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

Small-world brain networks

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

Efficiency and cost of economical brain functional networks

Brain anatomical networks are sparse, complex, and have economical small-world properties. We investigated the efficiency and cost of human brain functional networks measured using functional magnetic resonance imaging (fMRI) in a factorial design: two groups of healthy old (N = 11; mean age = 66.5 years) and healthy young (N = 15; mean age = 24.7 years) volunteers were each scanned twice in a no-task or "resting" state following placebo or a single dose of a dopamine receptor antagonist (sulpiride 400 mg). Functional connectivity between 90 cortical and subcortical regions was estimated by wavelet correlation analysis, in the frequency interval 0.06-0.11 Hz, and thresholded to construct undirected graphs. These brain functional networks were small-world and economical in the sense of providing high global and local efficiency of parallel information processing for low connection cost. Efficiency was reduced disproportionately to cost in older people, and the detrimental effects of age on efficiency were localised to frontal and temporal cortical and subcortical regions. Dopamine antagonism also impaired global and local efficiency of the network, but this effect was differentially localised and did not interact with the effect of age. Brain functional networks have economical small-world properties-supporting efficient parallel information transfer at relatively low cost-which are differently impaired by normal aging and pharmacological blockade of dopamine transmission.

Developmental increase of total cell numbers in the murine cerebellum

The cerebellum has been widely used as a paradigm to study basic mechanisms of brain development and cortical histogenesis. Its highly regular structure has always made it particularly attractive to approaches relying on, and yielding, quantitative information, which provide a cornerstone of systems-oriented integrative analyses. Astonishingly, though, a systematic quantification of cell generation during cerebellar development has so far not been provided. Here, we use the isotropic fractionator (i.e., cell counts based on tissue homogenates from anatomically defined regions; cf. Herculano-Houzel S, Lent R., J Neurosci. 2005;25:2518-21) to assess the developmental increase of total cell numbers in the murine cerebellum from embryonic day 17 into early adulthood. Our data show that the quantitative increase of cerebellar cell numbers follows a classical, S-shaped growth curve as described by the Hill-equation. The adult murine cerebellum was found to comprise a total of (44.03+/-0.42) * 10(6) cells, half of which are generated before postnatal day 12+/-0.18. Consistent results were obtained by using two approaches to cell counting, one based on manual assessment, the other on flow cytometry. These data provide a reliable quantitative description of cerebellar growth in the mouse and define a predictive model that should allow their integration with quantitative and qualitative descriptions of cerebellar development.

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

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

Cortical functional network organization from autoregressive modeling of local field potential oscillations

A framework is presented for quantifying functional network organization in the brain by spectral analysis based on autoregressive modeling. Local field potentials (LFPs), simultaneously recorded from distributed sites in the cerebral cortex of monkeys, are treated as signals generated by local neuronal assemblies. During the delay period of a visual pattern discrimination task, oscillatory assembly activity is manifested in the LFPs in the beta-frequency range (14-30 Hz). Coherence analysis has shown that these oscillations are phase synchronized in functional networks in the sensorimotor cortex in relation to maintenance of contralateral hand position, and in the visual cortex in relation to anticipation of the visual stimulus. Granger causality analysis has revealed information flow in the sensorimotor network that is consistent with a peripheral sensorimotor feedback loop, and in the visual network that is consistent with top-down anticipatory modulation of assemblies in the primary visual cortex.

Hierarchical organization unveiled by functional connectivity in complex brain networks

How do diverse dynamical patterns arise from the topology of complex networks? We study synchronization dynamics in the cortical brain network of the cat, which displays a hierarchically clustered organization, by modeling each node (cortical area) with a subnetwork of interacting excitable neurons. We find that in the biologically plausible regime the dynamics exhibits a hierarchical modular organization, in particular, revealing functional clusters coinciding with the anatomical communities at different scales. Our results provide insights into the relationship between network topology and functional organization of complex brain networks.

Mapping information flow in sensorimotor networks

Biological organisms continuously select and sample information used by their neural structures for perception and action, and for creating coherent cognitive states guiding their autonomous behavior. Information processing, however, is not solely an internal function of the nervous system. Here we show, instead, how sensorimotor interaction and body morphology can induce statistical regularities and information structure in sensory inputs and within the neural control architecture, and how the flow of information between sensors, neural units, and effectors is actively shaped by the interaction with the environment. We analyze sensory and motor data collected from real and simulated robots and reveal the presence of information structure and directed information flow induced by dynamically coupled sensorimotor activity, including effects of motor outputs on sensory inputs. We find that information structure and information flow in sensorimotor networks (a) is spatially and temporally specific; (b) can be affected by learning, and (c) can be affected by changes in body morphology. Our results suggest a fundamental link between physical embeddedness and information, highlighting the effects of embodied interactions on internal (neural) information processing, and illuminating the role of various system components on the generation of behavior.

