Theoretical neuroanatomy: relating anatomical and functional connectivity in graphs and cortical connection matrices

Dynamical reconnection and stability constraints on cortical network architecture

Stability under dynamical changes to network connectivity is invoked alongside previous criteria to constrain brain network architecture. A new hierarchical network is introduced that satisfies all these constraints, unlike more commonly studied regular, random, and small-world networks. It is shown that hierarchical networks can simultaneously have high clustering, short path lengths, and low wiring costs, while being robustly stable under large scale reconnection of substructures.

Tractography-based priors for dynamic causal models

Functional integration in the brain rests on anatomical connectivity (the presence of axonal connections) and effective connectivity (the causal influences mediated by these connections). The deployment of anatomical connections provides important constraints on effective connectivity, but does not fully determine it, because synaptic connections can be expressed functionally in a dynamic and context-dependent fashion. Although it is generally assumed that anatomical connectivity data is important to guide the construction of neurobiologically realistic models of effective connectivity; the degree to which these models actually profit from anatomical constraints has not yet been formally investigated. Here, we use diffusion weighted imaging and probabilistic tractography to specify anatomically informed priors for dynamic causal models (DCMs) of fMRI data. We constructed 64 alternative DCMs, which embodied different mappings between the probability of an anatomical connection and the prior variance of the corresponding of effective connectivity, and fitted them to empirical fMRI data from 12 healthy subjects. Using Bayesian model selection, we show that the best model is one in which anatomical probability increases the prior variance of effective connectivity parameters in a nonlinear and monotonic (sigmoidal) fashion. This means that the higher the likelihood that a given connection exists anatomically, the larger one should set the prior variance of the corresponding coupling parameter; hence making it easier for the parameter to deviate from zero and represent a strong effective connection. To our knowledge, this study provides the first formal evidence that probabilistic knowledge of anatomical connectivity can improve models of functional integration.

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.

Complex modular structure of large-scale brain networks

Modular structure is ubiquitous among real-world networks from related proteins to social groups. Here we analyze the modular organization of brain networks at a large scale (voxel level) extracted from functional magnetic resonance imaging signals. By using a random-walk-based method, we unveil the modularity of brain webs and show modules with a spatial distribution that matches anatomical structures with functional significance. The functional role of each node in the network is studied by analyzing its patterns of inter- and intramodular connections. Results suggest that the modular architecture constitutes the structural basis for the coexistence of functional integration of distant and specialized brain areas during normal brain activities at rest.

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.

Neural complexity and structural connectivity

Tononi [Proc. Natl. Acad. Sci. U.S.A. 91, 5033 (1994)] proposed a measure of neural complexity based on mutual information between complementary subsystems of a given neural network, which has attracted much interest in the neuroscience community and beyond. We develop an approximation of the measure for a popular Gaussian model which, applied to a continuous-time process, elucidates the relationship between the complexity of a neural system and its structural connectivity. Moreover, the approximation is accurate for weakly coupled systems and computationally cheap, scaling polynomially with system size in contrast to the full complexity measure, which scales exponentially. We also discuss connectivity normalization and resolve some issues stemming from an ambiguity in the original Gaussian model.

Uncovering intrinsic modular organization of spontaneous brain activity in humans

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

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

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

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.

Identifying neural drivers with functional MRI: an electrophysiological validation

Whether functional magnetic resonance imaging (fMRI) allows the identification of neural drivers remains an open question of particular importance to refine physiological and neuropsychological models of the brain, and/or to understand neurophysiopathology. Here, in a rat model of absence epilepsy showing spontaneous spike-and-wave discharges originating from the first somatosensory cortex (S1BF), we performed simultaneous electroencephalographic (EEG) and fMRI measurements, and subsequent intracerebral EEG (iEEG) recordings in regions strongly activated in fMRI (S1BF, thalamus, and striatum). fMRI connectivity was determined from fMRI time series directly and from hidden state variables using a measure of Granger causality and Dynamic Causal Modelling that relates synaptic activity to fMRI. fMRI connectivity was compared to directed functional coupling estimated from iEEG using asymmetry in generalised synchronisation metrics. The neural driver of spike-and-wave discharges was estimated in S1BF from iEEG, and from fMRI only when hemodynamic effects were explicitly removed. Functional connectivity analysis applied directly on fMRI signals failed because hemodynamics varied between regions, rendering temporal precedence irrelevant. This paper provides the first experimental substantiation of the theoretical possibility to improve interregional coupling estimation from hidden neural states of fMRI. As such, it has important implications for future studies on brain connectivity using functional neuroimaging.

