Frequency-based brain networks: From a multiplex framework to a full multilayer description

We explore how to study dynamical interactions between brain regions using functional multilayer networks whose layers represent the different frequency bands at which a brain operates. Specifically, we investigate the consequences of considering the brain as a multilayer network in which all brain regions can interact with each other at different frequency bands, instead of as a multiplex network, in which interactions between different frequency bands are only allowed within each brain region and not between them. We study the second smallest eigenvalue of the combinatorial supra-Laplacian matrix of the multilayer network in detail, and we thereby show that the heterogeneity of interlayer edges and, especially, the fraction of missing edges crucially modify the spectral properties of the multilayer network. We illustrate our results with both synthetic network models and real data sets obtained from resting state magnetoencephalography. Our work demonstrates an important issue in the construction of frequency-based multilayer brain networks.Comment: 13 pages, 8 figure

Persistent homology of time-dependent functional networks constructed from coupled time series

We use topological data analysis to study "functional networks" that we construct from time-series data from both experimental and synthetic sources. We use persistent homology with a weight rank clique filtration to gain insights into these functional networks, and we use persistence landscapes to interpret our results. Our first example uses time-series output from networks of coupled Kuramoto oscillators. Our second example consists of biological data in the form of functional magnetic resonance imaging (fMRI) data that was acquired from human subjects during a simple motor-learning task in which subjects were monitored on three days in a five-day period. With these examples, we demonstrate that (1) using persistent homology to study functional networks provides fascinating insights into their properties and (2) the position of the features in a filtration can sometimes play a more vital role than persistence in the interpretation of topological features, even though conventionally the latter is used to distinguish between signal and noise. We find that persistent homology can detect differences in synchronization patterns in our data sets over time, giving insight both on changes in community structure in the networks and on increased synchronization between brain regions that form loops in a functional network during motor learning. For the motor-learning data, persistence landscapes also reveal that on average the majority of changes in the network loops take place on the second of the three days of the learning process.Comment: 17 pages (+3 pages in Supplementary Information), 11 figures in many text (many with multiple parts) + others in SI, submitte

Functional connectivity analysis of cerebellum using spatially constrained spectral clustering

The human cerebellum contains almost 50% of the neurons in the brain, although its volume does not exceed 10% of the total brain volume. The goal of this study is to derive the functional network of the cerebellum during the resting-state and then compare the ensuing group networks between males and females. Toward this direction, a spatially constrained version of the classic spectral clustering algorithm is proposed and then compared against conventional spectral graph theory approaches, such as spectral clustering, and N-cut, on synthetic data as well as on resting-state fMRI data obtained from the Human Connectome Project (HCP). The extracted atlas was combined with the anatomical atlas of the cerebellum resulting in a functional atlas with 46 regions of interest. As a final step, a gender-based network analysis of the cerebellum was performed using the data-driven atlas along with the concept of the minimum spanning trees. The simulation analysis results confirm the dominance of the spatially constrained spectral clustering approach in discriminating activation patterns under noisy conditions. The network analysis results reveal statistically significant differences in the optimal tree organization between males and females. In addition, the dominance of the left VI lobule in both genders supports the results reported in a previous study of ours. To our knowledge, the extracted atlas comprises the first resting-state atlas of the cerebellum based on HCP data

Sparse DCM for whole-brain effective connectivity from resting-state fMRI data

Contemporary neuroscience has embraced network science and dynamical systems to study the complex and self-organized structure of the human brain. Despite the developments in non-invasive neuroimaging techniques, a full understanding of the directed interactions in whole brain networks, referred to as effective connectivity, as well as their role in the emergent brain dynamics is still lacking. The main reason is that estimating brain connectivity requires solving a formidable large-scale inverse problem from indirect and noisy measurements. Building on the dynamic causal modelling framework, the present study offers a novel method for estimating whole-brain effective connectivity from resting-state functional magnetic resonance data. To this purpose sparse estimation methods are adapted to infer the parameters of our novel model, which is based on a linearized, region-specific haemodynamic response function. The resulting algorithm, referred to as sparse DCM, is shown to compare favorably with state-of-the art methods when tested on both synthetic and real data. We also provide a graph-theoretical analysis on the whole-brain effective connectivity estimated using data from a cohort of healthy individuals, which reveals properties such as asymmetry in the connectivity structure as well as the different roles of brain areas in favoring segregation or integration