How neurons encode and process information is a key problem in computational biology and neuroscience. In this paper, Lungarella and Sporns present a novel application of computational methods to the integration of neural and sensorimotor processes at the systems-level scale. The central result of their study is that sensorimotor interaction and body morphology can induce statistical regularities and information structure in sensory inputs and within the neural control architecture. The informational content of inputs is thus not independent of output, and the authors suggest that neural coding needs to be considered in the context of the “embeddedness” of the organism within its eco-niche. Using robots and nonlinear time-series analysis techniques, they investigate how the flow of information between sensors, neural units, and effectors is actively shaped by interaction with the environment. This study represents a first step towards the development of an explicit quantitative framework that unifies neural and behavioral processes. Such a framework could also shed significant new light on key constraints shaping the evolution and development of nervous systems and their behavioral and cognitive capacities. In addition, it could provide an important design principle to guide the construction of more efficient artificial cognitive systems.

Small-world networks and disturbed functional connectivity in schizophrenia

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

An experimental study of the coloring problem on human subject networks

Theoretical work suggests that structural properties of naturally occurring networks are important in shaping behavior and dynamics. However, the relationships between structure and behavior are difficult to establish through empirical studies, because the networks in such studies are typically fixed. We studied networks of human subjects attempting to solve the graph or network coloring problem, which models settings in which it is desirable to distinguish one's behavior from that of one's network neighbors. Networks generated by preferential attachment made solving the coloring problem more difficult than did networks based on cyclical structures, and "small worlds" networks were easier still. We also showed that providing more information can have opposite effects on performance, depending on network structure.

Small-world connectivity, motif composition, and complexity of fractal neuronal connections

Connection patterns of the cerebral cortex consist of pathways linking neuronal populations across multiple levels of scale, from whole brain regions to local minicolumns. This nested interconnectivity suggests the hypothesis that cortical connections are arranged in fractal or self-similar patterns. We describe a simple procedure to generate fractal connection patterns that aim at capturing the potential self-similarity and hierarchical ordering of neuronal connections. We examine these connection patterns by calculating a broad range of structural measures, including small-world attributes and motif composition, as well as some global measures of functional connectivity, including complexity. As we vary fractal patterns by changing a critical control parameter, we find strongly correlated changes in several structural and functional measures, suggesting that they emerge together and are mutually linked. Measures obtained from some modeled fractal patterns closely resemble those of real neuroanatomical data sets, supporting the original hypothesis.

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

Previous studies demonstrated that intelligence is significantly related to an impressive array of psychological, social, biological and genetic factors and that working memory (WM) can be considered as a general cognitive resource strongly related with a wide variety of higher order cognitive competencies and intelligence. Also, evaluating the WM of subjects might allow one to test the neural efficiency hypothesis (NEH). WM typically involves functional interactions between frontal and parietal cortices. We recorded EEG signals to study neuronal interactions during one WM test in individuals who had few years of formal education (LE) as compared to individuals with university degrees (UE). The two groups of individuals differed in the scores they obtained in psychological tests. To quantify the synchronization between EEG channels in several frequency bands, we evaluated the "synchronization likelihood" (SL), which takes into consideration nonlinear processes as well as linear ones. SL was then converted into graphs to estimate the distance from "small-world network" (SWN) organization, i.e., an optimally organized network that would give rise to the data. In comparison to LE subjects, those with university degrees exhibited less prominent SWN properties in most frequency bands during the WM task. This finding supports the NEH and suggests that the connections between brain areas of well-educated subjects engaged in WM tasks are not as well-organized in the sense of SWN.

Operational principles of neurocognitive networks

Large-scale neural networks are thought to be an essential substrate for the implementation of cognitive function by the brain. If so, then a thorough understanding of cognition is not possible without knowledge of how the large-scale neural networks of cognition (neurocognitive networks) operate. Of necessity, such understanding requires insight into structural, functional, and dynamical aspects of network operation, the intimate interweaving of which may be responsible for the intricacies of cognition. Knowledge of anatomical structure is basic to understanding how neurocognitive networks operate. Phylogenetically and ontogenetically determined patterns of synaptic connectivity form a structural network of brain areas, allowing communication between widely distributed collections of areas. The function of neurocognitive networks depends on selective activation of anatomically linked cortical and subcortical areas in a wide variety of configurations. Large-scale functional networks provide the cooperative processing which gives expression to cognitive function. The dynamics of neurocognitive network function relates to the evolving patterns of interacting brain areas that express cognitive function in real time. This article considers the proposition that a basic similarity of the structural, functional, and dynamical features of all neurocognitive networks in the brain causes them to function according to common operational principles. The formation of neural context through the coordinated mutual constraint of multiple interacting cortical areas, is considered as a guiding principle underlying all cognitive functions. Increasing knowledge of the operational principles of neurocognitive networks is likely to promote the advancement of cognitive theories, and to seed strategies for the enhancement of cognitive abilities.