Our understanding of how the brain works relies on the development of neuropsychological models, which describe how brain activity is coordinated among different regions during the execution of a given task. Knowing the directionality of information transfer between connected regions, and in particular distinguishing neural drivers, or the source of forward connections in the brain, from other brain regions, is critical to refine models of the brain. However, whether functional magnetic resonance imaging (fMRI), the most common technique for imaging brain function, allows one to identify neural drivers remains an open question. Here, we used a rat model of absence epilepsy, a form of nonconvulsive epilepsy that occurs during childhood in humans, showing spontaneous spike-and-wave discharges (nonconvulsive seizures) originating from the first somatosensory cortex, to validate several functional connectivity measures derived from fMRI. Standard techniques estimating interactions directly from fMRI data failed because blood flow dynamics varied between regions. However, we were able to identify the neural driver of spike-and-wave discharges when hemodynamic effects were explicitly removed using appropriate modelling. This study thus provides the first experimental substantiation of the theoretical possibility to improve interregional coupling estimation from hidden neural states of fMRI. As such, it has important implications for future studies on connectivity in the functional neuroimaging literature.

Neural long-range interactions can be distinguished from hemodynamic confounds in functional magnetic resonance imaging using new data analysis techniques that will allow experimental validation of models of brain function.

Propofol induction reduces the capacity for neural information integration: implications for the mechanism of consciousness and general anesthesia

The cognitive unbinding paradigm suggests that the synthesis of neural information is attenuated by general anesthesia. Here, we analyzed the functional organization of brain activities in the conscious and anesthetized states, based on functional segregation and integration. Electroencephalography (EEG) recordings were obtained from 14 subjects undergoing induction of general anesthesia with propofol. We quantified changes in mean information integration capacity in each band of the EEG. After induction with propofol, mean information integration capacity was reduced most prominently in the gamma band of the EEG (p=.0001). Furthermore, we demonstrate that loss of consciousness is reflected by the breakdown of the spatiotemporal organization of gamma waves. We conclude that induction of general anesthesia with propofol reduces the capacity for information integration in the brain. These data directly support the information integration theory of consciousness and the cognitive unbinding paradigm of general anesthesia.

Consciousness as integrated information: a provisional manifesto

The integrated information theory (IIT) starts from phenomenology and makes use of thought experiments to claim that consciousness is integrated information. Specifically: (i) the quantity of consciousness corresponds to the amount of integrated information generated by a complex of elements; (ii) the quality of experience is specified by the set of informational relationships generated within that complex. Integrated information (Phi) is defined as the amount of information generated by a complex of elements, above and beyond the information generated by its parts. Qualia space (Q) is a space where each axis represents a possible state of the complex, each point is a probability distribution of its states, and arrows between points represent the informational relationships among its elements generated by causal mechanisms (connections). Together, the set of informational relationships within a complex constitute a shape in Q that completely and univocally specifies a particular experience. Several observations concerning the neural substrate of consciousness fall naturally into place within the IIT framework. Among them are the association of consciousness with certain neural systems rather than with others; the fact that neural processes underlying consciousness can influence or be influenced by neural processes that remain unconscious; the reduction of consciousness during dreamless sleep and generalized seizures; and the distinct role of different cortical architectures in affecting the quality of experience. Equating consciousness with integrated information carries several implications for our view of nature.

Hierarchical organization of human cortical networks in health and schizophrenia

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

Dynamical complexity in small-world networks of spiking neurons

A computer model is described which is used to assess the dynamical complexity of a class of networks of spiking neurons with small-world properties. Networks are constructed by forming an initially segregated set of highly intraconnected clusters and then applying a probabilistic rewiring method reminiscent of the Watts-Strogatz procedure to make intercluster connections. Causal density, which counts the number of independent significant interactions among a system's components, is used to assess dynamical complexity. This measure was chosen because it employs lagged observations, and is therefore more sensitive to temporally smeared evidence of segregation and integration than its alternatives. The results broadly support the hypothesis that small-world topology promotes dynamical complexity, but reveal a narrow parameter range within which this occurs for the network topology under investigation, and suggest an inverse correlation with phase synchrony inside this range.