Resolving Structure in Human Brain Organization: Identifying Mesoscale Organization in Weighted Network Representations

Human brain anatomy and function display a combination of modular and hierarchical organization, suggesting the importance of both cohesive structures and variable resolutions in the facilitation of healthy cognitive processes. However, tools to simultaneously probe these features of brain architecture require further development. We propose and apply a set of methods to extract cohesive structures in network representations of brain connectivity using multi-resolution techniques. We employ a combination of soft thresholding, windowed thresholding, and resolution in community detection, that enable us to identify and isolate structures associated with different weights. One such mesoscale structure is bipartivity, which quantifies the extent to which the brain is divided into two partitions with high connectivity between partitions and low connectivity within partitions. A second, complementary mesoscale structure is modularity, which quantifies the extent to which the brain is divided into multiple communities with strong connectivity within each community and weak connectivity between communities. Our methods lead to multi-resolution curves of these network diagnostics over a range of spatial, geometric, and structural scales. For statistical comparison, we contrast our results with those obtained for several benchmark null models. Our work demonstrates that multi-resolution diagnostic curves capture complex organizational profiles in weighted graphs. We apply these methods to the identification of resolution-specific characteristics of healthy weighted graph architecture and altered connectivity profiles in psychiatric disease.Comment: Comments welcom

Resolving structural variability in network models and the brain

Large-scale white matter pathways crisscrossing the cortex create a complex pattern of connectivity that underlies human cognitive function. Generative mechanisms for this architecture have been difficult to identify in part because little is known about mechanistic drivers of structured networks. Here we contrast network properties derived from diffusion spectrum imaging data of the human brain with 13 synthetic network models chosen to probe the roles of physical network embedding and temporal network growth. We characterize both the empirical and synthetic networks using familiar diagnostics presented in statistical form, as scatter plots and distributions, to reveal the full range of variability of each measure across scales in the network. We focus on the degree distribution, degree assortativity, hierarchy, topological Rentian scaling, and topological fractal scaling---in addition to several summary statistics, including the mean clustering coefficient, shortest path length, and network diameter. The models are investigated in a progressive, branching sequence, aimed at capturing different elements thought to be important in the brain, and range from simple random and regular networks, to models that incorporate specific growth rules and constraints. We find that synthetic models that constrain the network nodes to be embedded in anatomical brain regions tend to produce distributions that are similar to those extracted from the brain. We also find that network models hardcoded to display one network property do not in general also display a second, suggesting that multiple neurobiological mechanisms might be at play in the development of human brain network architecture. Together, the network models that we develop and employ provide a potentially useful starting point for the statistical inference of brain network structure from neuroimaging data.Comment: 24 pages, 11 figures, 1 table, supplementary material

Generative models of the human connectome

The human connectome represents a network map of the brain's wiring diagram and the pattern into which its connections are organized is thought to play an important role in cognitive function. The generative rules that shape the topology of the human connectome remain incompletely understood. Earlier work in model organisms has suggested that wiring rules based on geometric relationships (distance) can account for many but likely not all topological features. Here we systematically explore a family of generative models of the human connectome that yield synthetic networks designed according to different wiring rules combining geometric and a broad range of topological factors. We find that a combination of geometric constraints with a homophilic attachment mechanism can create synthetic networks that closely match many topological characteristics of individual human connectomes, including features that were not included in the optimization of the generative model itself. We use these models to investigate a lifespan dataset and show that, with age, the model parameters undergo progressive changes, suggesting a rebalancing of the generative factors underlying the connectome across the lifespan.Comment: 38 pages, 5 figures + 19 supplemental figures, 1 tabl