportional to educational level (less optimal functional connectivity in better educated individuals interpreted as the result of higher efficiency). In the present study, both groups succeeded in the 2-back WM task although healthy individuals had generally attained a higher level of education. First absolute power spectrum of the different frequency bands corresponding to the electrodes of each lobe was calculated. Then the mean values of coherence were calculated as an index of the average synchronization to construct graphs in order to characterize local and large scale topological patterns of cortico-cortical connectivity. The power spectra analyses showed signs of hypofrontality in schizophrenics with an asymmetry. Additionally, differences between the groups with greater changes during WM in healthy individuals were visible in all lobes more on the left side. The graph parameter results indicated decreased small-world architecture i.e. less optimal cortico-cortical functional organization in patients as compared to controls. These findings are consistent with the notion of aberrant neural organization in schizophrenics which is nevertheless sufficient in supporting adequate task performance.

Network analysis of intrinsic functional brain connectivity in Alzheimer's disease

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.

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.

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.

The functional connectivity of different EEG bands moves towards small-world network organization during sleep

OBJECTIVE

To analyze the functional connectivity patterns of the different EEG bands during wakefulness and sleep (different sleep stages and cyclic alternating pattern (CAP) conditions), using concepts derived from Graph Theory.

METHODS

We evaluated spatial patterns of EEG band synchronization between all possible pairs of electrodes (19) placed over the scalp of 10 sleeping healthy young normal subjects using two graph theoretical measures: the clustering coefficient (Cp) and the characteristic path length (Lp). The measures were obtained during wakefulness and the different sleep stages/CAP conditions from the real EEG connectivity networks and randomized control (surrogate) networks (Cp-s and Lp-s).

RESULTS

We found values of Cp and Lp compatible with a small-world network organization in all sleep stages and for all EEG bands. All bands below 15Hz showed an increase of these features during sleep (and during CAP-A phases in particular), compared to wakefulness.

CONCLUSIONS

The results of this study seem to confirm our initial hypothesis that during sleep there exists a clear trend for the functional connectivity of the EEG to move forward to an organization more similar to that of a small-world network, at least for the frequency bands lower than 15Hz.

SIGNIFICANCE

Sleep network "reconfiguration" might be one of the key mechanisms for the understanding of the "global" and "local" neural plasticity taking place during sleep.

Revealing modular architecture of human brain structural networks by using cortical thickness from MRI

Modularity, presumably shaped by evolutionary constraints, underlies the functionality of most complex networks ranged from social to biological networks. However, it remains largely unknown in human cortical networks. In a previous study, we demonstrated a network of correlations of cortical thickness among specific cortical areas and speculated that these correlations reflected an underlying structural connectivity among those brain regions. Here, we further investigated the intrinsic modular architecture of the human brain network derived from cortical thickness measurement. Modules were defined as groups of cortical regions that are connected morphologically to achieve the maximum network modularity. We show that the human cortical network is organized into 6 topological modules that closely overlap known functional domains such as auditory/language, strategic/executive, sensorimotor, visual, and mnemonic processing. The identified structure-based modular architecture may provide new insights into the functionality of cortical regions and connections between structural brain modules. This study provides the first report of modular architecture of the structural network in the human brain using cortical thickness measurements.

Cortical functional connectivity networks in normal and spinal cord injured patients: Evaluation by graph analysis

The present work aims at analyzing the structure of cortical connectivity during the attempt to move a paralyzed limb by a group of spinal cord injured (SCI) patients. Connectivity patterns were obtained by means of the Directed Transfer Function applied to the cortical signals estimated from high resolution EEG recordings. Electrical activity were estimated in normals (Healthy) and SCI patients on twelve regions of interest (ROIs) coincident with Brodmann areas. Degree distributions showed the presence of few cortical regions with a lot of outgoing connections in all the cortical networks estimated irrespectively of the frequency band investigated. For both of the groups (SCI and Healthy), bilateral cingulate motor area (CMA) acts as hub transmitting information flows. The efficiency index, allowed to assert the ordered properties of such estimated cortical networks in both populations. The comparison of such estimated networks with those obtained from random networks, elicited significant differences (P < 0.05, Bonferroni-corrected for multiple comparisons). A statistical comparison (ANOVA) between SCI patients and healthy subjects showed a significant difference (P < 0.05) between the local efficiency of their respective networks. For three frequency bands (theta 4-7 Hz, alpha 8-12 Hz, and beta 13-29 Hz) the higher value observed in the spinal cord injured population entails a larger level of internal organization and fault tolerance. This fact suggests a sort of compensative mechanism as local response to the alteration in their MIF areas, which is probably due to the indirect effects of the spinal injury.