Connection and Coordination: The Interplay Between Architecture and Dynamics in Evolved Model Pattern Generators

We undertake a systematic study of the role of neural architecture in shaping the dynamics of evolved model pattern generators for a walking task. First, we consider the minimum number of connections necessary to achieve high performance on this task. Next, we identify architectural motifs associated with high fitness. We then examine how high-fitness architectures differ in their ability to evolve. Finally, we demonstrate the existence of distinct parameter subgroups in some architectures and show that these subgroups are characterized by differences in neuron excitabilities and connection signs.

The neurocognitive bases of human multimodal food perception: sensory integration

This review addresses a fundamental neuroscientific question in food perception: how multimodal features of food are integrated. Much research and conceptualization has emerged related to multisensory integration in vision, audition and somatosensation, while it remains poorly understood and researched within the chemical and mouth feel senses. This review aims to bridge this gap. We discuss the main concepts in the fields of auditory, visual and somatosensory multisensory integration and relate them to oral-sensory (gustatory and somatosensory) and olfactory (orolfactory) interactions. We systematically review the psychophysical literature pertaining to intra- and intermodal interactions related to food perception, while making explicit distinctions between peripheral and central interactions. As the neural bases of crossmodal orolfaction currently are poorly understood, we introduce several plausible neuroscientific models, which provide a framework for further neuroscientific exploration in this area. We are guided by a new meta-analysis of the odor-taste neuroimaging literature, as well as by single-unit, anatomical and psychophysical studies. Finally, we propose strong involvement of recurrent neural networks in multisensory integration and make suggestions for future research.

Segregation of the brain into gray and white matter: a design minimizing conduction delays

A ubiquitous feature of the vertebrate anatomy is the segregation of the brain into white and gray matter. Assuming that evolution maximized brain functionality, what is the reason for such segregation? To answer this question, we posit that brain functionality requires high interconnectivity and short conduction delays. Based on this assumption we searched for the optimal brain architecture by comparing different candidate designs. We found that the optimal design depends on the number of neurons, interneuronal connectivity, and axon diameter. In particular, the requirement to connect neurons with many fast axons drives the segregation of the brain into white and gray matter. These results provide a possible explanation for the structure of various regions of the vertebrate brain, such as the mammalian neocortex and neostriatum, the avian telencephalon, and the spinal cord.

Estimating brain functional connectivity with sparse multivariate autoregression

There is much current interest in identifying the anatomical and functional circuits that are the basis of the brain's computations, with hope that functional neuroimaging techniques will allow the in vivo study of these neural processes through the statistical analysis of the time-series they produce. Ideally, the use of techniques such as multivariate autoregressive (MAR) modelling should allow the identification of effective connectivity by combining graphical modelling methods with the concept of Granger causality. Unfortunately, current time-series methods perform well only for the case that the length of the time-series Nt is much larger than p, the number of brain sites studied, which is exactly the reverse of the situation in neuroimaging for which relatively short time-series are measured over thousands of voxels. Methods are introduced for dealing with this situation by using sparse MAR models. These can be estimated in a two-stage process involving (i) penalized regression and (ii) pruning of unlikely connections by means of the local false discovery rate developed by Efron. Extensive simulations were performed with idealized cortical networks having small world topologies and stable dynamics. These show that the detection efficiency of connections of the proposed procedure is quite high. Application of the method to real data was illustrated by the identification of neural circuitry related to emotional processing as measured by BOLD.

Autism: a window onto the development of the social and the analytic brain

Although the neurobiological understanding of autism has been increasing exponentially, the diagnosis of autism spectrum conditions still rests entirely on behavioral criteria. Autism is therefore most productively approached using a combination of biological and psychological theory. The triad of behavioral abnormalities in social function, communication, and restricted and repetitive behaviors and interests can be explained psychologically by an impaired capacity for empathizing, or modeling the mental states governing the behavior of people, along with a superior capacity for systemizing, or inferring the rules governing the behavior of objects. This empathizing-systemizing theory explains other psychological models such as impairments of executive function or central coherence, and may have a neurobiological basis in abnormally low activity of brain regions subserving social cognition, along with abnormally high activity of regions subserving lower-level, perceptual processing--a pattern that may result from a skewed balance of local versus long-range functional connectivity.