Précis of neuroconstructivism: how the brain constructs cognition

Neuroconstructivism: How the Brain Constructs Cognition proposes a unifying framework for the study of cognitive development that brings together (1) constructivism (which views development as the progressive elaboration of increasingly complex structures), (2) cognitive neuroscience (which aims to understand the neural mechanisms underlying behavior), and (3) computational modeling (which proposes formal and explicit specifications of information processing). The guiding principle of our approach is context dependence, within and (in contrast to Marr [1982]) between levels of organization. We propose that three mechanisms guide the emergence of representations: competition, cooperation, and chronotopy; which themselves allow for two central processes: proactivity and progressive specialization. We suggest that the main outcome of development is partial representations, distributed across distinct functional circuits. This framework is derived by examining development at the level of single neurons, brain systems, and whole organisms. We use the terms encellment, embrainment, and embodiment to describe the higher-level contextual influences that act at each of these levels of organization. To illustrate these mechanisms in operation we provide case studies in early visual perception, infant habituation, phonological development, and object representations in infancy. Three further case studies are concerned with interactions between levels of explanation: social development, atypical development and within that, developmental dyslexia. We conclude that cognitive development arises from a dynamic, contextual change in embodied neural structures leading to partial representations across multiple brain regions and timescales, in response to proactively specified physical and social environment.

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.

Integrated information in discrete dynamical systems: motivation and theoretical framework

This paper introduces a time- and state-dependent measure of integrated information, phi, which captures the repertoire of causal states available to a system as a whole. Specifically, phi quantifies how much information is generated (uncertainty is reduced) when a system enters a particular state through causal interactions among its elements, above and beyond the information generated independently by its parts. Such mathematical characterization is motivated by the observation that integrated information captures two key phenomenological properties of consciousness: (i) there is a large repertoire of conscious experiences so that, when one particular experience occurs, it generates a large amount of information by ruling out all the others; and (ii) this information is integrated, in that each experience appears as a whole that cannot be decomposed into independent parts. This paper extends previous work on stationary systems and applies integrated information to discrete networks as a function of their dynamics and causal architecture. An analysis of basic examples indicates the following: (i) phi varies depending on the state entered by a network, being higher if active and inactive elements are balanced and lower if the network is inactive or hyperactive. (ii) phi varies for systems with identical or similar surface dynamics depending on the underlying causal architecture, being low for systems that merely copy or replay activity states. (iii) phi varies as a function of network architecture. High phi values can be obtained by architectures that conjoin functional specialization with functional integration. Strictly modular and homogeneous systems cannot generate high phi because the former lack integration, whereas the latter lack information. Feedforward and lattice architectures are capable of generating high phi but are inefficient. (iv) In Hopfield networks, phi is low for attractor states and neutral states, but increases if the networks are optimized to achieve tension between local and global interactions. These basic examples appear to match well against neurobiological evidence concerning the neural substrates of consciousness. More generally, phi appears to be a useful metric to characterize the capacity of any physical system to integrate information.

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.

On how network architecture determines the dominant patterns of spontaneous neural activity

In the absence of sensory stimulation, neocortical circuits display complex patterns of neural activity. These patterns are thought to reflect relevant properties of the network, including anatomical features like its modularity. It is also assumed that the synaptic connections of the network constrain the repertoire of emergent, spontaneous patterns. Although the link between network architecture and network activity has been extensively investigated in the last few years from different perspectives, our understanding of the relationship between the network connectivity and the structure of its spontaneous activity is still incomplete. Using a general mathematical model of neural dynamics we have studied the link between spontaneous activity and the underlying network architecture. In particular, here we show mathematically how the synaptic connections between neurons determine the repertoire of spatial patterns displayed in the spontaneous activity. To test our theoretical result, we have also used the model to simulate spontaneous activity of a neural network, whose architecture is inspired by the patchy organization of horizontal connections between cortical columns in the neocortex of primates and other mammals. The dominant spatial patterns of the spontaneous activity, calculated as its principal components, coincide remarkably well with those patterns predicted from the network connectivity using our theory. The equivalence between the concept of dominant pattern and the concept of attractor of the network dynamics is also demonstrated. This in turn suggests new ways of investigating encoding and storage capabilities of neural networks.

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.

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.

