Keywords

1 Functional Brain Networks

The brain consists of billions of interconnecting neurons, forming an extremely complex system (Tononi et al. 1998; Tononi and Edelman 1998) in which clusters of neurons are organized as functional units with more-or-less specific information processing capabilities (e.g. Born and Bradley 2005; Grodzinsky 2000). Yet, cognitive functions require the coordinated activity of these spatially separated units, where the oscillatory nature of neuronal activity may provide a possible mechanism (Buzsaki and Wang 2012; Engel et al. 2001; Fries 2005; Singer 1999; Varela et al. 2001). These interacting units form a large-scale complex network (Bullmore and Sporns 2012; Schnitzler and Gross 2005). The organization of such complex brain networks can be characterized using concepts from graph theory (Bullmore and Sporns 2009; Reijneveld et al. 2007; Stam and Reijneveld 2007; Watts and Strogatz 1998). Application of graph theoretical tools to human brain networks has shown that the brain is organized according to a highly efficient topology that combines a high level of local integration (i.e. dense local clustering of connections) with a high level of global efficiency (i.e. critical long-distance connections), forming a so-called small-world organization (Bassett and Bullmore 2006; Stam and van Straaten 2012b; Watts and Strogatz 1998). In addition, brain networks in healthy subjects contain a subset of relatively highly connected regions (‘hubs’) (Achard et al. 2006; Barabasi and Albert 1999). These hubs seem to be mutually and densely interconnected, forming a connectivity backbone or “rich club” crucial for efficient brain communication (van den Heuvel et al. 2012; van den Heuvel and Sporns 2011).

It has been shown that network topology is highly heritable (Smit et al. 2008, 2010), that the network configuration changes during the life span (Smit et al. 2012) and that there are gender differences (Smit et al. 2008; Tian et al. 2011). Moreover, an increasing number of studies has shown that various brain disorders disturb the optimal organization of the functional brain networks (for reviews see Reijneveld et al. 2007; Stam and van Straaten 2012b; van Straaten and Stam 2013), and that these network alterations correlate with cognitive performance, as well as with parameters of disease severity and/or progression.

2 Source-Space Analysis

Magnetoencephalography, with its high temporal resolution, can be used to characterize the functional brain networks that are formed by interacting sources of oscillatory activity. Although such an analysis can be performed directly at the sensor-level, there are several factors that should be considered. Firstly, multiple sensors pick up the signals from a single source due to the nature of the electromagnetic signals (Sarvas 1987), known as field spread, as well as due to volume conduction.Footnote 1 Both these phenomena may lead to erroneous estimates of functional connectivity. It is important to realize though that projection of the signals to source-level in itself does not eliminate these effects (Hillebrand et al. 2012). Secondly, the mixture of signals originating from spatially separated brain areas can result in under- or overestimation of functional connectivity (Schoffelen and Gross 2009). Demixing the contribution from spatially separate sources, as well as enabling a more straightforward interpretation of the functional data in relation to its underlying structure, are therefore the main reasons to perform an analysis in source-space. This requires the solution of the inverse problem, i.e. the problem of estimating the electrical current distribution that produced the recorded magnetic flux. This is an ill-posed problem, meaning that there is no unique solution, unless prior knowledge (or constraints) is added. We know, for example, that the cortical current density is small (the moment per unit area is typically of the order of 50 pAm/mm2; Lü and Williamson 1991), and solutions with estimated source strengths of several Ampere-meter can therefore safely be ignored. Different source reconstruction techniques exist (Baillet et al. 2001), and they vary in the type and number of constraints that are imposed (Hillebrand and Barnes 2005; Wipf and Nagarajan 2009). Constraints might be that there are only a small number of sources active at a specific instant in time (multi-dipole solutions; Supek and Aine 1993), that the whole cortex is active to some degree but with the minimum energy necessary to describe the measured data (minimum norm solutions; Hamalainen and Ilmoniemi 1994), or that there are no perfectly linearly correlated areas of activation within the brain (beamformers; Robinson and Vrba 1999; Sekihara and Nagarajan 2008; van Veen et al. 1997).

