Adaptive reconfiguration of fractal small-world human brain functional networks

A plastic temporal brain code for conscious state generation

Consciousness is known to be limited in processing capacity and often described in terms of a unique processing stream across a single dimension: time. In this paper, we discuss a purely temporal pattern code, functionally decoupled from spatial signals, for conscious state generation in the brain. Arguments in favour of such a code include Dehaene et al.'s long-distance reverberation postulate, Ramachandran's remapping hypothesis, evidence for a temporal coherence index and coincidence detectors, and Grossberg's Adaptive Resonance Theory. A time-bin resonance model is developed, where temporal signatures of conscious states are generated on the basis of signal reverberation across large distances in highly plastic neural circuits. The temporal signatures are delivered by neural activity patterns which, beyond a certain statistical threshold, activate, maintain, and terminate a conscious brain state like a bar code would activate, maintain, or inactivate the electronic locks of a safe. Such temporal resonance would reflect a higher level of neural processing, independent from sensorial or perceptual brain mechanisms.

Combining fMRI with EEG and MEG in order to relate patterns of brain activity to cognition

The common factor that underlies several types of functional brain imaging is the electric current of masses of dendrites. The prodigious demands for the energy that is required to drive the dendritic currents are met by hemodynamic and metabolic responses that are visualized with fMRI and PET techniques. The high current densities in parallel dendritic shafts and the broad distributions of the loop currents outside the dendrites generate both the scalp EEG and the magnetic fields seen in the MEG. The measurements of image intensities and potential fields provide state variables for modeling. The relationships between the intensities of current density and the electric, magnetic, and hemodynamic state variables are complex and far from proportionate. The state variables are complementary, because the information they convey comes from differing albeit overlapping neural populations, so that efforts to cross-validate localization of neural activity relating to specified cognitive behaviors have not always been successful. We propose an alternative way to use the three methods in combination through studies of hemisphere-wide, high-resolution spatiotemporal patterns of neural activity recorded non-invasively and analyzed with multivariate statistics. Success in this proposed endeavor requires specification of what patterns to look for. At the present level of understanding, an appropriate pattern is any significant departure from random noise in the spectral, temporal and spatial domains that can be scaled into the coarse-graining of time by fMRI/BOLD and the coarse-graining of space by EEG and MEG. Here the requisite patterns are predicted to be large-scale spatial amplitude modulation (AM) of synchronized neuronal signals in the beta and gamma ranges that are coordinated but not correlated with fMRI intensities.

Cognitive fitness of cost-efficient brain functional networks

The human brain's capacity for cognitive function is thought to depend on coordinated activity in sparsely connected, complex networks organized over many scales of space and time. Recent work has demonstrated that human brain networks constructed from neuroimaging data have economical small-world properties that confer high efficiency of information processing at relatively low connection cost. However, it has been unclear how the architecture of complex brain networks functioning at different frequencies can be related to behavioral performance on cognitive tasks. Here, we show that impaired accuracy of working memory could be related to suboptimal cost efficiency of brain functional networks operating in the classical beta frequency band, 15-30 Hz. We analyzed brain functional networks derived from magnetoencephalography data recorded during working-memory task performance in 29 healthy volunteers and 28 people with schizophrenia. Networks functioning at higher frequencies had greater global cost efficiency than low-frequency networks in both groups. Superior task performance was positively correlated with global cost efficiency of the beta-band network and specifically with cost efficiency of nodes in left lateral parietal and frontal areas. These results are consistent with biophysical models highlighting the importance of beta-band oscillations for long-distance functional connections in brain networks and with pathophysiological models of schizophrenia as a dysconnection syndrome. More generally, they echo the saying that "less is more": The information processing performance of a network can be enhanced by a sparse or low-cost configuration with disproportionately high efficiency.