Low-dimensional embedding of fMRI datasets

We propose a novel method to embed a functional magnetic resonance imaging (fMRI) dataset in a low-dimensional space. The embedding optimally preserves the local functional coupling between fMRI time series and provides a low-dimensional coordinate system for detecting activated voxels. To compute the embedding, we build a graph of functionally connected voxels. We use the commute time, instead of the geodesic distance, to measure functional distances on the graph. Because the commute time can be computed directly from the eigenvectors of (a symmetric version) the graph probability transition matrix, we use these eigenvectors to embed the dataset in low dimensions. After clustering the datasets in low dimensions, coherent structures emerge that can be easily interpreted. We performed an extensive evaluation of our method comparing it to linear and nonlinear techniques using synthetic datasets and in vivo datasets. We analyzed datasets from the EBC competition obtained with subjects interacting in an urban virtual reality environment. Our exploratory approach is able to detect independently visual areas (V1/V2, V5/MT), auditory areas, and language areas. Our method can be used to analyze fMRI collected during "natural stimuli".

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.

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.

Estimating interdependences in networks of weakly coupled deterministic systems

The extraction of information from measured data about the interactions taking place in a network of systems is a key topic in modern applied sciences. This topic has been traditionally addressed by considering bivariate time series, providing methods which are sometimes difficult to extend to multivariate data, the limiting factor being the computational complexity. Here, we present a computationally viable method based on black-box modeling which, while theoretically applicable only when a deterministic hypothesis about the processes behind the recordings is plausible, proves to work also when this assumption is severely affected. Conceptually, the method is very simple and is composed of three independent steps: in the first step a state-space reconstruction is performed separately on each measured signal; in the second step, a local model, i.e., a nonlinear dynamical system, is fitted separately on each (reconstructed) measured signal; afterward, a linear model of the dynamical interactions is obtained by cross-relating the (reconstructed) measured variables to the dynamics unexplained by the local models. The method is successfully validated on numerically generated data. An assessment of its sensitivity to data length and modeling and measurement noise intensity, and of its applicability to large-scale systems, is also provided.

A model of grid cells involving extra hippocampal path integration, and the hippocampal loop

In this paper, we present a model for the generation of grid cells and the emergence of place cells from multimodal input to the entorhinal cortex (EC). In this model, grid cell activity in the dorsocaudal medial entorhinal cortex (dMEC) [28] results from the operation of a long-distance path integration system located outside the hippocampal formation, presumably in retrosplenial and/or parietal cortex. If the connections between these structures and dMEC are organized as a modulo N operator, the resulting activity of dMEC neurons is a grid cell pattern. Furthermore, a robust high-resolution positional code can be built from a small set of different grid cells if the modulo factors are relatively prime. On the other hand, broad visual place cell activity in the MEC can result from the integration of visual information depending on the view-field of the visual input. The merging of entorhinal visual place cell information and grid cell information in the EC and/or in the dentate gyrus (DG) allows the building of precise and robust "place cells" (e.g., whose activity is maintained if light is suppressed for a short duration). Our model supports our previous proposition that hippocampal "place cell" activity code transitions between two successive states ("transition cells") rather than mere current locations. Furthermore, we discuss the possibility that the hippocampal loop participates in the emergence of grid cell activity but is not sufficient by itself. Finally, path integration at a short time scale (which is reset from one place to the next) would be merged in the subiculum with CA3/CA1 "transition cells" [22] to provide a robust feedback about current action to the deep layer of the entorhinal cortex in order to predict the recognition of the new animal location.

Identification of genetically mediated cortical networks: a multivariate study of pediatric twins and siblings

Structural magnetic resonance imaging data from 308 twins, 64 singleton siblings of twins, and 228 singletons were analyzed using structural equation modeling and selected multivariate methods to identify genetically mediated intracortical associations. Principal components analyses (PCA) of the genetic correlation matrix indicated a single factor accounting for over 60% of the genetic variability in cortical thickness. When covaried for mean global cortical thickness, PCA, cluster analyses, and graph models identified genetically mediated fronto-parietal and occipital networks. Graph theoretical models suggest that the observed genetically mediated relationships follow small world architectural rules. These findings are largely concordant with other multivariate studies of brain structure and function, the twin literature, and current understanding on the role of genes in cortical neurodevelopment.