Imaging functional brain connectivity patterns from high-resolution EEG and fMRI via graph theory

We describe a set of computational tools able to estimate cortical activity and connectivity from high-resolution EEG and fMRI recordings in humans. These methods comprise the estimation of cortical activity using realistic geometry head volume conductor models and distributed cortical source models, followed by the evaluation of cortical connectivity between regions of interest coincident with the Brodmann areas via the use of Partial Directed Coherence. Connectivity patterns estimated on the cortical surface in different frequency bands are then imaged and interpreted with measures based on graph theory. These computational tools were applied on a set of EEG and fMRI data from a Stroop task to demonstrate the potential of the proposed approach. The present findings suggest that the methodology is able to identify differences in functional connectivity patterns elicited by different experimental tasks or conditions.

Effects of Hebbian learning on the dynamics and structure of random networks with inhibitory and excitatory neurons

The aim of the present paper is to study the effects of Hebbian learning in random recurrent neural networks with biological connectivity, i.e. sparse connections and separate populations of excitatory and inhibitory neurons. We furthermore consider that the neuron dynamics may occur at a (shorter) time scale than synaptic plasticity and consider the possibility of learning rules with passive forgetting. We show that the application of such Hebbian learning leads to drastic changes in the network dynamics and structure. In particular, the learning rule contracts the norm of the weight matrix and yields a rapid decay of the dynamics complexity and entropy. In other words, the network is rewired by Hebbian learning into a new synaptic structure that emerges with learning on the basis of the correlations that progressively build up between neurons. We also observe that, within this emerging structure, the strongest synapses organize as a small-world network. The second effect of the decay of the weight matrix spectral radius consists in a rapid contraction of the spectral radius of the Jacobian matrix. This drives the system through the "edge of chaos" where sensitivity to the input pattern is maximal. Taken together, this scenario is remarkably predicted by theoretical arguments derived from dynamical systems and graph theory.

The application of graph theoretical analysis to complex networks in the brain

Considering the brain as a complex network of interacting dynamical systems offers new insights into higher level brain processes such as memory, planning, and abstract reasoning as well as various types of brain pathophysiology. This viewpoint provides the opportunity to apply new insights in network sciences, such as the discovery of small world and scale free networks, to data on anatomical and functional connectivity in the brain. In this review we start with some background knowledge on the history and recent advances in network theories in general. We emphasize the correlation between the structural properties of networks and the dynamics of these networks. We subsequently demonstrate through evidence from computational studies, in vivo experiments, and functional MRI, EEG and MEG studies in humans, that both the functional and anatomical connectivity of the healthy brain have many features of a small world network, but only to a limited extent of a scale free network. The small world structure of neural networks is hypothesized to reflect an optimal configuration associated with rapid synchronization and information transfer, minimal wiring costs, resilience to certain types of damage, as well as a balance between local processing and global integration. Eventually, we review the current knowledge on the effects of focal and diffuse brain disease on neural network characteristics, and demonstrate increasing evidence that both cognitive and psychiatric disturbances, as well as risk of epileptic seizures, are correlated with (changes in) functional network architectural features.

Graph theoretical analysis of complex networks in the brain

Since the discovery of small-world and scale-free networks the study of complex systems from a network perspective has taken an enormous flight. In recent years many important properties of complex networks have been delineated. In particular, significant progress has been made in understanding the relationship between the structural properties of networks and the nature of dynamics taking place on these networks. For instance, the 'synchronizability' of complex networks of coupled oscillators can be determined by graph spectral analysis. These developments in the theory of complex networks have inspired new applications in the field of neuroscience. Graph analysis has been used in the study of models of neural networks, anatomical connectivity, and functional connectivity based upon fMRI, EEG and MEG. These studies suggest that the human brain can be modelled as a complex network, and may have a small-world structure both at the level of anatomical as well as functional connectivity. This small-world structure is hypothesized to reflect an optimal situation associated with rapid synchronization and information transfer, minimal wiring costs, as well as a balance between local processing and global integration. The topological structure of functional networks is probably restrained by genetic and anatomical factors, but can be modified during tasks. There is also increasing evidence that various types of brain disease such as Alzheimer's disease, schizophrenia, brain tumours and epilepsy may be associated with deviations of the functional network topology from the optimal small-world pattern.