Investigating the properties of self-organization and synchronization in electronic systems

Nonlinear cooperative behavior appears naturally in many systems, such as cardiac cell oscillations; cellular calcium oscillations; oscillatory chemical reactions, and fireflies. Such systems have been studied in detail due to their inherent properties of robustness, adaptability, scalability, and emergence. In this paper, such nonlinear cooperative behaviors are considered within the domain of electronic system design. We investigate these desirable properties in a system composed of electronic oscillators. The paper presents a series of circuit simulation results showing that self-organizing principles, which can be emulated in an electronic circuit, enable the systems to show a phase transition to synchronization, in a manner similar to those of natural systems. Circuit simulation results presented here show that the circuits are robust to the unreliable performance of the electronic oscillators and tolerant to their run-time faults. These are important findings for future engineering applications in which the system's elements are likely to be unreliable and faulty, such as in molecular- and nanoelectronic systems.

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.

Ising-like dynamics in large-scale functional brain networks

Brain "rest" is defined--more or less unsuccessfully--as the state in which there is no explicit brain input or output. This work focuses on the question of whether such state can be comparable to any known dynamical state. For that purpose, correlation networks from human brain functional magnetic resonance imaging are contrasted with correlation networks extracted from numerical simulations of the Ising model in two dimensions at different temperatures. For the critical temperature Tc, striking similarities appear in the most relevant statistical properties, making the two networks indistinguishable from each other. These results are interpreted here as lending support to the conjecture that the dynamics of the functioning brain is near a critical point.

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.

Generic aspects of complexity in brain imaging data and other biological systems

A key challenge for systems neuroscience is the question of how to understand the complex network organization of the brain on the basis of neuroimaging data. Similar challenges exist in other specialist areas of systems biology because complex networks emerging from the interactions between multiple non-trivially interacting agents are found quite ubiquitously in nature, from protein interactomes to ecosystems. We suggest that one way forward for analysis of brain networks will be to quantify aspects of their organization which are likely to be generic properties of a broader class of biological systems. In this introductory review article we will highlight four important aspects of complex systems in general: fractality or scale-invariance; criticality; small-world and related topological attributes; and modularity. For each concept we will provide an accessible introduction, an illustrative data-based example of how it can be used to investigate aspects of brain organization in neuroimaging experiments, and a brief review of how this concept has been applied and developed in other fields of biomedical and physical science. The aim is to provide a didactic, focussed and user-friendly introduction to the concepts of complexity science for neuroscientists and neuroimagers.

Affective learning enhances activity and functional connectivity in early visual cortex

This study examined the impact of task-irrelevant affective information on early visual processing regions V1-V4. Fearful and neutral faces presented with rings of different colors were used as stimuli. During the conditioning phase, fearful faces presented with a certain ring color (e.g., black) were paired with mild electrical stimulation. Neutral faces shown with rings of that color, as well as fearful or neutral faces shown with another ring color (e.g., white), were never paired with shock. Our findings revealed that fearful faces evoked enhanced blood oxygen level dependent (BOLD) responses in V1 and V4 compared to neutral faces. Faces embedded in a color ring that was paired with shock (e.g., black) evoked greater BOLD responses in V1-V4 compared to a ring color that was never paired with shock (e.g., white). Finally, BOLD responses in early visual cortex were tightly interrelated (i.e., correlated) during an affectively potent context (i.e., ring color) but not during a neutral one, suggesting that increased functional integration was present with affective learning. Taken together, the results suggest that task-irrelevant affective information not only influences evoked responses in early, retinotopically organized visual cortex, but also determines the pattern of responses across early visual cortex.

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.

