Brain connectivity refers to a pattern of anatomical links ("anatomical connectivity"), of statistical dependencies ("functional connectivity") or of causal interactions ("effective connectivity") between distinct units within a nervous system. The units correspond to individual neurons, neuronal populations, or anatomically segregated brain regions. The connectivity pattern is formed by structural links such as synapses or fiber pathways, or it represents statistical or causal relationships measured as cross-correlations, coherence, or information flow. Neural activity, and by extension neural codes, are constrained by connectivity. Brain connectivity is thus crucial to elucidating how neurons and neural networks process information.
Levels of Brain Connectivity
A major aspect of the complexity of nervous systems relates to their intricate morphology, especially the interconnectivity of their neuronal processing elements. Neural connectivity patterns have long attracted the attention of neuroanatomists (Cajal, 1909; Brodmann, 1909; Swanson, 2003) and play crucial roles in determining the functional properties of neurons and neuronal systems. In more highly evolved nervous systems, brain connectivity can be described at several levels of scale. These levels include individual synaptic connections that link individual neurons at the microscale, networks connecting neuronal populations at the mesoscale, as well as brain regions linked by fiber pathways at the macroscale. At the microscale, detailed anatomical and physiological studies have revealed many of the basic components and interconnections of microcircuits in the mammalian cerebral cortex. At the mesoscale, they are arranged into networks of columns and minicolumns. At the macroscale, very large numbers of neurons and neuronal populations forming distinct brain regions are interconnected by inter-regional pathways, forming large-scale patterns of anatomical connectivity.
Anatomical connections at all levels of scale are both specific and variable. Specificity is found in the arrangement of individual synaptic connections between morphologically and physiologically distinct neuronal types, in the spatial extent and branching pattern of axonal arborizations, and in long-range connectivity between neural structures such as cell nuclei or brain regions. Variability is found in the shape of individual neurons and their processes, as well as in the size, placement and interconnection of large-scale structures. Variability may be measured between corresponding structures in brains of individuals of the same species. In addition, neural structures within the same individual vary across time, as a result of experiential and developmental processes of growth, plasticity and repair. It is likely that anatomical variability is one of the main sources for functional variability, expressed in neural dynamics and behavioral performance.
The remainder of this article will focus on brain connectivity at the large-scale, i.e. connectivity patterns that span across functionally diverse and structurally widely distributed components of a nervous system. The subject is more extensively covered in Jirsa and McIntosh (2007) and Sporns (2010).
Modes of Brain Connectivity
When applied to the brain, the term connectivity refers to several different and interrelated aspects of brain organization (Horwitz, 2003). A fundamental distinction is that between structural connectivity, functional connectivity and effective connectivity (Friston, 1994). Although this distinction is often cited in the context of functional neuroimaging, it is equally applicable to neuronal networks at other levels of organization, e.g. the mapping of "functional input connectivity" of individual neurons in vitro (Schubert et al., 2007).
Anatomical connectivity refers to a network of physical or structural (synaptic) connections linking sets of neurons or neuronal elements, as well as their associated structural biophysical attributes encapsulated in parameters such as synaptic strength or effectiveness. The physical pattern of anatomical connections is relatively stable at shorter time scales (seconds to minutes). At longer time scales (hours to days), structural connectivity patterns are likely to be subject to significant morphological change and plasticity. It is important to note that currently only invasive tracing studies are capable of unanimously demonstrating direct axonal connections. By contrast, diffusion weighted imaging techniques, such as DTI, have an insufficient spatial resolution, but are useful as whole brain in vivo markers of temporal changes in fibre tracts.
Functional connectivity, in contrast, is fundamentally a statistical concept. In general, functional connectivity captures deviations from statistical independence between distributed and often spatially remote neuronal units. Statistical dependence may be estimated by measuring correlation or covariance, spectral coherence or phase-locking. Functional connectivity is often calculated between all elements of a system, regardless of whether these elements are connected by direct structural links. Unlike structural connectivity, functional connectivity is highly time-dependent. Statistical patterns between neuronal elements fluctuate on multiple time scales, some as short as tens or hundreds of milliseconds. It should be noted that functional connectivity does not make any explicit reference to specific directional effects or to an underlying structural model.