The human connectome: A structural description of the human brain

The connection matrix of the human brain (the human "connectome") represents an indispensable foundation for basic and applied neurobiological research. However, the network of anatomical connections linking the neuronal elements of the human brain is still largely unknown. While some databases or collations of large-scale anatomical connection patterns exist for other mammalian species, there is currently no connection matrix of the human brain, nor is there a coordinated research effort to collect, archive, and disseminate this important information. We propose a research strategy to achieve this goal, and discuss its potential impact.

Dynamics of a neural system with a multiscale architecture

The architecture of the brain is characterized by a modular organization repeated across a hierarchy of spatial scales-neurons, minicolumns, cortical columns, functional brain regions, and so on. It is important to consider that the processes governing neural dynamics at any given scale are not only determined by the behaviour of other neural structures at that scale, but also by the emergent behaviour of smaller scales, and the constraining influence of activity at larger scales. In this paper, we introduce a theoretical framework for neural systems in which the dynamics are nested within a multiscale architecture. In essence, the dynamics at each scale are determined by a coupled ensemble of nonlinear oscillators, which embody the principle scale-specific neurobiological processes. The dynamics at larger scales are 'slaved' to the emergent behaviour of smaller scales through a coupling function that depends on a multiscale wavelet decomposition. The approach is first explicated mathematically. Numerical examples are then given to illustrate phenomena such as between-scale bifurcations, and how synchronization in small-scale structures influences the dynamics in larger structures in an intuitive manner that cannot be captured by existing modelling approaches. A framework for relating the dynamical behaviour of the system to measured observables is presented and further extensions to capture wave phenomena and mode coupling are suggested.

Neural Darwinism and consciousness

Neural Darwinism (ND) is a large scale selectionist theory of brain development and function that has been hypothesized to relate to consciousness. According to ND, consciousness is entailed by reentrant interactions among neuronal populations in the thalamocortical system (the 'dynamic core'). These interactions, which permit high-order discriminations among possible core states, confer selective advantages on organisms possessing them by linking current perceptual events to a past history of value-dependent learning. Here, we assess the consistency of ND with 16 widely recognized properties of consciousness, both physiological (for example, consciousness is associated with widespread, relatively fast, low amplitude interactions in the thalamocortical system), and phenomenal (for example, consciousness involves the existence of a private flow of events available only to the experiencing subject). While no theory accounts fully for all of these properties at present, we find that ND and its recent extensions fare well.

Causal connectivity of evolved neural networks during behavior

To show how causal interactions in neural dynamics are modulated by behavior, it is valuable to analyze these interactions without perturbing or lesioning the neural mechanism. This paper proposes a method, based on a graph-theoretic extension of vector autoregressive modeling and 'Granger causality,' for characterizing causal interactions generated within intact neural mechanisms. This method, called 'causal connectivity analysis' is illustrated via model neural networks optimized for controlling target fixation in a simulated head-eye system, in which the structure of the environment can be experimentally varied. Causal connectivity analysis of this model yields novel insights into neural mechanisms underlying sensorimotor coordination. In contrast to networks supporting comparatively simple behavior, networks supporting rich adaptive behavior show a higher density of causal interactions, as well as a stronger causal flow from sensory inputs to motor outputs. They also show different arrangements of 'causal sources' and 'causal sinks': nodes that differentially affect, or are affected by, the remainder of the network. Finally, analysis of causal connectivity can predict the functional consequences of network lesions. These results suggest that causal connectivity analysis may have useful applications in the analysis of neural dynamics.

Local design principles of mammalian cortical networks

To understand global and local design principles of mammalian cerebral cortical networks, we applied network-theoretical approaches to connectivity data from macaque and cat cortical networks. We first confirmed "small-world" properties and searched for the evidence of hierarchical modularity. To elucidate their local design principles, we then compared these cortical networks, based on the significance profile (SP) of network motifs in the real network compared to randomized networks. We found that SPs of different mammalian cortical networks are highly conserved and robust, suggesting constraints of neocortical development and evolution.

Graph theoretical characterization and tracking of the effective neural connectivity during episodes of mesial temporal epileptic seizure

Via a detailed case study of mesial temporal lobe epilepsy, we show that a method of determining the direction of information flow among signals is able to provide focal localization via the simultaneous analysis of multiple EEG channels. This determination is accomplished by representing information flow direction via directed graphs, where focal electrodes are associated with high observed rates of pertinence to strongly connected subgraphs. Further clinical support to this finding is provided by results for an additional 9 cases of focal epilepsy cases. The graph theoretical approach is a tool for describing and analyzing the effective connectivity dynamics behind epileptic seizures and may provide a common language for studying other complex dynamic relationships between neural structures.

