Functional brain networks are shaped and constrained by the underlying structural network. However, functional networks are not merely a one-to-one reflection of the structural network. Several theories have been put forward to understand the relationship between structural and functional networks. However, it remains unclear how these theories can be unified. Two existing recent theories state that 1) functional networks can be explained by all possible walks in the structural network, which we will refer to as the series expansion approach, and 2) functional networks can be explained by a weighted combination of the eigenmodes of the structural network, the so-called eigenmode approach. To elucidate the unique or common explanatory power of these approaches to estimate functional networks from the structural network, we analysed the relationship between these two existing views. Using linear algebra, we first show that the eigenmode approach can be written in terms of the series expansion approach, i.e., walks on the structural network associated with different hop counts correspond to different weightings of the eigenvectors of this network. Second, we provide explicit expressions for the coefficients for both the eigenmode and series expansion approach. These theoretical results were verified by empirical data from Diffusion Tensor Imaging (DTI) and functional Magnetic Resonance Imaging (fMRI), demonstrating a strong correlation between the mappings based on both approaches. Third, we analytically and empirically demonstrate that the fit of the eigenmode approach to measured functional data is always at least as good as the fit of the series expansion approach, and that errors in the structural data lead to large errors of the estimated coefficients for the series expansion approach. Therefore, we argue that the eigenmode approach should be preferred over the series expansion approach. Results hold for eigenmodes of the weighted adjacency matrices as well as eigenmodes of the graph Laplacian. ​Taken together, these results provide an important step towards unification of existing theories regarding the structure-function relationships in brain networks.

There is an increasing awareness of the advantages of multi-modal neuroimaging. Networks obtained from different modalities are usually treated in isolation, which is however contradictory to accumulating evidence that these networks show non-trivial interdependencies. Even networks obtained from a single modality, such as frequency-band specific functional networks measured from magnetoencephalography (MEG) are often treated independently. Here, we discuss how a multilayer network framework allows for integration of multiple networks into a single network description and how graph metrics can be applied to quantify multilayer network organisation for group comparison. We analyse how well-known biases for single layer networks, such as effects of group differences in link density and/or average connectivity, influence multilayer networks, and we compare four schemes that aim to correct for such biases: the minimum spanning tree (MST), effective graph resistance cost minimisation, efficiency cost optimisation (ECO) and a normalisation scheme based on singular value decomposition (SVD). These schemes can be applied to the layers independently or to the multilayer network as a whole. For correction applied to whole multilayer networks, only the SVD showed sufficient bias correction. For correction applied to individual layers, three schemes (ECO, MST, SVD) could correct for biases. By using generative models as well as empirical MEG and functional magnetic resonance imaging (fMRI) data, we further demonstrated that all schemes were sensitive to identify network topology when the original networks were perturbed. In conclusion, uncorrected multilayer network analysis leads to biases. These biases may differ between centres and studies and could consequently lead to unreproducible results in a similar manner as for single layer networks. We therefore recommend using correction schemes prior to multilayer network analysis for group comparisons.

Network science has widely studied the properties of brain networks. Recent work has observed a global back-to-front pattern of information flow for higher frequency bands in magnetoencephalography data. However, the effective connectivity at a local level remains yet to be analyzed. On a local level, the building blocks of all networks are motifs. In this study, we exploit the measure of dPTE to analyze motifs of the estimated effective connectivity networks. We find that some 3- and 4-motifs, the bidirectional two-hop path and its extended 4-node versions, are significantly overexpressed in the analyzed networks in comparison with random networks. With a recently developed motif-based clustering algorithm we separate the effective connectivity network in two main clusters which reveal its higher-order organization with a strong information flow between posterior hubs and anterior regions.