Effective connectivity may be viewed as the union of structural and functional connectivity, as it describes networks of directional effects of one neural element over another. In principle, causal effects can be inferred through systematic perturbations of the system, or, since causes must precede effects in time, through time series analysis. Some techniques for extracting effective connectivity require the specification of a model including structural parameters. Other techniques are largely “model-free”, for example those that involve the application of time series causality measures such as Granger causality or transfer entropy.
Formally, brain connectivity patterns can be represented in graph or matrix format ( Figure 1). Structural brain connectivity forms a sparse and directed graph. The graph may be weighted, with weights representing connection densities or efficacies, or binary, with binary elements indicating the presence or absence of a connection. Functional brain connectivity forms a full symmetric matrix, with each of the elements encoding statistical dependence or proximity between two system elements (neurons, recording sites, voxels). Such matrices may be thresholded to yield binary undirected graphs, with the setting of the threshold controlling the degree of sparsity. Effective brain connectivity yields a full non-symmetric matrix. Applying a threshold to such matrices yields binary directed graphs.
Analysis of Brain Connectivity
Brain connectivity may be studied and analyzed using a broad range of network analysis approaches, many of which are also applied in parallel efforts to map and describe other biological networks, e.g. those of cellular metabolism, gene regulation, or ecology.
Graph theory, especially the theory of directed graphs, is of special interest as it applies to structural, functional and effective brain connectivity at all levels. Graphs are composed of vertices (corresponding to neurons or brain regions) and edges (corresponding to synapses or pathways, or statistical dependencies between neural elements). In their simplest form, graphs can be described by a connection matrix or adjacency matrix with binary elements that represent the presence or absence of a directed edge between pairs of vertices. Vertices can interact through direct connections, or indirectly via paths composed of multiple edges. The functional efficacy of these indirect interactions depends on the path length. The distance between two vertices corresponds to the length of the shortest path between them. The global average of all distances is also called the characteristic path length.
Graphs of brain networks can be quantitatively examined for vertex degrees and strengths, degree correlations (assortativity), subgraphs (motifs), clustering coefficients, path lengths (distances), and vertex and edge centrality, among many other graph theory measures (e.g. Brandes and Erlebach, 2005). In many cases, the statistical evaluation of these measures requires the design of appropriate null hypotheses, involving the choice of suitable random graph models. Such models are not uniquely defined, as statistical comparisons may be carried out relative to a number of different random models that preserve various subsets of structural parameters. Currently, one of the most frequently used random models involves edge randomization techniques that preserve vertex degrees. Clustering techniques are another important set of techniques that can be applied to all types of brain connectivity data sets. A multitude of clustering algorithms exists, including those based on principal components analysis or multidimensional scaling. Other approaches allow the detection of network communities or modules through an examination of the graph’s eigenspectrum.
In addition to graph theoretical and community detection tools, which enable the analysis of the network’s topological features, other analysis approaches focus on the three-dimensional (metrically embedded) structure of brain networks. These approaches include morphometric methods, for example those for measuring wiring length or volume (e.g. Wen and Chklovskii, 2005).
Functional brain connectivity can be estimated in a variety of ways, for example through computing cross-correlations in the time or frequency domain, mutual information or spectral coherence. Computing effective connectivity is more challenging. As defined above, functional connectivity captures patterns of statistical dependence, while effective connectivity attempts to extract networks of causal influences of one neural element over another. Various techniques for extracting effective connectivity have been pursued. One technique called “covariance structural equation modeling” assigns effective connection strengths to anatomical pathways that best match observed covariances in a given task (McIntosh and Gonzalez-Lima, 1994). A generalization of this approach called “dynamic causal modeling” (Friston et al., 2003) operates in a Bayesian framework to estimate and make inferences about directed influences between variables. Yet another approach to identifying highly interactive brain regions and their directional interactions involves the use of effective information, a measure that uses a perturbational approach to capture the degree to which two brain regions or systems causally influence each other (Tononi and Sporns, 2003). Effective connectivity may also be estimated on the basis of time-series analysis. Some of these methods are based on interpretations or adaptations of the concept of Granger causality. Another measure called transfer entropy (Schreiber, 2000) was designed to detect directed exchange of information between two systems by considering the effects of the state of one element on the state transition probabilities of the other element. The extraction of causality from time series data is sensitive to choices of sampling rates, windowing parameters, or state spaces.