Scale-free brain functional networks

Functional magnetic resonance imaging is used to extract functional networks connecting correlated human brain sites. Analysis of the resulting networks in different tasks shows that (a) the distribution of functional connections, and the probability of finding a link versus distance are both scale-free, (b) the characteristic path length is small and comparable with those of equivalent random networks, and (c) the clustering coefficient is orders of magnitude larger than those of equivalent random networks. All these properties, typical of scale-free small-world networks, reflect important functional information about brain states.

Neurophysiological Architecture of Functional Magnetic Resonance Images of Human Brain

We investigated large-scale systems organization of the whole human brain using functional magnetic resonance imaging (fMRI) data acquired from healthy volunteers in a no-task or 'resting' state. Images were parcellated using a prior anatomical template, yielding regional mean time series for each of 90 regions (major cortical gyri and subcortical nuclei) in each subject. Significant pairwise functional connections, defined by the group mean inter-regional partial correlation matrix, were mostly either local and intrahemispheric or symmetrically interhemispheric. Low-frequency components in the time series subtended stronger inter-regional correlations than high-frequency components. Intrahemispheric connectivity was generally related to anatomical distance by an inverse square law; many symmetrical interhemispheric connections were stronger than predicted by the anatomical distance between bilaterally homologous regions. Strong interhemispheric connectivity was notably absent in data acquired from a single patient, minimally conscious following a brainstem lesion. Multivariate analysis by hierarchical clustering and multidimensional scaling consistently defined six major systems in healthy volunteers-- corresponding approximately to four neocortical lobes, medial temporal lobe and subcortical nuclei- - that could be further decomposed into anatomically and functionally plausible subsystems, e.g. dorsal and ventral divisions of occipital cortex. An undirected graph derived by thresholding the healthy group mean partial correlation matrix demonstrated local clustering or cliquishness of connectivity and short mean path length compatible with prior data on small world characteristics of non-human cortical anatomy. Functional MRI demonstrates a neurophysiological architecture of the normal human brain that is anatomically sensible, strongly symmetrical, disrupted by acute brain injury, subtended predominantly by low frequencies and consistent with a small world network topology.

Semantic graphs and associative memories

Graphs have been increasingly utilized in the characterization of complex networks from diverse origins, including different kinds of semantic networks. Human memories are associative and are known to support complex semantic nets; these nets are represented by graphs. However, it is not known how the brain can sustain these semantic graphs. The vision of cognitive brain activities, shown by modern functional imaging techniques, assigns renewed value to classical distributed associative memory models. Here we show that these neural network models, also known as correlation matrix memories, naturally support a graph representation of the stored semantic structure. We demonstrate that the adjacency matrix of this graph of associations is just the memory coded with the standard basis of the concept vector space, and that the spectrum of the graph is a code invariant of the memory. As long as the assumptions of the model remain valid this result provides a practical method to predict and modify the evolution of the cognitive dynamics. Also, it could provide us with a way to comprehend how individual brains that map the external reality, almost surely with different particular vector representations, are nevertheless able to communicate and share a common knowledge of the world. We finish presenting adaptive association graphs, an extension of the model that makes use of the tensor product, which provides a solution to the known problem of branching in semantic nets.

Motifs in brain networks

Complex brains have evolved a highly efficient network architecture whose structural connectivity is capable of generating a large repertoire of functional states. We detect characteristic network building blocks (structural and functional motifs) in neuroanatomical data sets and identify a small set of structural motifs that occur in significantly increased numbers. Our analysis suggests the hypothesis that brain networks maximize both the number and the diversity of functional motifs, while the repertoire of structural motifs remains small. Using functional motif number as a cost function in an optimization algorithm, we obtain network topologies that resemble real brain networks across a broad spectrum of structural measures, including small-world attributes. These results are consistent with the hypothesis that highly evolved neural architectures are organized to maximize functional repertoires and to support highly efficient integration of information.

Interneuron Diversity series: Circuit complexity and axon wiring economy of cortical interneurons

The performance of the brain is constrained by wiring length and maintenance costs. The apparently inverse relationship between number of neurons in the various interneuron classes and the spatial extent of their axon trees suggests a mathematically definable organization, reminiscent of 'small-world' or scale-free networks observed in other complex systems. The wiring-economy-based classification of cortical inhibitory interneurons is supported by the distinct physiological patterns of class members in the intact brain. The complex wiring of diverse interneuron classes could represent an economic solution for supporting global synchrony and oscillations at multiple timescales with minimum axon length.