In recent years, beamforming has become one of the main source reconstruction approaches for MEG. It has been argued that the uncorrelated-source assumption may be realistic for many empirical datasets (Hillebrand and Barnes 2005), and violations of this assumption can be tolerated to some extent (Hadjipapas et al. 2005). For those cases where strongly correlated sources are encountered, for example during auditory stimulation or parallel processing of visual stimuli, the beamformer formulism can be adapted (Brookes et al. 2007; Dalal et al. 2006; Diwakar et al. 2011; Hui et al. 2010; Quraan and Cheyne 2010). From a practical point of view, there are few parameters to set when performing beamformer analysis, the main ones being the time-frequency window(s) in the data for which to perform the source reconstruction (Dalal et al. 2008). Source reconstruction is achieved in a sequential manner, where for each target location in the brain (typically a grid consisting of 5 × 5 × 5 mm voxels is used; Barnes et al. 2004) neuronal activity is estimated using an optimal set of beamformer weights, W:

$${\hat{\text{Q}}}= {\mathbf{ W B}},$$
(1)

where \({\hat{\text{Q}}}\)is the estimated source strength in nAm for a source at a given target voxel, and with a certain orientation; B is a vector containing the recorded magnetic flux at a given latency.

These weights are optimal in the sense that the values of the weights are chosen such that activity would be fully reconstructed for a target location, if this target location happens to be active (this is called the unit-gain constraint), whilst rejecting the contribution from all other sources, be it within or outside the brain. For a mathematical description we refer the reader to Robinson and Vrba 1999; Sekihara and Nagarajan 2008; van Veen et al. 1997, and for a review see Hillebrand et al. 2005).

Although other source reconstruction approaches also require accurate MEG/MRI co-registration and modeling of the volume conductor, beamforming is particularly sensitive to inaccuracies in the forward solution (Hillebrand and Barnes 2011, 2003; Vrba 2002): the unit-gain constraint described above results in a suppression of source activity if there is a deviation from the correct forward solution.

Equation 1 assumes that the orientation of a source is known. In practice, this is not the case, and the orientation can be set to the one that gives the maximum beamformer output (scalar beamformer; Robinson and Vrba 1999; Sekihara et al. 2004), the orientation of the cortical surface could be used (but see Hillebrand and Barnes 2003), or one could estimate the beamformer output for three orthogonal directions (vector beamformer; van Veen et al. 1997). Finally, the decrease in sensitivity for deeper sources (Hillebrand and Barnes 2002) results in an increase in the (norm of) the beamformer weights with source depth, and a disproportionate amplification of white sensor noise for deeper sources. To compensate for this depth bias, the beamformer weights, or equivalently the reconstructed beamformer image (Cheyne et al. 2006), are typically rescaled using a (projection of) the sensor noise. An estimate of the sensor noise therefore has to be provided. The effects of noise can further be reduced through regularization (Vrba 2002).

Once the beamformer weights have been estimated, one can reconstruct a three-dimensional volumetric image of activity (or of a change in activity in case experimental conditions are contrasted; see also Brookes et al. 2005). The statistical significance of these individual images is difficult to determine (but see Barnes and Hillebrand 2003), yet one can readily perform group-level statistics using tools developed for functional Magnetic Resonance Imaging (fMRI; Singh et al. 2002, 2003). We have recently introduced an atlas-based approach that also enables the comparison of beamformer results across individuals (see Fig. 1; Hillebrand et al. 2012). For each individual, the neuronal activity is reconstructed for a limited set of regions-of-interest (ROIs) that covers almost the entire brain, where the ROIs can be obtained from a standard atlas (Collins et al. 1995; Evans et al. 2012; Lancaster et al. 1997, 2000; Tzourio-Mazoyer et al., 2002). This approach has two main advantages: i) it enables the comparison between different modalities (Bullmore and Sporns 2009); ii) the number of ROIs is always the same across individuals, such that functional networks can more readily be compared (but see below).

