CHAPTER 1 INTRODUCTION
2. Functional Connectivity (FC)
2.3. Network modelling
Independently from the technique used (EEG/MEG, fMRI, DTI), the inferred communication between brain regions can be graphically represented as a set of connections constituting networks. Usually, schematic illustration of a network implies nodes and connections between them (links, edges), altogether creating a complex functional or structural system. We can describe this system by characterizing different aspects like, e.g., its size, lengths and number of its connections, the architecture of its connections (topology) or its dynamics (the development of interactions between nodes over time) (Sporns et al., 2004). Accordingly, a range of theoretical models have been proposed attempting to reconstruct real-life neural network approximations through linear or non-linear statistical dependencies (functional connectivity) (Zhou et al., 2009a) or causal interactions (effective connectivity) (Friston et al., 2003).
Graph Theoretical Approach
Graph theory describes the topology of complex networks as a group of nodes or vertices and the edges between them (Bullmore & Sporns, 2009). Within this approach, two global properties of networks can be studied: the level of their segregation (organization in groups or clusters) and their integration (global efficiency). Diverse parameters of node centrality assess the importance of individual nodes for the communication within a network. To name only few of numerous existing measures, degree of centrality is a common measure of the number of connections linking a node to
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the rest of the network; closeness centrality describes the average shortest path length from one node to all other nodes in the network; average path length across nodes indicates the expected distance.(Rubinov & Sporns, 2010). Depending on the connectivity measure (e.g. coherence, Granger causality) the final graphs can be represented as undirected, directed or weighted graphs.
Each step of graph analysis entails choices that can have an impact on the final results and must be carefully informed by the experimental question (Bullmore & Sporns, 2009). First of all, the reconstruction of nodes in individual brain networks is highly determined by the data set and by a choice of brain mapping method or atlas. For example, in EEG or MEG studies nodes can be defined either on the sensor level (as single electrodes) or on the source level (voxels). Second, the choice of connectivity metric is important. Only effective connectivity measures will permit estimation of the graph directionality. Finally, different parameters of network can be calculated. They must be subsequently statistically tested against the distribution of the same parameters in random networks containing the same number of nodes and connections (Bullmore & Sporns, 2009).
Small-world Network
The small-world model represents a global concept useful for studying complex natural and artificial networks (Bassett & Bullmore, 2006; Sporns & Honey, 2006). It has been analyzed extensively in social and life sciences and seems to be highly widespread in natural systems. Indeed, small-world networks have provided important insights into cellular metabolism and transcriptional regulation (Sporns and Honey, 2006); small-world organization has been found in structural neural networks in animal models (Sporns & Kotter, 2004). Structural and functional brain networks in humans were also shown to have small-world attributes (Sporns et al., 2004; Bassett & Bullmore, 2006; Honey et al., 2009). Characteristic for such networks is an optimal balance of integrated and segregated information processing (Sporns & Honey, 2006). Furthermore, they are generally associated with global and local parallel information processing and low wiring costs (Bassett & Bullmore, 2006).
In context of the graph theory, small-world network can be seen as a special case of highly efficient network organization.
Granger Causality
Activity in a brain region can directly or indirectly exert influence on the activity of another brain region.
Such causal interactions are described by the concept of effective connectivity (Friston & Dolan, 2010). Granger causality (GC) analysis is a statistical approach introduced first in econometrics. It determines causal dependencies by measuring whether one time series can be used to forecast another. In other words, GC tries to establish a statistical dependence between a local measurement of neural activity and measurement of activity elsewhere in the past (Friston et al., 2013). Hence, this approach can be used to extract information about the dynamics and directionality of the signal. GC can be used in task-evoked and resting state studies, revealing the directionality of the information
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flow in a stimulated (Chen et al., 2009; Zhou et al., 2009b) or unstimulated setting (Stevens et al., 2009; Liao et al., 2010).This approach has provided useful descriptions of directed FC in many electrophysiological studies (Bernasconi & Konig, 1999; Brovelli et al., 2004). It can be applied to standard EEG and MEG signals, either at the source (Barrett et al., 2012) or at the sensor level (Brovelli et al., 2004). The application of GC to fMRI data is more controversial due to the slow dynamics and regional variability of the haemodynamic response. However, careful consideration of methodological issues has permitted some valuable applications (Wen et al., 2012).
Dynamic Causal Modelling
Dynamic casual modelling (DCM) is probably the most popular approach for the assessment of effective connectivity. It incorporates the basic assumption that neural activity propagates through brain networks as in an input-state-output system with causal interactions mediating unobservable (hidden) neuronal dynamics (Friston et al., 2013). The modelling of both, neuronal and observed input-output dependencies is then reconciled in a generative model which can be optimized post hoc according to the data and studied question (Friston & Dolan, 2010). The neuronal models are formulated so as to explain how data are caused in terms of a network of distributed sources. These sources are assumed to communicate with each other through connections and influence the dynamics of hidden states (Friston & Dolan, 2010). In this way a number of models can be created.
Subsequently each model is tested on how well it explains the data. Finally, the models are compared against each other. Model comparison permits to explore a wide diversity of explanatory designs and to find optimal architectures or networks. Having selected the best model (or subset of models), it is possible then to get access to the coupling parameters defining the network (Friston et al., 2003).
Even though DCM is best known through its application to fMRI, recent advances have focused on modeling of electrophysiological dynamics. Different data (e.g. ERPs or induced responses) can be modelled by using DCM (Kiebel et al., 2006), and it has even been applied to resting state recordings (Friston et al., 2011) demonstrating its universality.
In contrast to GC, which models dependency among observed responses, DCM models couplings among the hidden states generating observations. Despite this fundamental difference, the two approaches can be viewed as complementary (Friston et al., 2013).
Network modeling is a useful and efficient tool to study neural networks. It yields informative, intuitive and highly visually compelling results. A wealth of studies in healthy and diseased participants as well as studies combining different models reported meaningful and important findings on the network structure and information flow. In spite of these advantages, it is important to be aware that the choice and application of the mathematical models to the real data represent a simplified approximation and is highly dependent on a priori assumptions. Therefore, the interpretation of the results should be made only with necessary carefulness.