Edge vulnerability in neural and metabolic networks

Biological networks, such as cellular metabolic pathways or networks of corticocortical connections in the brain, are intricately organized, yet remarkably robust toward structural damage. Whereas many studies have investigated specific aspects of robustness, such as molecular mechanisms of repair, this article focuses more generally on how local structural features in networks may give rise to their global stability. In many networks the failure of single connections may be more likely than the extinction of entire nodes, yet no analysis of edge importance (edge vulnerability) has been provided so far for biological networks. We tested several measures for identifying vulnerable edges and compared their prediction performance in biological and artificial networks. Among the tested measures, edge frequency in all shortest paths of a network yielded a particularly high correlation with vulnerability and identified intercluster connections in biological but not in random and scale-free benchmark networks. We discuss different local and global network patterns and the edge vulnerability resulting from them.

Performance of networks of artificial neurons: the role of clustering

The performance of the Hopfield neural network model is numerically studied on various complex networks, such as the Watts-Strogatz network, the Barabási-Albert network, and the neuronal network of Caenorhabditis elegans. Through the use of a systematic way of controlling the clustering coefficient, with the degree of each neuron kept unchanged, we find that the networks with the lower clustering exhibit much better performance. The results are discussed in the practical viewpoint of application, and the biological implications are also suggested.

Network participation indices: characterizing component roles for information processing in neural networks

We propose a set of indices that characterize-on the basis of connectivity data-how a network node participates in a larger network and what roles it may take given the specific sub-network of interest. These Network Participation Indices are derived from simple graph theoretic measures and have the interesting property of linking local features of individual network components to distributed properties that arise within the network as a whole. We use connectivity data on large-scale cortical networks to demonstrate the virtues of this approach and highlight some interesting features that had not been brought up in previously published material. Some implications of our approach for defining network characteristics relevant to functional segregation and functional integration, for example, from functional imaging studies are discussed.

How does connectivity between cortical areas depend on brain size? Implications for efficient computation

A formula for an average connectivity between cortical areas in mammals is derived. Based on comparative neuroanatomical data, it is found, surprisingly, that this connectivity is either only weakly dependent or independent of brain size. It is discussed how this formula can be used to estimate the average length of axons in white matter. Other allometric relations, such as cortical patches and area sizes vs. brain size, are also provided. Finally, some functional implications, with an emphasis on efficient cortical computation, are discussed as well.

Cortex, Countercurrent Context, and Dimensional Integration of Lifetime Memory

The correlation between relative neocortex size and longevity in mammals encourages a search for a cortical function specifically related to the life-span. A candidate in the domain of permanent and cumulative memory storage is proposed and explored in relation to basic aspects of cortical organization. The pattern of cortico-cortical connectivity between functionally specialized areas and the laminar organization of that connectivity converges on a globally coherent representational space in which contextual embedding of information emerges as an obligatory feature of cortical function. This brings a powerful mode of inductive knowledge within reach of mammalian adaptations. It combines item specificity with classificatory generality, as embodied in "latent semantic analysis" algorithms. Its neural implementation is proposed to depend on an obligatory interaction between the oppositely directed feedforward and feedback currents of cortical activity, in countercurrent fashion. Direct interaction of the two streams along their cortex-wide local interface supports a scheme of "contextual capture" for information storage responsible for the lifelong cumulative growth of a uniquely cortical form of memory termed "personal history." This approach to cortical function helps elucidate key features of cortical organization as well as cognitive aspects of mammalian life history strategies.

Measuring information integration

BACKGROUND

To understand the functioning of distributed networks such as the brain, it is important to characterize their ability to integrate information. The paper considers a measure based on effective information, a quantity capturing all causal interactions that can occur between two parts of a system.

RESULTS

The capacity to integrate information, or Phi, is given by the minimum amount of effective information that can be exchanged between two complementary parts of a subset. It is shown that this measure can be used to identify the subsets of a system that can integrate information, or complexes. The analysis is applied to idealized neural systems that differ in the organization of their connections. The results indicate that Phi is maximized by having each element develop a different connection pattern with the rest of the complex (functional specialization) while ensuring that a large amount of information can be exchanged across any bipartition of the network (functional integration).

CONCLUSION

Based on this analysis, the connectional organization of certain neural architectures, such as the thalamocortical system, are well suited to information integration, while that of others, such as the cerebellum, are not, with significant functional consequences. The proposed analysis of information integration should be applicable to other systems and networks.

Topology and computational performance of attractor neural networks

To explore the relation between network structure and function, we studied the computational performance of Hopfield-type attractor neural nets with regular lattice, random, small-world, and scale-free topologies. The random configuration is the most efficient for storage and retrieval of patterns by the network as a whole. However, in the scale-free case retrieval errors are not distributed uniformly among the nodes. The portion of a pattern encoded by the subset of highly connected nodes is more robust and efficiently recognized than the rest of the pattern. The scale-free network thus achieves a very strong partial recognition. The implications of these findings for brain function and social dynamics are suggestive.

