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In recent years there has been a shift in focus from the study of local, mostly task-related activation to the exploration of the organization and functioning of large-scale structural and functional complex brain networks. Progress in the interdisciplinary field of modern network science has introduced many new concepts, analytical tools and models which allow a systematic interpretation of multivariate data obtained from structural and functional MRI, EEG and MEG. However, progress in this field has been hampered by the absence of a simple, unbiased method to represent the essential features of brain networks, and to compare these across different conditions, behavioural states and neuropsychiatric/neurological diseases. One promising solution to this problem is to represent brain networks by a minimum spanning tree (MST), a unique acyclic subgraph that connects all nodes and maximizes a property of interest such as synchronization between brain areas. We explain how the global and local properties of an MST can be characterized. We then review early and more recent applications of the MST to EEG and MEG in epilepsy, development, schizophrenia, brain tumours, multiple sclerosis and Parkinson's disease, and show how MST characterization performs compared to more conventional graph analysis. Finally, we illustrate how MST characterization allows representation of observed brain networks in a space of all possible tree configurations and discuss how this may simplify the construction of simple generative models of normal and abnormal brain network organization.

Normal brain function requires interactions between spatially separated, and functionally specialized, macroscopic regions, yet the directionality of these interactions in large-scale functional networks is unknown. Magnetoencephalography was used to determine the directionality of these interactions, where directionality was inferred from time series of beamformer-reconstructed estimates of neuronal activation, using a recently proposed measure of phase transfer entropy. We observed well-organized posterior-to-anterior patterns of information flow in the higher-frequency bands (alpha1, alpha2, and beta band), dominated by regions in the visual cortex and posterior default mode network. Opposite patterns of anterior-toposterior flow were found in the theta band, involving mainly regions in the frontal lobe that were sending information to a more distributed network. Many strong information senders in the theta band were also frequent receivers in the alpha2 band, and vice versa. Our results provide evidence that large-scale resting-state patterns of information flow in the human brain form frequencydependent reentry loops that are dominated by flow from parietooccipital cortex to integrative frontal areas in the higher-frequency bands, which is mirrored by a theta band anterior-to-posterior flow.information flow | phase transfer entropy | resting-state networks | magnetoencephalography | atlas-based beamforming T he brain is an extremely complex system (1-3) containing, at the macroscopic scale, interconnected functional units (4) with more-or-less specific information processing capabilities (5). However, cognitive functions require the coordinated activity of these spatially separated units, where the oscillatory nature of neuronal activity may provide a possible mechanism (6-9). A complete description of these interactions, in terms of both strength and directionality, is therefore necessary for the understanding of both normal and abnormal brain functioning.Functional interactions may be inferred from statistical dependencies between the time series of neuronal activity at different sites, so-called functional connectivity (10). Indeed, interactions in large-scale functional networks have been observed using Electroencephalography, Magnetoencephalography (EEG/MEG) and functional Magnetic Resonance Imaging (fMRI) (e.g., refs. 11-14). However, as yet, little is known about the directionality of these interactions in large-scale functional networks during the resting state. Estimating directionality from fMRI is challenging due to its limited temporal resolution and indirect relation to neuronal activity (15, 16). In contrast, EEG studies in healthy controls have revealed a front-to-back pattern of directed connectivity, particularly in the alpha band (17-22), consistent with modeling studies that have shown that such patterns may arise due to differences in the number of anatomical connections (the degree) of anterior and posterior regions (22, 23). However, modeled patterns of information flow depend on the a...

Brain connectivity studies have revealed that highly connected ‘hub’ regions are particularly vulnerable to Alzheimer pathology: they show marked amyloid-β deposition at an early stage. Recently, excessive local neuronal activity has been shown to increase amyloid deposition. In this study we use a computational model to test the hypothesis that hub regions possess the highest level of activity and that hub vulnerability in Alzheimer's disease is due to this feature. Cortical brain regions were modeled as neural masses, each describing the average activity (spike density and spectral power) of a large number of interconnected excitatory and inhibitory neurons. The large-scale network consisted of 78 neural masses, connected according to a human DTI-based cortical topology. Spike density and spectral power were positively correlated with structural and functional node degrees, confirming the high activity of hub regions, also offering a possible explanation for high resting state Default Mode Network activity. ‘Activity dependent degeneration’ (ADD) was simulated by lowering synaptic strength as a function of the spike density of the main excitatory neurons, and compared to random degeneration. Resulting structural and functional network changes were assessed with graph theoretical analysis. Effects of ADD included oscillatory slowing, loss of spectral power and long-range synchronization, hub vulnerability, and disrupted functional network topology. Observed transient increases in spike density and functional connectivity match reports in Mild Cognitive Impairment (MCI) patients, and may not be compensatory but pathological. In conclusion, the assumption of excessive neuronal activity leading to degeneration provides a possible explanation for hub vulnerability in Alzheimer's disease, supported by the observed relation between connectivity and activity and the reproduction of several neurophysiologic hallmarks. The insight that neuronal activity might play a causal role in Alzheimer's disease can have implications for early detection and interventional strategies.