Small-world properties of nonlinear brain activity in schizophrenia

A disturbance in the interactions between distributed cortical regions may underlie the cognitive and perceptual dysfunction associated with schizophrenia. In this article, nonlinear measures of cortical interactions and graph-theoretical metrics of network topography are combined to investigate this schizophrenia "disconnection hypothesis." This is achieved by analyzing the spatiotemporal structure of resting state scalp EEG data previously acquired from 40 young subjects with a recent first episode of schizophrenia and 40 healthy matched controls. In each subject, a method of mapping the topography of nonlinear interactions between cortical regions was applied to a widely distributed array of these data. The resulting nonlinear correlation matrices were converted to weighted graphs. The path length (a measure of large-scale network integration), clustering coefficient (a measure of "cliquishness"), and hub structure of these graphs were used as metrics of the underlying brain network activity. The graphs of both groups exhibited high levels of local clustering combined with comparatively short path lengths--features consistent with a "small-world" topology--as well as the presence of strong, central hubs. The graphs in the schizophrenia group displayed lower clustering and shorter path lengths in comparison to the healthy group. Whilst still "small-world," these effects are consistent with a subtle randomization in the underlying network architecture--likely associated with a greater number of links connecting disparate clusters. This randomization may underlie the cognitive disturbances characteristic of schizophrenia.

Broadband criticality of human brain network synchronization

Self-organized criticality is an attractive model for human brain dynamics, but there has been little direct evidence for its existence in large-scale systems measured by neuroimaging. In general, critical systems are associated with fractal or power law scaling, long-range correlations in space and time, and rapid reconfiguration in response to external inputs. Here, we consider two measures of phase synchronization: the phase-lock interval, or duration of coupling between a pair of (neurophysiological) processes, and the lability of global synchronization of a (brain functional) network. Using computational simulations of two mechanistically distinct systems displaying complex dynamics, the Ising model and the Kuramoto model, we show that both synchronization metrics have power law probability distributions specifically when these systems are in a critical state. We then demonstrate power law scaling of both pairwise and global synchronization metrics in functional MRI and magnetoencephalographic data recorded from normal volunteers under resting conditions. These results strongly suggest that human brain functional systems exist in an endogenous state of dynamical criticality, characterized by a greater than random probability of both prolonged periods of phase-locking and occurrence of large rapid changes in the state of global synchronization, analogous to the neuronal “avalanches” previously described in cellular systems. Moreover, evidence for critical dynamics was identified consistently in neurophysiological systems operating at frequency intervals ranging from 0.05–0.11 to 62.5–125 Hz, confirming that criticality is a property of human brain functional network organization at all frequency intervals in the brain's physiological bandwidth.

Systems in a critical state are poised on the cusp of a transition between ordered and random behavior. At this point, they demonstrate complex patterning of fluctuations at all scales of space and time. Criticality is an attractive model for brain dynamics because it optimizes information transfer, storage capacity, and sensitivity to external stimuli in computational models. However, to date there has been little direct experimental evidence for critical dynamics of human brain networks. Here, we considered two measures of functional coupling or phase synchronization between components of a dynamic system: the phase lock interval or duration of synchronization between a specific pair of time series or processes in the system and the lability of global synchronization among all pairs of processes. We confirmed that both synchronization metrics demonstrated scale invariant behaviors in two computational models of critical dynamics as well as in human brain functional systems oscillating at low frequencies (<0.5 Hz, measured using functional MRI) and at higher frequencies (1–125 Hz, measured using magnetoencephalography). We conclude that human brain functional networks demonstrate critical dynamics in all frequency intervals, a phenomenon we have described as broadband criticality.

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.

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

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

Predicting human resting-state functional connectivity from structural connectivity

In the cerebral cortex, the activity levels of neuronal populations are continuously fluctuating. When neuronal activity, as measured using functional MRI (fMRI), is temporally coherent across 2 populations, those populations are said to be functionally connected. Functional connectivity has previously been shown to correlate with structural (anatomical) connectivity patterns at an aggregate level. In the present study we investigate, with the aid of computational modeling, whether systems-level properties of functional networks--including their spatial statistics and their persistence across time--can be accounted for by properties of the underlying anatomical network. We measured resting state functional connectivity (using fMRI) and structural connectivity (using diffusion spectrum imaging tractography) in the same individuals at high resolution. Structural connectivity then provided the couplings for a model of macroscopic cortical dynamics. In both model and data, we observed (i) that strong functional connections commonly exist between regions with no direct structural connection, rendering the inference of structural connectivity from functional connectivity impractical; (ii) that indirect connections and interregional distance accounted for some of the variance in functional connectivity that was unexplained by direct structural connectivity; and (iii) that resting-state functional connectivity exhibits variability within and across both scanning sessions and model runs. These empirical and modeling results demonstrate that although resting state functional connectivity is variable and is frequently present between regions without direct structural linkage, its strength, persistence, and spatial statistics are nevertheless constrained by the large-scale anatomical structure of the human cerebral cortex.