Extracting information from cortical connectivity patterns estimated from high resolution EEG recordings: a theoretical graph approach

Over the last 20 years, a body of techniques known as high resolution EEG has allowed precise estimation of cortical activity from non-invasive EEG measurements. The availability of cortical waveforms from non-invasive EEG recordings allows to have not only the level of activation within a single region of interest (ROI) during a particular task, but also to estimate the causal relationships among activities of several cortical regions. However, interpreting resulting connectivity patterns is still an open issue, due to the difficulty to provide an objective measure of their properties across different subjects or groups. A novel approach addressed to solve this difficulty consists in manipulating these functional brain networks as graph objects for which a large body of indexes and tools are available in literature and already tested for complex networks at different levels of scale (Social, WorldWide-Web and Proteomics). In the present work, we would like to show the suitability of such approach, showing results obtained comparing separately two groups of subjects during the same motor task and two different motor tasks performed by the same group. In the first experiment two groups of subjects (healthy and spinal cord injured patients) were compared when they moved and attempted to move simultaneously their right foot and lips, respectively. The contrast between the foot-lips movement and the simple foot movement was addressed in the second experiment for the population of the healthy subjects. For both the experiments, the main question is whether the "architecture" of the functional connectivity networks obtained could show properties that are different in the two groups or in the two tasks. All the functional connectivity networks gathered in the two experiments showed ordered properties and significant differences from "random" networks having the same characteristic sizes. The proposed approach, based on the use of indexes derived from graph theory, can apply to cerebral connectivity patterns estimated not only from the EEG signals but also from different brain imaging methods.

Small-world networks and epilepsy: Graph theoretical analysis of intracerebrally recorded mesial temporal lobe seizures

OBJECTIVE

Neuronal networks with a so-called "small-world" topography (characterized by strong clustering in combination with short path lengths) are known to facilitate synchronization, and possibly seizure generation. We tested the hypothesis that real functional brain networks during seizures display small-world features, using intracerebral recordings of mesial temporal lobe seizures.

METHODS

We used synchronization likelihood (SL) to characterize synchronization patterns in intracerebral EEG recordings of 7 patients for 5 periods of interest: interictal, before-, during- and after rapid discharges (in which the last two periods are ictal) and postictal. For each period, graphs (abstract network representations) were reconstructed from the synchronization matrix and characterized by a clustering coefficient C (measure of local connectedness) and a shortest path length L (measure of overall network integration). Results were also compared with those obtained from random networks.

RESULTS

The neuronal network changed during seizure activity, with an increase of C and L most prominent in the alpha, theta and delta frequency bands during and after the seizure.

CONCLUSIONS

During seizures, the neuronal network moves in the direction of a more ordered configuration (higher C combined with a slightly, but significantly, higher L) compared to the more randomly organized interictal network, even after correcting for changes in synchronization strength.

SIGNIFICANCE

Analysis of neuronal networks during seizures may provide insight into seizure genesis and development.

Dynamic functional connectivity

Recent studies show that anatomical and functional brain networks exhibit similar small-world properties. However, the networks that are compared often differ in what the nodes represent (e.g. sensors or brain areas), what kind of connectivity is measured, and what temporal and spatial scales are probed. Here, I review studies of large-scale connectivity and recent results from a variety of real-time recording techniques, which together suggest that an adequate description of brain organization requires a hierarchy of networks rather than the single, binary networks that are currently in vogue. Pattern analysis methods now offer a principled way for constructing such network hierarchies. As shown at the end of this review, a correspondence principle can be formulated to guide the interpretation across network levels and to relate nodes to well defined anatomical entities.

Small-world brain networks

Many complex networks have a small-world topology characterized by dense local clustering or cliquishness of connections between neighboring nodes yet a short path length between any (distant) pair of nodes due to the existence of relatively few long-range connections. This is an attractive model for the organization of brain anatomical and functional networks because a small-world topology can support both segregated/specialized and distributed/integrated information processing. Moreover, small-world networks are economical, tending to minimize wiring costs while supporting high dynamical complexity. The authors introduce some of the key mathematical concepts in graph theory required for small-world analysis and review how these methods have been applied to quantification of cortical connectivity matrices derived from anatomical tract-tracing studies in the macaque monkey and the cat. The evolution of small-world networks is discussed in terms of a selection pressure to deliver cost-effective information-processing systems. The authors illustrate how these techniques and concepts are increasingly being applied to the analysis of human brain functional networks derived from electroencephalography/magnetoencephalography and fMRI experiments. Finally, the authors consider the relevance of small-world models for understanding the emergence of complex behaviors and the resilience of brain systems to pathological attack by disease or aberrant development. They conclude that small-world models provide a powerful and versatile approach to understanding the structure and function of human brain systems.

Small-world network organization of functional connectivity of EEG slow-wave activity during sleep

OBJECTIVE

To analyze the functional connectivity patterns of the EEG slow-wave activity during the different sleep stages and Cyclic Alternating Pattern (CAP) conditions, using concepts derived from Graph Theory.

METHODS

We evaluated spatial patterns of EEG slow-wave synchronization between all possible pairs of electrodes (19) placed over the scalp of 10 sleeping healthy young normal subjects using two graph theoretical measures: the clustering coefficient (Cp) and the characteristic path length (Lp). The measures were obtained during the different sleep stages and CAP conditions from the real EEG connectivity networks and randomized control (surrogate) networks (Cp-s and Lp-s).