Coemergence of regularity and complexity during neural network development

With the growing recognition that rhythmic and oscillatory patterns are widespread in the brain and play important roles in all aspects of the function of our nervous system, there has been a resurgence of interest in neuronal synchronized bursting activity. Here, we were interested in understanding the development of synchronized bursts as information-bearing neuronal activity patterns. For that, we have monitored the morphological organization and spontaneous activity of neuronal networks cultured on multielectrode-arrays during their self-executed evolvement from a mixture of dissociated cells into an active network. Complex collective network electrical activity evolved from sporadic firing patterns of the single neurons. On the system (network) level, the activity was marked by bursting events with interneuronal synchronization and nonarbitrary temporal ordering. We quantified these individual-to-collective activity transitions using newly-developed system level quantitative measures of time series regularity and complexity. We found that individual neuronal activity before synchronization was characterized by high regularity and low complexity. During neuronal wiring, there was a transient period of reorganization marked by low regularity, which then leads to coemergence of elevated regularity and functional (nonstochastic) complexity. We further investigated the morphology-activity interplay by modeling artificial neuronal networks with different topological organizations and connectivity schemes. The simulations support our experimental results by showing increased levels of complexity of neuronal activity patterns when neurons are wired up and organized in clusters (similar to mature real networks), as well as network-level activity regulation once collective activity forms.

Brain architecture: a design for natural computation

Fifty years ago, John von Neumann compared the architecture of the brain with that of the computers he invented and which are still in use today. In those days, the organization of computers was based on concepts of brain organization. Here, we give an update on current results on the global organization of neural systems. For neural systems, we outline how the spatial and topological architecture of neuronal and cortical networks facilitates robustness against failures, fast processing and balanced network activation. Finally, we discuss mechanisms of self-organization for such architectures. After all, the organization of the brain might again inspire computer architecture.

Combining structural and functional neuroimaging data for studying brain connectivity: a review

Different brain areas are thought to be integrated into large-scale networks to support cognitive function. Recent approaches for investigating structural organization and functional coordination within these networks involve measures of connectivity among brain areas. We review studies combining in vivo structural and functional brain connectivity data, where (a) structural connectivity analysis, mostly based on diffusion tensor imaging is paired with voxel-wise analysis of functional neuroimaging data or (b) the measurement of functional connectivity based on covariance analysis is guided/aided by structural connectivity data. These studies provide insights into the relationships between brain structure and function. Promising trends involve (a) studies where both functional and anatomical connectivity data are collected using high-resolution neuroimaging methods and (b) the development of advanced quantitative models of integration.

Identification and classification of hubs in brain networks

Brain regions in the mammalian cerebral cortex are linked by a complex network of fiber bundles. These inter-regional networks have previously been analyzed in terms of their node degree, structural motif, path length and clustering coefficient distributions. In this paper we focus on the identification and classification of hub regions, which are thought to play pivotal roles in the coordination of information flow. We identify hubs and characterize their network contributions by examining motif fingerprints and centrality indices for all regions within the cerebral cortices of both the cat and the macaque. Motif fingerprints capture the statistics of local connection patterns, while measures of centrality identify regions that lie on many of the shortest paths between parts of the network. Within both cat and macaque networks, we find that a combination of degree, motif participation, betweenness centrality and closeness centrality allows for reliable identification of hub regions, many of which have previously been functionally classified as polysensory or multimodal. We then classify hubs as either provincial (intra-cluster) hubs or connector (inter-cluster) hubs, and proceed to show that lesioning hubs of each type from the network produces opposite effects on the small-world index. Our study presents an approach to the identification and classification of putative hub regions in brain networks on the basis of multiple network attributes and charts potential links between the structural embedding of such regions and their functional roles.

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.

Simulation of robustness against lesions of cortical networks

Structure entails function, and thus a structural description of the brain will help to understand its function and may provide insights into many properties of brain systems, from their robustness and recovery from damage to their dynamics and even their evolution. Advances in the analysis of complex networks provide useful new approaches to understanding structural and functional properties of brain networks. Structural properties of networks recently described allow their characterization as small-world, random (exponential) and scale-free. They complement the set of other properties that have been explored in the context of brain connectivity, such as topology, hodology, clustering and hierarchical organization. Here we apply new network analysis methods to cortical interareal connectivity networks for the cat and macaque brains. We compare these corticocortical fibre networks to benchmark rewired, small-world, scale-free and random networks using two analysis strategies, in which we measure the effects of the removal of nodes and connections on the structural properties of the cortical networks. The structural decay of the brain networks is in most respects similar to that of scale-free networks. The results implicate highly connected hub-nodes and bottleneck connections as a structural basis for some of the conditional robustness of brain systems. This informs the understanding of the development of connectivity of the brain networks.