The interplay between structural connections and emerging information flow in the human brain remains an open research problem. A recent study observed global patterns of directional information flow in empirical data using the measure of transfer entropy. For higher frequency bands, the overall direction of information flow was from posterior to anterior regions whereas an anterior-to-posterior pattern was observed in lower frequency bands. In this study, we applied a simple Susceptible-Infected-Susceptible (SIS) epidemic spreading model on the human connectome with the aim to reveal the topological properties of the structural network that give rise to these global patterns. We found that direct structural connections induced higher transfer entropy between two brain regions and that transfer entropy decreased with increasing distance between nodes (in terms of hops in the structural network). Applying the SIS model, we were able to confirm the empirically observed opposite information flow patterns and posterior hubs in the structural network seem to play a dominant role in the network dynamics. For small time scales, when these hubs acted as strong receivers of information, the global pattern of information flow was in the posterior-to-anterior direction and in the opposite direction when they were strong senders. Our analysis suggests that these global patterns of directional information flow are the result of an unequal spatial distribution of the structural degree between posterior and anterior regions and their directions seem to be linked to different time scales of the spreading process.

The relationship between structural and functional brain networks is still highly debated. Most previous studies have used a single functional imaging modality to analyze this relationship. In this work, we use multimodal data, from functional MRI, magnetoencephalography, and diffusion tensor imaging, and assume that there exists a mapping between the connectivity matrices of the resting-state functional and structural networks. We investigate this mapping employing group averaged as well as individual data. We indeed find a significantly high goodness of fit level for this structure-function mapping. Our analysis suggests that a functional connection is shaped by all walks up to the diameter in the structural network in both modality cases. When analyzing the inverse mapping, from function to structure, longer walks in the functional network also seem to possess minor influence on the structural connection strengths. Even though similar overall properties for the structure-function mapping are found for different functional modalities, our results indicate that the structure-function relationship is modality dependent.

Neuronal oscillations exist across a broad frequency spectrum, and are thought to provide a mechanism of interaction between spatially separated brain regions. Since ongoing mental activity necessitates the simultaneous formation of multiple networks, it seems likely that the brain employs interactions within multiple frequency bands, as well as cross-frequency coupling, to support such networks. Here, we propose a multi-layer network framework that elucidates this pan-spectral picture of network interactions. Our network consists of multiple layers (frequency-band specific networks) that influence each other via inter-layer (cross-frequency) coupling. Applying this model to MEG resting-state data and using envelope correlations as connectivity metric, we demonstrate strong dependency between within layer structure and inter-layer coupling, indicating that networks obtained in different frequency bands do not act as independent entities. More specifically, our results suggest that frequency band specific networks are characterised by a common structure seen across all layers, superimposed by layer specific connectivity, and inter-layer coupling is most strongly associated with this common mode. Finally, using a biophysical model, we demonstrate that there are two regimes of multi-layer network behaviour; one in which different layers are independent and a second in which they operate highly dependent. Results suggest that the healthy human brain operates at the transition point between these regimes, allowing for integration and segregation between layers. Overall, our observations show that a complete picture of global brain network connectivity requires integration of connectivity patterns across the full frequency spectrum.

Understanding the electrophysiological basis of resting state networks (RSNs) in the human brain is a critical step towards elucidating how inter-areal connectivity supports healthy brain function. In recent years, the relationship between RSNs (typically measured using haemodynamic signals) and electrophysiology has been explored using functional Magnetic Resonance Imaging (fMRI) and magnetoencephalography (MEG). Significant progress has been made, with similar spatial structure observable in both modalities. However, there is a pressing need to understand this relationship beyond simple visual similarity of RSN patterns. Here, we introduce a mathematical model to predict fMRI-based RSNs using MEG. Our unique model, based upon a multivariate Taylor series, incorporates both phase and amplitude based MEG connectivity metrics, as well as linear and non-linear interactions within and between neural oscillations measured in multiple frequency bands. We show that including non-linear interactions, multiple frequency bands and cross-frequency terms significantly improves fMRI network prediction. This shows that fMRI connectivity is not only the result of direct electrophysiological connections, but is also driven by the overlap of connectivity profiles between separate regions. Our results indicate that a complete understanding of the electrophysiological basis of RSNs goes beyond simple frequency-specific analysis, and further exploration of non-linear and cross-frequency interactions will shed new light on distributed network connectivity, and its perturbation in pathology.