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

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

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

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.

Investigations of human EEG response to viewing fractal patterns

Owing to the prevalence of fractal patterns in natural scenery and their growing impact on cultures around the world, fractals constitute a common feature of our daily visual experiences, raising an important question: what responses do fractals induce in the observer? We monitored subjects' EEG while they were viewing fractals with different fractal dimensions, and the results show that significant effects could be found in the EEG even by employing relatively simple silhouette images. Patterns with a fractal dimension of 1.3 elicited the most interesting EEG, with the highest alpha in the frontal lobes but also the highest beta in the parietal area, pointing to a complicated interplay between different parts of the brain when experiencing this pattern.

Viewing brain processes as Critical State Transitions across levels of organization: Neural events in Cognition and Consciousness, and general principles

In this theoretical and speculative essay, I propose that insights into certain aspects of neural system functions can be gained from viewing brain function in terms of the branch of Statistical Mechanics currently referred to as "Modern Critical Theory" [Stanley, H.E., 1987. Introduction to Phase Transitions and Critical Phenomena. Oxford University Press; Marro, J., Dickman, R., 1999. Nonequilibrium Phase Transitions in Lattice Models. Cambridge University Press, Cambridge, UK]. The application of this framework is here explored in two stages: in the first place, its principles are applied to state transitions in global brain dynamics, with benchmarks of Cognitive Neuroscience providing the relevant empirical reference points. The second stage generalizes to suggest in more detail how the same principles could also apply to the relation between other levels of the structural-functional hierarchy of the nervous system and between neural assemblies. In this view, state transitions resulting from the processing at one level are the input to the next, in the image of a 'bucket brigade', with the content of each bucket being passed on along the chain, after having undergone a state transition. The unique features of a process of this kind will be discussed and illustrated.

Age-related changes in modular organization of human brain functional networks

Graph theory allows us to quantify any complex system, e.g., in social sciences, biology or technology, that can be abstractly described as a set of nodes and links. Here we derived human brain functional networks from fMRI measurements of endogenous, low frequency, correlated oscillations in 90 cortical and subcortical regions for two groups of healthy (young and older) participants. We investigated the modular structure of these networks and tested the hypothesis that normal brain aging might be associated with changes in modularity of sparse networks. Newman's modularity metric was maximised and topological roles were assigned to brain regions depending on their specific contributions to intra- and inter-modular connectivity. Both young and older brain networks demonstrated significantly non-random modularity. The young brain network was decomposed into 3 major modules: central and posterior modules, which comprised mainly nodes with few inter-modular connections, and a dorsal fronto-cingulo-parietal module, which comprised mainly nodes with extensive inter-modular connections. The mean network in the older group also included posterior, superior central and dorsal fronto-striato-thalamic modules but the number of intermodular connections to frontal modular regions was significantly reduced, whereas the number of connector nodes in posterior and central modules was increased.

Hierarchical organization of human cortical networks in health and schizophrenia

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

Imaging structural and functional connectivity: towards a unified definition of human brain organization?

PURPOSE OF REVIEW

Diffusion tractography and functional/effective connectivity MRI provide a better understanding of the structural and functional human brain connectivity. This review will underline the major recent methodological developments and their exceptional respective contributions to physiological and pathophysiological studies in vivo. We will also emphasize the benefits provided by computational models of complex networks such as graph theory.

