This short survey the reviews recent literature on brain connectivity studies. It encompasses all forms of static and dynamic connectivity whether anatomical, functional, or effective. The last decade has seen an ever increasing number of studies devoted to deduce functional or effective connectivity, mostly from functional neuroimaging experiments. Resting state conditions have become a dominant experimental paradigm, and a number of resting state networks, among them the prominent default mode network, have been identified. Graphical models represent a convenient vehicle to formalize experimental findings and to closely and quantitatively characterize the various networks identified. Underlying these abstract concepts are anatomical networks, the so-called connectome, which can be investigated by functional imaging techniques as well. Future studies have to bridge the gap between anatomical neuronal connections and related functional or effective connectivities.
The functional organization of the brain is characterized by segregation and integration of information being processed. A central paradigm in modern neuroscience is that anatomical and functional connections between brain regions are organized in a way such that information processing is near optimal. Functional interactions seem to be provided by synchronized activity, both locally and between distant brain regions. Brain networks thus consist of spatially distributed but functionally connected regions that process information. Brain connectivity analysis rests upon three different but related forms of connectivity [ Anatomical connectivity (AC), also called structural connectivity, which forms the Functional connectivity (FC) which is defined as the temporal dependency of neuronal activation patterns of anatomically separated brain regions. It reflects statistical dependencies between distinct and distant regions of information processing neuronal populations. Hence, it is basically a statistical concept which relies on such statistical measures as correlation, covariance, spectral coherence, or phase locking. Statistical dependencies are highly time dependent and fluctuate on multiple time scales ranging form milliseconds to seconds. Effective connectivity (EC) describes the influence one neuronal system exerts upon another, thus reflecting causal interactions between activated brain areas. It combines structural and effective connectivity into a wiring diagram which reflects directional effects within a neuronal network. Causality can be inferred from network perturbations or time series analysis (TSA). Techniques based on network perturbations generally need structural information as input, while TSA-based techniques, like Granger causality, may be considered model-free.
A synthesis of the latter two concepts of connectivity, mainly applied to and deduced from functional neuroimaging modalities, has been provided by Friston [
In 2003, Horwitz [
Still connectivity analysis studies created the notion of complex brain networks characterized by densely connected nodes of information processing which are distant in anatomical space and only sparsely connected via long-range connections between different functionally interacting brain regions. These network topologies reflect two basic principles underlying information processing in the brain: functional segregation and functional integration. Experimental evidence for such network topologies mainly comes from neuroimaging techniques (EEG, MEG, fMRI, PET, and SPECT) and neuroanatomical methods.
Signal transmission between distinct brain regions requires connecting fiber tracts, thus forming the structural basis of the human connectome. Diffusion-weighted magnetic resonance imaging and its variant called diffusion tensor imaging (DTI) represent the most promising approaches for fiber tracking [
Brain connectivity can be quantified by encoding neighborhood relations into a connectivity matrix, whose rows and columns correspond to different brain regions. This representation lends itself to be mapped to a graphical model which provides means to quantify different topological aspects of the connectome. Graphical models represent a versatile mathematical framework for a generic study of pairwise relations between interacting brain regions. Recent years have witnessed an exponential growth of studies related to the application of graph theory to unravel characteristic features of structural, functional, and effective connectivity from neuroimaging investigations [
The survey is organized in the following way. First, some recent studies and reviews about experimental studies of functional connectivity are reported. This is not meant to be comprehensive, rather it should illustrate some prototypical studies in this field. Next, recent computational methods dealing with functional connectivity and some illustrative applications are collected. This is followed by a short survey of recent studies on effective connectivity. Finally, the important concept of graphical models applied to such complex brain networks as well as some applications to connectivity analysis is discussed.