RESULTS

Cp and Cp/Cp-s increased significantly from wakefulness to sleep while Lp and Lp/Lp-s did not show changes. Cp/Cp-s was higher for A1 phases, compared to B phases of CAP.

CONCLUSIONS

The network organization of the EEG slow-wave synchronization during sleep shows features characteristic of small-world networks (high Cp combined with low Lp); this type of organization is slightly but significantly more evident during the CAP A1 subtypes.

SIGNIFICANCE

Our results show feasibility of using graph theoretical measures to characterize the complexity of brain networks during sleep and might indicate sleep, and the A1 phases of CAP in particular, as a period during which slow-wave synchronization shows optimal network organization for information processing.

Small-World Anatomical Networks in the Human Brain Revealed by Cortical Thickness from MRI

An important issue in neuroscience is the characterization for the underlying architectures of complex brain networks. However, little is known about the network of anatomical connections in the human brain. Here, we investigated large-scale anatomical connection patterns of the human cerebral cortex using cortical thickness measurements from magnetic resonance images. Two areas were considered anatomically connected if they showed statistically significant correlations in cortical thickness and we constructed the network of such connections using 124 brains from the International Consortium for Brain Mapping database. Significant short- and long-range connections were found in both intra- and interhemispheric regions, many of which were consistent with known neuroanatomical pathways measured by human diffusion imaging. More importantly, we showed that the human brain anatomical network had robust small-world properties with cohesive neighborhoods and short mean distances between regions that were insensitive to the selection of correlation thresholds. Additionally, we also found that this network and the probability of finding a connection between 2 regions for a given anatomical distance had both exponentially truncated power-law distributions. Our results demonstrated the basic organizational principles for the anatomical network in the human brain compatible with previous functional networks studies, which provides important implications of how functional brain states originate from their structural underpinnings. To our knowledge, this study provides the first report of small-world properties and degree distribution of anatomical networks in the human brain using cortical thickness measurements.

Adaptive reconfiguration of fractal small-world human brain functional networks

Brain function depends on adaptive self-organization of large-scale neural assemblies, but little is known about quantitative network parameters governing these processes in humans. Here, we describe the topology and synchronizability of frequency-specific brain functional networks using wavelet decomposition of magnetoencephalographic time series, followed by construction and analysis of undirected graphs. Magnetoencephalographic data were acquired from 22 subjects, half of whom performed a finger-tapping task, whereas the other half were studied at rest. We found that brain functional networks were characterized by small-world properties at all six wavelet scales considered, corresponding approximately to classical delta (low and high), , alpha, beta, and gamma frequency bands. Global topological parameters (path length, clustering) were conserved across scales, most consistently in the frequency range 2-37 Hz, implying a scale-invariant or fractal small-world organization. Dynamical analysis showed that networks were located close to the threshold of order/disorder transition in all frequency bands. The highest-frequency gamma network had greater synchronizability, greater clustering of connections, and shorter path length than networks in the scaling regime of (lower) frequencies. Behavioral state did not strongly influence global topology or synchronizability; however, motor task performance was associated with emergence of long-range connections in both beta and gamma networks. Long-range connectivity, e.g., between frontal and parietal cortex, at high frequencies during a motor task may facilitate sensorimotor binding. Human brain functional networks demonstrate a fractal small-world architecture that supports critical dynamics and task-related spatial reconfiguration while preserving global topological parameters.

Small-world networks and disturbed functional connectivity in schizophrenia

Disturbances in "functional connectivity" have been proposed as a major pathophysiological mechanism for schizophrenia, and in particular, for cognitive disorganization. Detection and estimation of these disturbances would be of clinical interest. Here we characterize the spatial pattern of functional connectivity by computing the "synchronization likelihood" (SL) of EEG at rest and during performance of a 2Back working memory task using letters of the alphabet presented on a PC screen in subjects with schizophrenia and healthy controls. The spatial patterns of functional connectivity were then characterized with graph theoretical measures to test whether a disruption of an optimal spatial pattern ("small-world") of the functional connectivity network underlies schizophrenia. Twenty stabilized patients with schizophrenia, who were able to work, and 20 healthy controls participated in the study. During the working memory (WM) task healthy subjects exhibited small-world properties (a combination of local clustering and high overall integration of the functional networks) in the alpha, beta and gamma bands. These properties were not present in the schizophrenia group. These findings are in accordance with a partially inadequate organization of neuronal networks in subjects with schizophrenia. This method could be helpful for diagnosis and evaluation of the severity of the disease, as well as understanding the pathophysiologic mechanisms underlying cognitive dysfunction in schizophrenia.