RECENT FINDINGS

Imaging structural and functional brain connectivity has revealed the complex brain organization into large-scale networks. Such an organization not only permits the complex information segregation and integration during high cognitive processes but also determines the clinical consequences of alterations encountered in development, ageing, or neurological diseases. Recently, it has also been demonstrated that human brain networks shared topological properties with the so-called 'small-world' mathematical model, allowing a maximal efficiency with a minimal energy and wiring cost.

SUMMARY

Separately, magnetic resonance tractography and functional MRI connectivity have both brought new insights into brain organization and the impact of injuries. The small-world topology of structural and functional human brain networks offers a common framework to merge structural and functional imaging as well as dynamical data from electrophysiology that might allow a comprehensive definition of the brain organization and plasticity.

Evolving functional network properties and synchronizability during human epileptic seizures

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

The systems view in addiction research

The wealth of information accessible on the molecular level after completion of sequencing of the human and other genomes and in conjunction with new high-throughput technologies like microarrays has paved the way for research on whole systems rather than on single components. Here we describe exemplarily the construction of a rather complex molecular network involved in alcohol addiction by using information from DNA-microarray studies in alcohol-dependent animals. In this network, haemoglobin downregulation in different parts of the brain reward system plays a central role in affecting synaptic plasticity, circadian rhythmicity and opioid receptors. Furthermore, we discuss the dynamic aspect of biological systems with respect to repeated and intermittent drug intake. This aspect can best be captured by the allostatic model on the molecular level. Using a molecular oscillator model where levels of oscillations are changed by repetitive drug administration, changes in set point adjustment are described that underlay allostatic shifts in drug reinforcement processes.

Dynamical consequences of lesions in cortical networks

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

Mapping the structural core of human cerebral cortex

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

Endogenous brain fluctuations and diagnostic imaging

Much of the rising health care costs in aging populations can be attributed to congenital disease and psychiatric and neurologic disorders. Early detection of changes related to these diseases can promote the development of new therapeutic strategies and effective treatments. Changes in tissue, such as damage resulting from continued functional abnormality, often exhibit a time-delay before detection is possible. Methods for detecting functional alterations in endogenous brain fluctuations allow for an early diagnosis before tissue damage occurs, enabling early treatment and a more likely positive outcome. A literature review and comprehensive overview of the current state of knowledge about endogenous brain fluctuations is presented here. Recent findings of the association between various pathological conditions and endogenous fluctuations are discussed. A particular emphasis is placed on research showing the relationship between clinical measures and pathological findings to the dynamics of endogenous fluctuations of the brain. Recent discoveries of methods for detecting abnormal functional connectivity are discussed and future research directions explored.

Structural insights into aberrant topological patterns of large-scale cortical networks in Alzheimer's disease

Recent research on Alzheimer's disease (AD) has shown that cognitive and memory decline in this disease is accompanied by disrupted changes in the coordination of large-scale brain functional networks. However, alterations in coordinated patterns of structural brain networks in AD are still poorly understood. Here, we used cortical thickness measurement from magnetic resonance imaging to investigate large-scale structural brain networks in 92 AD patients and 97 normal controls. Brain networks were constructed by thresholding cortical thickness correlation matrices of 54 regions and analyzed using graph theoretical approaches. Compared with controls, AD patients showed decreased cortical thickness intercorrelations between the bilateral parietal regions and increased intercorrelations in several selective regions involving the lateral temporal and parietal cortex as well as the cingulate and medial frontal cortex regions. Specially, AD patients showed abnormal small-world architecture in the structural cortical networks (increased clustering and shortest paths linking individual regions), implying a less optimal topological organization in AD. Moreover, AD patients were associated with reduced nodal centrality predominantly in the temporal and parietal heteromodal association cortex regions and increased nodal centrality in the occipital cortex regions. Finally, the brain networks of AD were about equally as robust to random failures as those of controls, but more vulnerable against targeted attacks, presumably because of the effects of pathological topological organization. Our findings suggest that the coordinated patterns of cortical morphology are widely altered in AD patients, thus providing structural evidence for disrupted integrity in large-scale brain networks that underlie cognition. This work has implications for our understanding of how functional deficits in patients are associated with their underlying structural (morphological) basis.