Fig. 1
figure 1

Examples of recent applications of an atlas-based beamformer in combination with functional network analysis. Panel a shows data from a group of 13 healthy controls. The mean alpha band PLI, also known as the weighted degree or node strength in terms of graph theory, for each ROI, is displayed as a color-coded map (thresholded at p = 0.05) on a schematic of the parcellated template brain (modified from Hillebrand et al. 2012). Note that the regions in the occipital lobe are most strongly connected, as can be expected for the alpha band. Panel b shows the connections (PLI) from each ROI to all other ROIs, using an arbitrary threshold. Again, there is a clear pattern of strong connections between regions in the occipital lobe, with additional connections to areas in the temporal and frontal lobes. The upper panel in c shows a similar patterns for data from 17 healthy controls from a different study (Tewarie et al. 2014). This figure displays only the connections that formed part of the MST in the alpha2 band (the colorbar indicates PLI values). Interestingly, there seems to be a shift from an occipital to frontal pattern in healthy controls to a more diffuse pattern in 21 patients with Multiple Sclerosis (lower panel) (modified from Tewarie et al. 2014). Moreover, Tewarie and colleagues showed that this change in network topology correlated with reduced overall cognitive performance

Full size image

3 Functional Connectivity in Source-Space

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):

$${\text{PLI }} = \left| {\left\langle {{\text{sign}}\left[ {\sin \left( {\Delta \phi ({\text{t}}_{\text{k}} )} \right)} \right]} \right\rangle } \right|,$$
(2)

where Δϕ is the difference between the instantaneous phases for two time-series defined in the interval [−π, π], tk 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).

4 Topology of the Functional Network

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 coefficientFootnote 2 and the (average shortest) path length,Footnote 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 valuesFootnote 4) 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.

5 Applications in Neurology

5.1 Glioma

In a recent MEG study we revealed a relationship between resting-state functional network properties and protein expression patterns in tumor tissue collected during neurosurgery (Douw et al. 2013). In particular, between-module connectivity was selectively associated with two epilepsy-related proteins, namely synaptic vesicle protein 2A (SV2A) and poly-glycoprotein (P-gp), yet only for the ROIs that contained tumor tissue. Moreover, receiver operator characteristic (ROC) analysis revealed that SV2A expression could be classified with 100 % accuracy on the basis of the between-module connectivity, indicating that the role of the tumor area in the brain network may be an excellent marker for molecular features of brain tissue, which may be used clinically to monitor the efficacy of the anti-epileptic drug levetiracetam (de Groot et al. 2011). Moreover, lower between-module connectivity in the tumor area and higher number of seizures significantly predicted higher P-gp expression, which is in line with previous research showing that high seizure proneness is related to increased P-gp expression (Miller et al. 2008), and suggests that local network topology is an intermediate level between molecular tissue features and clinical patient status. A separate study (van Dellen et al. 2014) examined the link between functional network organization and seizure status further in a longitudinal study. Resting-state MEG recordings were obtained for 20 lesional epilepsy patients at baseline (preoperatively; T0), and at 3–7 (T1) and 9–15 months after resection (T2). Functional connectivity in the lower alpha band correlated positively with seizure frequency at baseline, especially in regions where lesions were located. MST leaf fraction, a measure of integration of information in the network, was significantly increased between T0 and T2, yet only for the seizure-free patients. Moreover, MST-based eccentricity and betweenness centrality, which are measures of node importance and hub-status, decreased between T0 and T2 in seizure free patients, also in regions that were anatomically close to lesion locations and resection cavities. These results demonstrate that there is a link between successful epilepsy surgery and changes in functional network topology. These insights may eventually be utilized for optimization of neurosurgical approaches.