Bright spots: correlations of gray matter volume with IQ in a normal pediatric population

The localization of brain areas related to cognitive functions has yet to be thoroughly explored in children. We therefore set out to apply volumetric, voxel-based, and structural connectivity analyses to magnetic resonance images from a large sample of healthy children. We could confirm a strong correlation of whole-brain gray matter volume and the individual intelligence quotient; however, this correlation only developed with age in our sample, in that it was not present in the younger children. With the application of an optimized protocol for voxel-based morphometry, the anterior cingulate was shown to be directly correlated with a measure of human intelligence. Furthermore, an analysis of structural connectivity identified gray matter volume in several distinct brain areas to be related to cognitive functions. The implications of our findings for normal development, pathological processes, and our understanding of cognition are discussed and related to previous findings.

Theoretical neuroanatomy and the connectivity of the cerebral cortex

Over recent years, a wealth of neuroanatomical information on the pattern of interconnections between segregated areas of the cerebral cortex has become available. Here, we describe a set of structural measures, based on graph theory, which can be used to analyze these anatomical patterns. We describe relationships between these structural measures and measures based on patterns of functional connectivity, i.e. patterns of correlations in neural activity. We find that networks capable of producing highly complex functional dynamics share common structural motifs. These motifs are also found in cortical connection matrices, which are characterized by the existence of densely linked groups of areas, low potential wiring length, and a high abundance of reciprocal connections and short cycles. An analysis of cortical functional connectivity demonstrates the existence of functional clusters of highly interactive areas, producing highly complex dynamics. The combined structural and functional analysis outlined in this chapter provides insight into the large-scale functional organization of distributed cortical systems.

Edge overload breakdown in evolving networks

We investigate growing networks based on Barabási and Albert's algorithm for generating scale-free networks, but with edges sensitive to overload breakdown. The load is defined through edge betweenness centrality. We focus on the situation where the average number of connections per vertex is, like the number of vertices, linearly increasing in time. After an initial stage of growth, the network undergoes avalanching breakdowns to a fragmented state from which it never recovers. This breakdown is much less violent if the growth is by random rather than by preferential attachment (as defines the Barabási and Albert model). We briefly discuss the case where the average number of connections per vertex is constant. In this case no breakdown avalanches occur. Implications to the growth of real-world communication networks are discussed.

Relation between structure and size in social networks

In the context of complex network systems, we model social networks with the property that there is certain degradation of the information flowing through the network. We analyze different kinds of networks, from regular lattices to random graphs. We define an average coordination degree for the network, which can be associated with a certain notion of efficiency. Assuming that there is a limit to the information a person may handle, we show that there exists a close relationship between the structure of the network and its maximum size.

Connection from cortical area V2 to MT in macaque monkey

The extrastriate visual area of the macaque monkey called MT or V5, receives its input from multiple sources. We have previously examined the synaptic connections made by V1 cells that project to MT (Anderson et al., 1998). Here, we provide a similar analysis of the projection from V2 to MT. The major target of the V2 projection in MT is layer 4, where it forms clusters of asymmetric (excitatory) synapses. Unlike the V1 projection, it also forms synapses in layers 1 and 2 and does not form synapses in layer 6. The most frequently encountered targets of boutons labeled from V2 were spines (67% in layer 4; 82% in layer 2/3). Unusually, only 5/12 boutons examined in layer 1 actually formed synapses. Unlike the V1 projection, multisynaptic boutons were rare (mean, 1.1 synapses per bouton vs. 1.7 for the V1 projection). Like the V1 projection, the input to MT from any point in V2 is sparse (contributing approximately 4-6% of the asymmetric synapses in the densest clusters in layer 4). The synapses of the V2 projection were similar in size to those of the V1 projection (0.1 microm(2) vs. 0.09 microm(2)) and both formed more complex postsynaptic densities on spines than on dendritic shafts. The clear differences between the V1 and V2 projection to MT indicate that their functions are complementary rather than completely overlapping.