Functional connectivity is a statistical concept which refers to statistical dependencies between voxel activity time courses. More generally, functional connectivity between two given regions is considered in terms of the temporal coherence or correlation between the oscillatory firing rate of neuronal assemblies [
Functional connectivity (FC), though deduced from intervoxel cross-correlations only, is nonetheless often assumed to also reflect interregional coherence of fluctuations in activity of the underlying neuronal networks in the brain. It thus is considered to refer to interregional synchrony of low-frequency fluctuations where low denotes frequencies
In a recent review, Broyd et al. [ Regional task-non-specific deactivations during goal-directed activity. Activity in the DMN becomes attenuated during task performance [ Coherence and functional connectivity within the DMN. In the context of fMRI data, functional connectivity simply refers to the temporal correlation between fluctuations in the BOLD signal of discrete anatomical regions [ A Low-frequency BOLD signal. Very Low-frequency neuronal oscillations provide temporal synchrony between functionally specific and diverse regions in the DMN [ Anticorrelated task-positive and task-negative resting state networks. In the resting state, brain activity is characterized by task-positive as well as task-negative components. The latter are characteristic for the DMN as originally defined. The second network of spontaneous Low-frequency activity, the so-called task-positive network, includes the dorsolateral prefrontal cortex (DLPFC), inferior parietal cortex (IPC), and supplementary motor area (SMA). It appears to be associated with task-related patterns of increased alertness and has also been related to response preparation and selection [ Functions subserved by the DMN. Broyd et al. [
Greicius et al. [
A recent review of van den Heuvel and Hulshoff Pol [
Alzheimer’s disease (AD) causes strong alterations of the structure and function of cerebral networks. Spontaneous brain activity is organized by synchronized activities across distinct spatial and temporal scales, thereby reflecting the complex structure of the resting state network. The latter can be studied through temporal correlations of the fMRI signals. AD-induced changes of network structure and function can thus be characterized through studying such temporal correlations at different levels of brain organization: the regional (microscopic), interregional (mesoscopic), and large-scale (macroscopic) level. Especially the PCC in the brain of patients suffering from Alzheimer’s disease (AD) is vulnerable to isolation from the rest of the brain. Zhang et al. [
White matter fibre tracts represent anatomical connectivity and provide the physical substrate for functional connectivity. In a recent review, Yo et al. [
Combining functional and anatomical connectivity is therefore needed to reveal the relation of the former abstract concept to the physical substrate of the latter. Greicius et al. [
Intrinsic neural networks can best be identified by measuring correlations between brain regions in resting state activity. The studies discussed above, and numerous others not mentioned here, focus on static aspects of functional connectivity. Traditionally, the analysis of resting state functional connectivity studies, employing correlation or data-driven exploratory decomposition techniques, generally assumes temporal stationarity of the recorded signals. However, recent experiments showed that functional connectivity networks may exhibit dynamic changes on short time scales. Chang and Glover [
Though fMRI is a popular technique to determine functional connectivity in the brain, it is limited by its indirect nature in measuring a BOLD response rather than electrical neuronal activity directly. Brookes et al. [
Resting state networks are characterized by slow fluctuations which seem to be highly structured by anatomical connections. However, the relation of these slow dynamics to fluctuating neuronal activities, particularly in the
Functionally connected regions synchronize their activities. Measuring such oscillatory dynamics requires methods with high temporal resolution like EEG, or MEG. Considering the dynamics of brain connectivity, EEG coherence is often used to measure functional connectivity in human brain [
In an attempt to quantify remediation of subjects suffering from schizophrenia, Weiss et al. [
Ghuman et al. [
Considering prototypical studies of functional brain connectivity as discussed above, functional neuroimaging during resting state conditions seems especially interesting in that it explores spontaneous brain activity. The latter has been shown to organize itself into reproducible activity patterns. Hence, it displays structure which reflects the underlying brain architecture and carries markers of brain pathologies. An important view of modern neuroscience is that such large-scale structure of coherent activity reflects modularity properties of brain connectivity graphs. Learning such models entails two main challenges. Modeling full brain connectivity is a difficult estimation problem that has to face the curse of dimensionality. Variability between subjects, coupled with variability of functional signals between experimental trials, makes the use of multiple data sets challenging.
Concerning computational methods for functional brain connectivity studies, two broad classes may be identified, namely knowledge-based, also called supervised methods, as well as data-driven, also called exploratory or unsupervised methods. The latter can be subdivided further into decomposition methods and clustering techniques [
Supervised methods afford prior knowledge about the spatial and temporal patterns of activation involved, as well as a model for the data generation process. Usually methods employ specific cognitive tasks the volunteers are supposed to perform. However, recently, they have been applied also to resting state conditions. They are widely used because of their easy implementation and straightforward interpretation. Basically, knowledge-based data analysis methods select some regions of interest (ROI) as seeds and generate a connectivity map of the human brain by determining whether other regions are functionally connected to these seeds according to predefined metrics. A convenient method to define such a metric is based on cross-correlation analysis (CCA) between the BOLD time courses of the seed region and any other brain region under consideration. Correlation is measured by the Pearson correlation coefficient
An alternative metric is based on coherence rather than correlation. The former operates in the frequency domain and is defined as