Dynamic small-world behavior in functional brain networks unveiled by an event-related networks approach

There is growing interest in studying the role of connectivity patterns in brain functions. In recent years, functional brain networks were found to exhibit small-world properties during different brain states. In previous studies, time-independent networks were recovered from long time periods of brain activity. In this paper, we propose an approach, the event-related networks, that allows one to characterize the dynamical evolution of functional brain networks in time-frequency space. We illustrate this approach by characterizing connectivity patterns in magnetoencephalographic signals recorded during a visual stimulus paradigm. When compared with equivalent random and regular networks, the results reveal that functional connectivity varies with time and frequency during the processing of the stimulus, while maintaining a small-world structure. This approach may provide insights into the connectivity of other complex and spatially extended nonstationary systems.

Network 'small-world-ness': a quantitative method for determining canonical network equivalence

BACKGROUND

Many technological, biological, social, and information networks fall into the broad class of 'small-world' networks: they have tightly interconnected clusters of nodes, and a shortest mean path length that is similar to a matched random graph (same number of nodes and edges). This semi-quantitative definition leads to a categorical distinction ('small/not-small') rather than a quantitative, continuous grading of networks, and can lead to uncertainty about a network's small-world status. Moreover, systems described by small-world networks are often studied using an equivalent canonical network model--the Watts-Strogatz (WS) model. However, the process of establishing an equivalent WS model is imprecise and there is a pressing need to discover ways in which this equivalence may be quantified.

METHODOLOGY/PRINCIPAL FINDINGS

We defined a precise measure of 'small-world-ness' S based on the trade off between high local clustering and short path length. A network is now deemed a 'small-world' if S>1--an assertion which may be tested statistically. We then examined the behavior of S on a large data-set of real-world systems. We found that all these systems were linked by a linear relationship between their S values and the network size n. Moreover, we show a method for assigning a unique Watts-Strogatz (WS) model to any real-world network, and show analytically that the WS models associated with our sample of networks also show linearity between S and n. Linearity between S and n is not, however, inevitable, and neither is S maximal for an arbitrary network of given size. Linearity may, however, be explained by a common limiting growth process.

CONCLUSIONS/SIGNIFICANCE

We have shown how the notion of a small-world network may be quantified. Several key properties of the metric are described and the use of WS canonical models is placed on a more secure footing.

Consciousness related neural events viewed as brain state space transitions

This theoretical and speculative essay addresses a categorical distinction between neural events of sensory-motor cognition and those presumably associated with consciousness. It proposes to view this distinction in the framework of the branch of Statistical Physics currently referred to as Modern Critical Theory (Stanley, Introduction to phase transitions and critical phenomena, 1987; Marro and Dickman, Nonequilibrium phase transitions in lattice, 1999). Based on established landmarks of brain dynamics, network configurations and their role for conveying oscillatory activity of certain frequencies bands, the question is examined: what kind of state space transitions can systems with these properties undergo, and could the relation between neural processes of sensory-motor cognition and those of events in consciousness be of the same category as is characterized by state transitions in non-equilibrium physical systems? Approaches for empirical validation of this view by suitably designed brain imaging studies, and for computational simulations of the proposed principle are discussed.

A small world of neuronal synchrony

A small-world network has been suggested to be an efficient solution for achieving both modular and global processing-a property highly desirable for brain computations. Here, we investigated functional networks of cortical neurons using correlation analysis to identify functional connectivity. To reconstruct the interaction network, we applied the Ising model based on the principle of maximum entropy. This allowed us to assess the interactions by measuring pairwise correlations and to assess the strength of coupling from the degree of synchrony. Visual responses were recorded in visual cortex of anesthetized cats, simultaneously from up to 24 neurons. First, pairwise correlations captured most of the patterns in the population's activity and, therefore, provided a reliable basis for the reconstruction of the interaction networks. Second, and most importantly, the resulting networks had small-world properties; the average path lengths were as short as in simulated random networks, but the clustering coefficients were larger. Neurons differed considerably with respect to the number and strength of interactions, suggesting the existence of "hubs" in the network. Notably, there was no evidence for scale-free properties. These results suggest that cortical networks are optimized for the coexistence of local and global computations: feature detection and feature integration or binding.