On the marriage of cognition and neuroscience

This paper summarizes five major themes of discussion stemming from a recent workshop at the University of Toronto. The focus of the workshop was whether the phenomenology of cognition has a direct translation to the biological processes of the brain. The study of this translation is the goal of cognitive neuroscience. The themes were: (1) the influence of context on the understanding of brain function, in which regional activity may have different functional relevance depending on activity in the rest of the brain; (2) the merger of anatomy and function, emphasizing how interfacing at the systems level can have the potential to aid in the understanding of how anatomy constrains function; (3) the development of mathematical measures that take advantage of organizing principles of the nervous system; (4) the observation that the relation between "top-down" and "bottom-up" both neurally and conceptually could be better appreciated through a more principled mathematical approach; and (5) a central role for large-scale neural modeling to bridge basic neurophysiology and anatomy. Despite the consensus on these themes, there are several challenges for the field. Significant obstacles arise from the multidisciplinary nature of cognitive neuroscience, in which terms do not mean the same thing across disciplines (e.g., networks and systems). The imprecision of explanations for cognitive neuroscience data was also seen as a significant problem, suggesting that more principled attempts at explicit model specifications and prediction will be necessary for the field to develop.

Efficient behavior of small-world networks

We introduce the concept of efficiency of a network as a measure of how efficiently it exchanges information. By using this simple measure, small-world networks are seen as systems that are both globally and locally efficient. This gives a clear physical meaning to the concept of "small world," and also a precise quantitative analysis of both weighted and unweighted networks. We study neural networks and man-made communication and transportation systems and we show that the underlying general principle of their construction is in fact a small-world principle of high efficiency.

Advanced database methodology for the Collation of Connectivity data on the Macaque brain (CoCoMac)

The need to integrate massively increasing amounts of data on the mammalian brain has driven several ambitious neuroscientific database projects that were started during the last decade. Databasing the brain's anatomical connectivity as delivered by tracing studies is of particular importance as these data characterize fundamental structural constraints of the complex and poorly understood functional interactions between the components of real neural systems. Previous connectivity databases have been crucial for analysing anatomical brain circuitry in various species and have opened exciting new ways to interpret functional data, both from electrophysiological and from functional imaging studies. The eventual impact and success of connectivity databases, however, will require the resolution of several methodological problems that currently limit their use. These problems comprise four main points: (i) objective representation of coordinate-free, parcellation-based data, (ii) assessment of the reliability and precision of individual data, especially in the presence of contradictory reports, (iii) data mining and integration of large sets of partially redundant and contradictory data, and (iv) automatic and reproducible transformation of data between incongruent brain maps. Here, we present the specific implementation of the 'collation of connectivity data on the macaque brain' (CoCoMac) database (http://www.cocomac.org). The design of this database addresses the methodological challenges listed above, and focuses on experimental and computational neuroscientists' needs to flexibly analyse and process the large amount of published experimental data from tracing studies. In this article, we explain step-by-step the conceptual rationale and methodology of CoCoMac and demonstrate its practical use by an analysis of connectivity in the prefrontal cortex.

The portable UNIX programming system (PUPS) and CANTOR: a computational environment for dynamical representation and analysis of complex neurobiological data

Many problems in analytical biology, such as the classification of organisms, the modelling of macromolecules, or the structural analysis of metabolic or neural networks, involve complex relational data. Here, we describe a software environment, the portable UNIX programming system (PUPS), which has been developed to allow efficient computational representation and analysis of such data. The system can also be used as a general development tool for database and classification applications. As the complexity of analytical biology problems may lead to computation times of several days or weeks even on powerful computer hardware, the PUPS environment gives support for persistent computations by providing mechanisms for dynamic interaction and homeostatic protection of processes. Biological objects and their interrelations are also represented in a homeostatic way in PUPS. Object relationships are maintained and updated by the objects themselves, thus providing a flexible, scalable and current data representation. Based on the PUPS environment, we have developed an optimization package, CANTOR, which can be applied to a wide range of relational data and which has been employed in different analyses of neuroanatomical connectivity. The CANTOR package makes use of the PUPS system features by modifying candidate arrangements of objects within the system's database. This restructuring is carried out via optimization algorithms that are based on user-defined cost functions, thus providing flexible and powerful tools for the structural analysis of the database content. The use of stochastic optimization also enables the CANTOR system to deal effectively with incomplete and inconsistent data. Prototypical forms of PUPS and CANTOR have been coded and used successfully in the analysis of anatomical and functional mammalian brain connectivity, involving complex and inconsistent experimental data. In addition, PUPS has been used for solving multivariate engineering optimization problems and to implement the digital identification system (DAISY), a system for the automated classification of biological objects. PUPS is implemented in ANSI-C under the POSIX.1 standard and is to a great extent architecture- and operating-system independent. The software is supported by systems libraries that allow multi-threading (the concurrent processing of several database operations), as well as the distribution of the dynamic data objects and library operations over clusters of computers. These attributes make the system easily scalable, and in principle allow the representation and analysis of arbitrarily large sets of relational data. PUPS and CANTOR are freely distributed (http://www.pups.org.uk) as open-source software under the GNU license agreement.