5.2 Parkinson’s Disease

A longitudinal study involving patients with Parkinson’s disease (PD) also revealed a relationship between disease progression and functional brain network topology (Olde Dubbelink et al. 2014). MST analysis revealed a decentralized and less integrated network configuration in early stage untreated PD, which progressed over time. Conventional analysis of clustering and path length also revealed an initial impaired local efficiency, which continued to progress over time, together with reductions in global efficiency. Importantly, these longitudinal changes in network topology were associated with deteriorating motor function and cognitive performance.

6 Future Developments

Excitingly, network analysis, particularly in combination with a standard parcellation of the brain (e.g. through the use of an anatomical atlas), provides a principled way to compare results across different modalities (Bullmore and Sporns 2009). For example, in recent years there has been an insurgence of research into the functional and cognitive relevance of resting-state functional connectivity as determined using fMRI (van den Heuvel and Hulshoff Pol 2010). Although it is already becoming clear that there is a close link between resting-state networks based on hemodynamic phenomena and the underlying electrophysiological networks (e.g. Brookes et al. 2011b; Niu et al. 2012), we envisage that a bias-free network approach allows for an even more accurate integration of these modalities, leading to a better understanding of brain function. Similarly, this approach enables us to directly link the properties of, and dynamics on, functional networks to the topology of the underlying structural network (Guye et al. 2008; Honey et al. 2007). An interesting direction for future work is the study of the interaction between these two types of networks, i.e. to study how functional plasticity affects the structural network, and vice versa (Assenza et al. 2011). Additionally, the same framework can be used to create anatomically and functionally realistic models that can simulate MEG signals. That is, neural mass models can be placed at each location of the anatomical parcellation scheme, where the anatomical connections between the neural masses can be based on experimental DTI data that were obtained for the same atlas. The parameters in these simulated structural/functional networks can subsequently be adjusted in order to test hypotheses (based on observations in experimental data) about disease mechanisms, or to generate new hypotheses about disease effects that we should be able to observe in experimental studies (de Haan et al. 2012; van Dellen et al. 2013).

The atlas-based beamforming approach itself may be developed further in several aspects. We have proposed to use the voxel with maximum power as representative for a ROI, which can introduce some biases, for example for ROIs that cover a large area of cortex. Indeed, the spatial resolution that is obtainable with MEG varies from millimeters to centimeters across the brain (Hillebrand and Barnes 2002), and depends on factors such as location of the neuronal activity, orientation of the cortex and signal-to-noise ratio (Barnes et al. 2004; Hillebrand and Barnes 2002). Our current hypothesis is that the AAL atlas has a resolution that matches the spatial resolution of MEG resting-state data. However, future research should test whether this hypothesis is valid for all cortical regions, for example through the use of atlases with higher spatial resolution (Evans et al. 2012; Seibert and Brewer 2011). In addition, selection of a single representative voxel might be prone to noise and outliers. However, the optimal method of dealing with multiple voxels within a ROI has not been defined yet, and using for instance an averaging method presents other biases, such as introducing artificial differences in signal-to-noise ratios for different sized ROIs. Similarly, one could argue that a priori selection of a target location within a ROI would speed up the computations. However, beamformer reconstructions vary most around peak activations (Barnes and Hillebrand 2003) and as a consequence, the a priori selection of a target voxel could have the effect that the activity for a ROI is completely missed (Barnes et al. 2004).

Another interesting direction for new research is to study the dynamics of functional networks in more detail (de Pasquale et al. 2012), thereby taking advantage of the strongest attribute of MEG, namely its high temporal resolution. A prerequisite is the development of measures of functional connectivity that have high temporal resolution, yet are insensitive to the effects of volume conduction. This would allow us to study functional networks in more detail, and examine the importance of the evolution of functional networks on short time-scales. For example, it was described above that functional brain networks can be divided into modules; are these modules stable over time (Bassett et al. 2013)? And hubs play an important role in the network, is this also reflected in their dynamics, i.e., do hubs evolve differently than non-hubs? Similarly, how does the formation and re-configuration over time of functional networks relate to cognitive performance? Are these dynamics altered in the diseased brain? And if so, is there a phase-transition that distinguishes the healthy from the diseased brain?