Mapping functional connectivity in patients with brain lesions

OBJECTIVE

The spatial distribution of functional connectivity between brain areas and the disturbance introduced by focal brain lesions are poorly understood. Based on the rationale that damaged brain tissue is disconnected from the physiological interactions among healthy areas, this study aimed to map the functionality of brain areas according to their connectivity with other areas.

METHODS

Magnetoencephalography recordings of spontaneous cortical activity during resting state were obtained from 15 consecutive patients with focal brain lesions and from 14 healthy control subjects. Neural activity in the brain was estimated using an adaptive spatial filtering technique. The mean imaginary coherence between brain voxels was then calculated as an index of functional connectivity.

RESULTS

Imaginary coherence was greatest in the alpha frequency range corresponding to the human cortical idling rhythm. In healthy subjects, functionally critical brain areas such as the somatosensory and language cortices had the highest alpha coherence. When compared with healthy control subjects, all lesion patients had diffuse or scattered brain areas with decreased alpha coherence. Patients with lesion-induced neurological deficits displayed decreased connectivity estimates in the corresponding brain area compared with intact contralateral regions. In tumor patients without preoperative neurological deficits, brain areas showing decreased coherence could be surgically resected without the occurrence of postoperative deficits.

INTERPRETATION

Resting state coherence measured with magnetoencephalography is capable of mapping the functional connectivity of the brain, and can therefore offer valuable information for use in planning resective surgeries in patients with brain lesions, as well as investigations into structural-functional relationships in healthy subjects.

Fractal connectivity of long-memory networks

Using the multivariate long memory (LM) model and Taylor expansions, we find the conditions for convergence of the wavelet correlations between two LM processes on an asymptotic value at low frequencies. These mathematical results, and a least squares estimator of LM parameters, are validated in simulations and applied to neurophysiological (human brain) and financial market time series. Both brain and market systems had multivariate LM properties including a "fractal connectivity" regime of scales over which wavelet correlations were invariantly close to their asymptotic value. This analysis provides efficient and unbiased estimation of long-term correlations in diverse dynamic networks.

On the relationship between emotion and cognition

The current view of brain organization supports the notion that there is a considerable degree of functional specialization and that many regions can be conceptualized as either 'affective' or 'cognitive'. Popular examples are the amygdala in the domain of emotion and the lateral prefrontal cortex in the case of cognition. This prevalent view is problematic for a number of reasons. Here, I will argue that complex cognitive-emotional behaviours have their basis in dynamic coalitions of networks of brain areas, none of which should be conceptualized as specifically affective or cognitive. Central to cognitive-emotional interactions are brain areas with a high degree of connectivity, called hubs, which are critical for regulating the flow and integration of information between regions.

A dual-networks architecture of top-down control

Complex systems ensure resilience through multiple controllers acting at rapid and slower timescales. The need for efficient information flow through complex systems encourages small-world network structures. On the basis of these principles, a group of regions associated with top-down control was examined. Functional magnetic resonance imaging showed that each region had a specific combination of control signals; resting-state functional connectivity grouped the regions into distinct 'fronto-parietal' and 'cingulo-opercular' components. The fronto-parietal component seems to initiate and adjust control; the cingulo-opercular component provides stable 'set-maintenance' over entire task epochs. Graph analysis showed dense local connections within components and weaker 'long-range' connections between components, suggesting a small-world architecture. The control systems of the brain seem to embody the principles of complex systems, encouraging resilient performance.

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.