Exploratory data analysis techniques, predominantly decomposition and clustering techniques, represent global methods which do not rely on prior knowledge. Hence, they are able to reveal unexpected correlations in the data. These methods rely on the assumption that the brain is organized in a finite set of functional networks. Exploratory matrix factorization (EMF) techniques address such blind source separation problems by extracting, from the observations, distinct components with predefined properties from only a minimal set of constraints. Such data-driven methods deem most suitable for resting state studies exploring, beneath others, so-called default mode networks (DMN). Decomposition-based techniques such as singular value decomposition (SVD), principal component analysis (PCA), independent component analysis (ICA), and nonnegative matrix and tensor factorization (NMF/NTF) consider any observation as a linear superposition of underlying features. The latter are supposed to capture the essence of the information buried in the functional images; hence, they can also be considered feature-generating techniques. Which features are to be extracted is, however, unknown, and different methods yield different features which expose the relevant information in a more or less transparent way to the analyzer. SVD and PCA transform the functional images in a way that uncorrelated, orthogonal eigenimages result. The decomposition can be written as
In recent years, other decomposition techniques which alleviate such constraints have been considered, most notably independent component analysis (ICA) [
If the data matrix
Considering EMF as an unsupervised data analysis tool and the number of extracted components as an unconstrained parameter of the model, these techniques may also be categorized as clustering methods which achieve an unsupervised partitioning of the data set into subsets according to a predefined metric or nonmetric [
Closely related to clustering are classification problems, especially when functional connectivity is to be compared between certain disease states and their normal counterparts. The latter comparison is especially interesting when images are acquired under resting state conditions. With specific stimuli presented, such multivoxel pattern analysis has been named brain reading [
Assessing functional connectivity from neuroimaging recordings essentially follows two strategies: seed based versus ICA based. The two methodologies can be combined estimating temporal correlations with a specified seed voxel or small region of interest and spatially independent components (sICs). Independent component analysis (ICA) and related exploratory decomposition techniques set out to approximate any observed activity distribution
Recent evidence from several neuroimaging studies suggests that the human brain has a modular hierarchical organization which resembles the hierarchy depicted by different ICA model orders (the number of columns of the mixing matrix
Estimating functional or effective connectivity relies on the correlational or causal structure of activity distributions in distant brain areas. Such activity patterns, however, are subject to intra- and intersubject variations. Hence, it is generally of interest to identify sources of variation for fMRI connectivity. Rogers and Gore [
Varoquaux et al. [
Sofar functional connectivity has been discussed only in a static perspective. A dynamic system’s perspective needs to deal with time dependencies of functional connectivities and has to consider studies of functional network features across a broad range of frequencies. Hence, instead of employing matrix factorization techniques, functional connectivity can also be modeled in the frequency domain using multivariate autoregressive models (MVAR). Traditionally, such estimates based on MVAR models neglect instantaneous effects. Erla et al. [
Dynamic neuronal activity can be characterized locally by employing EEG or MEG recordings. However, the large-scale structure of synchronized cortical networks remains poorly characterized still. Palva et al. [
Deco et al. [
Another important aspect which recently came into the focus of current research is the development of functional connectivity in the developing brain. Fair et al. [
Effective connectivity is directed and dynamically changes according to a given context or a task performed. Therefore, one important aspect of effective connectivity analysis is to dig out the directionality of causal influences. If an observation of temporal fluctuations in the neuronal activity in one brain region allows to better predict future temporal fluctuations in the neuronal activity in another region, then the former region is said to influence the latter. Understanding brain connectivity generally follows two different routes: dynamic causal modeling (DCM) [
Zhou et al. [
While these methods do not entail temporal aspects, Rajapakse et al. [
Roebroeck et al. [
Contrary to anatomical connectivity, effective connectivity flexibly depends on contexts and tasks. Battaglia et al. [
Despite the potential usefulness of the concept of effective connectivity, it remains a source of constant concern and ongoing discussion, mainly because of the temporal blurring induced by the hemodynamical response.
Graph-theoretical concepts experience increasing attention in recent years in characterizing static and dynamic structures of complex brain networks [
Considering the functional organization of the brain into local interactions performing low-level information processing, called regions of interest (ROI) or modules, and long-range couplings supporting distributed information processing and providing control and high-level information fusion, brain networks form graphs intermediate between regular graphs where only nearest neighbor nodes are connected and random graphs where all nodes are connected randomly. Functional networks thus form graphs
A simple global measure of a graph is its degree distribution
A couple of recent reviews deal with graph-theoretical concepts applied to complex brain networks. Reijneveld et al. [
In summary, the small worldness and modularity of the structural connectivity of brain networks have been elucidated through diffusion tensor imaging (DTI) [
Sofar single-graph theoretical descriptions of complex brain networks were confined to single-subject studies. Group-based brain connectivity networks have great appeal for researchers interested in gaining further insight into complex brain function and how it changes across different mental states and disease conditions. Accurately constructing these networks presents a daunting challenge given the difficulties associated with accounting for intersubject topological variability. The conventional approach has been to use a mean or median correlation network [
Although a number of graph theoretical characterizations of functional connectivity networks of the brain have been reported since, test-retest (TRT) reliability of topological metrics of functional brain networks has hardly been studied. Recently, Deuker et al. [
Finally, methodological issues yet to be solved have been discussed in [ Node selection criteria. Alternative ways of parcellation of the cortex may explain discrepancies in topological parameters extracted from graphical models [ Threshold selection of connection metrics. A standardization of statistical methods seems most needed for comparative studies, especially when weighted graphs are employed [ Relationship between anatomical structure and cognitive function. Functional connectivity between two regions of the brain does not entail a direct structural connectivity. Especially in pathological situations, more combined studies are clearly needed still [
Support by the DAAD-FCT, the BFHZ-CCUFB, and the GENIL-SPR project at CITIC-UGR is gratefully acknowledged.