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.

Network structure of cerebral cortex shapes functional connectivity on multiple time scales

Neuronal dynamics unfolding within the cerebral cortex exhibit complex spatial and temporal patterns even in the absence of external input. Here we use a computational approach in an attempt to relate these features of spontaneous cortical dynamics to the underlying anatomical connectivity. Simulating nonlinear neuronal dynamics on a network that captures the large-scale interregional connections of macaque neocortex, and applying information theoretic measures to identify functional networks, we find structure-function relations at multiple temporal scales. Functional networks recovered from long windows of neural activity (minutes) largely overlap with the underlying structural network. As a result, hubs in these long-run functional networks correspond to structural hubs. In contrast, significant fluctuations in functional topology are observed across the sequence of networks recovered from consecutive shorter (seconds) time windows. The functional centrality of individual nodes varies across time as interregional couplings shift. Furthermore, the transient couplings between brain regions are coordinated in a manner that reveals the existence of two anticorrelated clusters. These clusters are linked by prefrontal and parietal regions that are hub nodes in the underlying structural network. At an even faster time scale (hundreds of milliseconds) we detect individual episodes of interregional phase-locking and find that slow variations in the statistics of these transient episodes, contingent on the underlying anatomical structure, produce the transfer entropy functional connectivity and simulated blood oxygenation level-dependent correlation patterns observed on slower time scales.

Parametric spherical deconvolution: inferring anatomical connectivity using diffusion MR imaging

The human brain forms a complex neural network with a connectional architecture that is still far from being known in full detail, even at the macroscopic level. The advent of diffusion MR imaging has enabled the exploration of the structural properties of white matter in vivo. In this article we propose a new forward model that maps the microscopic geometry of nervous tissue onto the water diffusion process and further onto the measured MR signals. Our spherical deconvolution approach completely parameterizes the fiber orientation density by a finite mixture of Bingham distributions. In addition, we define the term anatomical connectivity, taking the underlying image modality into account. This neurophysiological metric may represent the proportion of the nerve fibers originating in the source area which intersect a given target region. The specified inverse problem is solved by Bayesian statistics. Posterior probability maps denote the probability that the connectivity value exceeds a chosen threshold, conditional upon the noisy observations. These maps allow us to draw inferences about the structural organization of the cerebral cortex. Moreover, we will demonstrate the proposed approach with diffusion-weighted data sets featuring high angular resolution.

Properties of the organization of memory for people: Evidence from dream reports

Steyvers and Tenenbaum (2005) showed that semantic networks for words have three organizational properties: short average path lengths, high clustering, and power law degree distribution. If these are general properties of memory organization, they would apply to memory for other complex material, including people and relations between them. In addition, if during dreaming, characters are generated via knowledge in the dreamer's memory, the three properties would be found in a relational network of characters in dreams. In dream reports from three individuals, two characters in the same dream were considered affiliated. Resulting social networks have the three properties, with the power law holding when low degrees are omitted. One network with a tree-like outline is different from the other two. Results suggest associative memory has the three properties, and demonstrate that dream reports are a potentially valuable source for information about social networks.

Exploring the assortativity-clustering space of a network's degree sequence

Nowadays there is a multitude of measures designed to capture different aspects of network structure. To be able to say if a measured value is expected or not, one needs to compare it with a reference model (null model). One frequently used null model is the ensemble of graphs with the same set of degrees as the original network. Here, we argue that this ensemble can give more information about the original network than effective values of network structural quantities. By mapping out this ensemble in the space of some low-level network structure--in our case, those measured by the assortativity and clustering coefficients--one can, for example, study where in the valid region of the parameter space the observed networks are. Such analysis suggests which quantities (or combination of quantities) are actively optimized during the evolution of the network. We use four very different biological networks to exemplify our method. Among other things, we find that high clustering might be a force in the evolution of protein interaction networks. We also find that all four networks are conspicuously robust to both random errors and targeted attacks.