Patterns of Brain Connectivity
Structural Brain Networks
Analyses of structural brain connectivity patterns, for example of large-scale connectivity matrices of the cerebral cortex, allow the quantification of a broad range of network characteristics (Sporns et al., 2004). Results demonstrate that the cerebral cortex is comprised of clusters of densely and reciprocally coupled cortical areas that are globally interconnected. These connectivity patterns are neither completely regular nor completely random, but combine structural aspects of both of these extremes. Large-scale cortical networks share some attributes of small-world networks, including high values for clustering coefficients and short characteristic path lengths, and they are composed of specific sets of structural motifs. An analysis of the structural contributions of individual areas allows the identification and classification of network hubs, defined as highly connected and highly central brain regions, which include areas of parietal and prefrontal cortex. The structural networks of the human cerebral cortex have not yet been comprehensively mapped (Sporns et al., 2005). The use of noninvasive diffusion imaging methodologies has opened new and promising avenues towards achieving this important goal.
Functional and Effective Brain Networks
Studies of patterns of functional connectivity (based on coherence or correlation) among cortical regions have demonstrated that functional brain networks exhibit small-world attributes (Achard et al., 2006) possibly reflecting the underlying structural organization of anatomical connections. More detailed graph theoretic analysis of functional brain connectivity has helped to identify functional hubs, which are highly connected and central to information flow and integration. Functional connectivity studies in the frequency domain have provided evidence for a fractal organization of functional brain networks.
Effective brain connectivity has been studied using various techniques. Covariance modeling has allowed the identification of significant differences in effective connectivity between a given set of brain regions when estimated in different cognitive tasks, thus illustrating the time- and task-dependent nature of these patterns (McIntosh and Gonzalez-Lima, 1994). Granger causality has been applied to EEG as well as fMRI time series and has provided information about directed interactions between neural elements in the course of behavioral and cognitive tasks (Brovelli et al., 2004). The combination of transcranial magnetic stimulation (TMS) with functional neuroimaging allows the use of localized perturbations of brain networks while they are engaged in the performance of specific tasks. For example, a combination of TMS and high-density electroencephalography has revealed a striking reduction in the extent of cortical effective connectivity during non-REM sleep compared to waking (Massimini et al., 2005).
Interrelations between Structural, Functional and Effective Connectivity
The relationship between anatomical, functional and effective connectivity in the cortex represents a significant challenge to present-day theoretical neuroscience. Two potential principles that link these different modes of brain connectivity are segregation and integration (Tononi et al., 1994). Segregation refers to the existence of specialized neurons and brain areas, organized into distinct neuronal populations and grouped together to form segregated cortical areas. The complementary principle, integration, gives rise to the coordinated activation of distributed neuronal populations thus enabling the emergence of coherent cognitive and behavioral states. The interplay of segregation and integration in brain networks generates information that is simultaneously highly diversified and highly integrated, thus creating patterns of high complexity.
The application of network analysis techniques allows the comparison of brain connectivity patterns obtained form structural and functional studies. For example, the discovery of small-world attributes in functional connectivity patterns derived from fMRI, EEG and MEG studies raises the question of how closely functional connections map onto structural connections. An emerging view suggests that structural connection patterns are indeed major constraints for the dynamics of cortical circuits and systems, which are captured by functional and effective connectivity. In addition to the constraining influence of structural connections, rapid temporal fluctuations in functional or effective connectivity may reflect additional changes in physiological variables or input. Given these links between structural and functional connectivity, it is likely that at least some structural characteristics of brain regions are reflected in their functional interactions. For example, structural hub regions should maintain larger numbers of functional relations. A computational model of the large-scale structure of cerebral cortex (Honey et al., 2007) suggested a partial correspondence between structural and functional hubs even at very short time scales (msecs. to secs.). Future work will likely involve the parallel analysis of structural connectivity maps of the human brain and of patterns of functional and effective connectivity recorded in various conditions of rest or cognitive activation.
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Brain, Complexity, Computational Neuroanatomy, Connectome, Diffusion Tensor Imaging, Functional Magnetic Resonance Imaging, Granger Causality, Graph Theory, Large-Scale Brain Models, Neuroimaging, Neocortex, Neuroanatomy, Neurocognitive Networks, Scale-Free Networks, Small-World Networks, Thalamocortical Circuit