The recorded MEG data can be projected through the estimated beamformer weights in order to obtain the time-series for each voxel in the brain (Eq. 1), which are often referred to as virtual electrodes. In order to obtain a single time-series for each ROI, we subsequently select the voxel with maximum power as representative for the ROI. These ROI time-series can then be used as input for functional connectivity analysis. A wide range of functional connectivity estimators are available (Pereda et al. 2005), yet most of these measures are sensitive to the effects of volume conduction and field spread. One could remove these biases before performing connectivity analysis (Brookes et al. 2012; Hipp et al. 2012), or estimate the extent of the bias through simulations (Brookes et al. 2011a). Perhaps more straightforward is the use of measures such as the imaginary part of coherency (Nolte et al. 2004), phase-slope index (Nolte et al. 2008), the Phase Lag Index (PLI; Stam et al. 2007) and related lagged phase synchronization (Pascual-Marqui 2007), as these are inherently insensitive to these biases, where the PLI has the additional advantage that it does not directly depend on the amplitude of the signals (but see Muthukumaraswamy and Singh 2011). These measures have therefore gained popularity in recent years (Canuet et al. 2011, 2012; Guggisberg et al. 2008; Ioannides et al. 2012; Martino et al. 2011; Nolte and Muller 2010; Ponsen et al. 2013; Sekihara et al. 2011; Shahbazi et al. 2012; Tarapore et al. 2012).

The PLI is defined as (Stam et al. 2007):
where Δϕ is the difference between the instantaneous phases for two time-series defined in the interval [−π, π], t_k are discrete time-steps, and < > denotes the mean. In short, PLI is a measure for the asymmetry of the distribution of phase differences between two signals, and ranges between 0 and 1. A PLI value of 0 indicates no coupling, coupling with a phase difference of 0 ± nπ radians (with n an integer), or an equal distribution of positive and negative phase differences. Common sources lead to a phase difference of 0 ± nπ radians between two signals, hence the PLI is insensitive to the influence of common sources (Stam et al. 2007). A PLI > 0 is obtained when the distribution of phase differences is asymmetric and is indicative of functional coupling between two signals. Note that PLI does not indicate which of the signals is leading in phase (but see Stam and van Straaten 2012a) and that it potentially discards true interactions with zero-phase lag. Moreover, a value of 0 for uncoupled sources is only achieved for (infinitely) long time-series, hence the PLI is affected by the length of the time-series. PLI also underestimates connectivity between sources with small-lag interactions. A modification of PLI addresses this issue, albeit at the expense of introducing an arbitrary bias favoring large phase differences and mixing of the estimation of consistency of phase differences with the estimation of the magnitude of the phase difference (Vinck et al. 2011).

Graph theory provides the mathematical framework to characterize the topology of the functional network that is formed by the interacting sources. For this purpose, each ROI is denoted as a vertex (node) and each connection (e.g. the PLI value) is denoted as an edge between the vertices (see Fig. 1). Various graph-theoretical measures can subsequently be used to characterize the network (e.g. Rubinov and Sporns 2010). Two such measures, the clustering coefficient² and the (average shortest) path length,³ can be used to explain how the brain can fulfill two seemingly contradictory requirements, namely the processing of information in local functional units on the one hand (‘segregation’) and simultaneous coordination of activity in and between these spatially separated units (‘integration’) (Sporns et al. 2002, 2004). Watts and Strogatz (1998) famously demonstrated with a simple rewiring model that adding a few long distance connections to a network with many local interconnections results in a high clustering yet small average path length. Many large networks, including the brain, have such a so-called small-world configuration (Bassett and Bullmore 2006; Stam 2004). However, this model does not provide a completely satisfactory description of functional brain networks, since it can not explain the occurrence of hubs (Eguiluz et al. 2005). Similarly, the scale-free growth model by Barabasi and Albert (1999), which explains the occurrence of hubs, does not capture the high level of clustering and (hierarchical) modularity observed in experimental data (Meunier et al. 2009). Obviously, we currently lack a model that integrates small-world and scale-free models and fully and elegantly explains the observed functional brain network characteristics (Bullmore and Bassett 2011; Clune et al. 2013; Stam and van Straaten 2012b).

From a practical point of view, although the application of graph theory at the source-level already aids the interpretation of results and the comparability across studies, it is not trivial to compare network topology across individuals, groups, studies or modalities, as was elegantly shown by van Wijk et al. (2010). At the heart of the problem lies the observation that many network properties depend on the size, sparsity (percentage of all possible edges that are present), and the average degree (i.e. the average number of connections per node) of the network. Fixing the number of nodes and average degree in the network (by setting a threshold) does eliminate size effects but may introduce spurious connections or ignore strong connections in the network, and using random surrogates for normalization does not solve this problem either (and may even exuberate it; van Wijk et al. 2010).

A novel approach is to construct the minimum spanning tree (MST) of the original graphs (Boersma et al. 2013; Jackson and Read 2010a, b; Wang et al. 2008). A tree is a sub-graph that does not contain circles or loops and connects all nodes in the original graph, and the MST is the tree that has the minimum total weight (i.e. the sum of all edge values⁴) of all possible spanning trees of the original graph. If the original graph contains N nodes than the MST always has N nodes and M = N−1 edges, therefore enabling direct comparison of MSTs between groups and avoiding aforementioned methodological difficulties. Furthermore, if the original network can be interpreted as a kind of transport network, and if edge weights in the original graph possess strong fluctuations, also called the strong disorder limit, then all transport in the original graph flows over the MST (van Mieghem and van Langen 2005), forming the critical backbone of the original graph (van Mieghem and Magdalena 2005; Wang et al. 2008). Interestingly, it seems that for source-reconstructed MEG data for patients with Multiple Sclerosis, as well as for healthy controls, there is a tendency of the weight distribution towards the strong disorder limit (Tewarie et al. 2014). This implies that there is a high probability that the MSTs for both patients and healthy controls can be considered as the critical backbone of the original functional brain networks. Hence, analysis of the minimum spanning tree not only provides a bias free approach to network analysis, but also captures important properties of the original network.