Degree Distribution of Human Brain Functional Connectivity is Generalized Pareto: A Multi-Scale Analysis

Are degree distributions of human brain functional connectivity networks heavy-tailed? Initial claims based on least-square fitting suggested that brain functional connectivity networks obey power law scaling in their degree distributions. This interpretation has been challenged on methodological grounds. Subsequently, estimators based on maximum-likelihood and non-parametric tests involving surrogate data have been proposed. No clear consensus has emerged as results especially depended on data resolution. To identify the underlying topological distribution of brain functional connectivity calls for a closer examination of the relationship between resolution and statistics of model fitting. In this study, we analyze high-resolution functional magnetic resonance imaging (fMRI) data from the Human Connectome Project to assess its degree distribution across resolutions. We consider resolutions from one thousand to eighty thousand regions of interest (ROIs) and test whether they follow a heavy or short-tailed distribution. We analyze power law, exponential, truncated power law, log-normal, Weibull and generalized Pareto probability distributions. Notably, the Generalized Pareto distribution is of particular interest since it interpolates between heavy-tailed and short-tailed distributions, and it provides a handle on estimating the tail's heaviness or shortness directly from the data. Our results show that the statistics support the short-tailed limit of the generalized Pareto distribution, rather than a power law or any other heavy-tailed distribution. Working across resolutions of the data and performing cross-model comparisons, we further establish the overall robustness of the generalized Pareto model in explaining the data. Moreover, we account for earlier ambiguities by showing that down-sampling the data systematically affects statistical results. At lower resolutions models cannot easily be differentiated on statistical grounds while their plausibility consistently increases up to an upper bound. Indeed, more power law distributions are reported at low resolutions (5K) than at higher ones (50K or 80K). However, we show that these positive identifications at low resolutions fail cross-model comparisons and that down-sampling data introduces the risk of detecting spurious heavy-tailed distributions. This dependence of the statistics of degree distributions on sampling resolution has broader implications for neuroinformatic methodology, especially, when several analyses rely on down-sampled data, for instance, due to a choice of anatomical parcellations or measurement technique. Our findings that node degrees of human brain functional networks follow a short-tailed distribution have important implications for claims of brain organization and function. Our findings do not support common simplistic representations of the brain as a generic complex system with optimally efficient architecture and function, modeled with simple growth mechanisms. Instead these findings reflect a more nuanced picture of a biological system that has been shaped by longstanding and pervasive developmental and architectural constraints, including wiring-cost constraints on the centrality architecture of individual nodes

Robust self-replication of combinatorial information via crystal growth and scission

Understanding how a simple chemical system can accurately replicate combinatorial information, such as a sequence, is an important question for both the study of life in the universe and for the development of evolutionary molecular design techniques. During biological sequence replication, a nucleic acid polymer serves as a template for the enzyme-catalyzed assembly of a complementary sequence. Enzymes then separate the template and complement before the next round of replication. Attempts to understand how replication could occur more simply, such as without enzymes, have largely focused on developing minimal versions of this replication process. Here we describe how a different mechanism, crystal growth and scission, can accurately replicate chemical sequences without enzymes. Crystal growth propagates a sequence of bits while mechanically-induced scission creates new growth fronts. Together, these processes exponentially increase the number of crystal sequences. In the system we describe, sequences are arrangements of DNA tile monomers within ribbon-shaped crystals. 99.98% of bits are copied correctly and 78% of 4-bit sequences are correct after two generations; roughly 40 sequence copies are made per growth front per generation. In principle, this process is accurate enough for 1,000-fold replication of 4-bit sequences with 50% yield, replication of longer sequences, and Darwinian evolution. We thus demonstrate that neither enzymes nor covalent bond formation are required for robust chemical sequence replication. The form of the replicated information is also compatible with the replication and evolution of a wide class of materials with precise nanoscale geometry such as plasmonic nanostructures or heterogeneous protein assemblies

Exponential Random Graph Modeling for Complex Brain Networks

Exponential random graph models (ERGMs), also known as p* models, have been utilized extensively in the social science literature to study complex networks and how their global structure depends on underlying structural components. However, the literature on their use in biological networks (especially brain networks) has remained sparse. Descriptive models based on a specific feature of the graph (clustering coefficient, degree distribution, etc.) have dominated connectivity research in neuroscience. Corresponding generative models have been developed to reproduce one of these features. However, the complexity inherent in whole-brain network data necessitates the development and use of tools that allow the systematic exploration of several features simultaneously and how they interact to form the global network architecture. ERGMs provide a statistically principled approach to the assessment of how a set of interacting local brain network features gives rise to the global structure. We illustrate the utility of ERGMs for modeling, analyzing, and simulating complex whole-brain networks with network data from normal subjects. We also provide a foundation for the selection of important local features through the implementation and assessment of three selection approaches: a traditional p-value based backward selection approach, an information criterion approach (AIC), and a graphical goodness of fit (GOF) approach. The graphical GOF approach serves as the best method given the scientific interest in being able to capture and reproduce the structure of fitted brain networks

Intra- and inter-hemispheric structural connectome in agenesis of the corpus callosum

Agenesis of the corpus callosum (AgCC) is a congenital brain malformation characterized by the complete or partial failure to develop the corpus callosum. Despite missing the largest white matter bundle connecting the left and right hemispheres of the brain, studies have shown preserved inter-hemispheric communication in individuals with AgCC. It is likely that plasticity provides mechanisms for the brain to adjust in the context of AgCC, as the malformation disrupts programmed developmental brain processes very early on. A proposed candidate for neuroplastic response in individuals with AgCC is strengthening of intra-hemispheric structural connections. In the present study, we explore this hypothesis using a graph-based approach of the structural connectome, which enables intra- and inter-hemispheric analyses at multiple resolutions and quantification of structural characteristics through graph metrics. Structural graph metrics of 19 children with AgCC (13 with complete, 6 with partial AgCC) were compared to those of 29 typically developing controls (TDC). Associations between structural graph metrics and a wide range of neurobehavioral outcomes were examined using a multivariate data-driven approach (Partial Least Squares Correlation, PLSC). Our results provide new evidence suggesting structural strengthening of intra-hemispheric pathways as a neuroplastic response in the acallosal brain, and highlight regional variability in structural connectivity in children with AgCC compared to TDC. There was little evidence that structural graph properties in children with AgCC were associated with neurobehavioral outcomes. To our knowledge, this is the first report leveraging graph theory tools to explicitly characterize whole-brain intra- and inter-hemispheric structural connectivity in AgCC, opening avenues for future research on neuroplastic responses in AgCC

Alterations in the microstructure of white matter in children and adolescents with Tourette syndrome measured using tract-based spatial statistics and probabilistic tractography

Tourette syndrome (TS) is a neurodevelopmental disorder characterised by repetitive and intermittent motor and vocal tics. TS is thought to reflect fronto-striatal dysfunction and the aetiology of the disorder has been linked to widespread alterations in the functional and structural integrity of the brain. The aim of this study was to assess white matter (WM) abnormalities in a large sample of young patients with TS in comparison to a sample of matched typically developing control individuals (CS) using diffusion MRI. The study included 35 patients with TS (3 females; mean age: 14.0 ± 3.3) and 35 CS (3 females; mean age: 13.9 ± 3.3). Diffusion MRI data was analysed using tract-based spatial statistics (TBSS) and probabilistic tractography. Patients with TS demonstrated both marked and widespread decreases in axial diffusivity (AD) together with altered WM connectivity. Moreover, we showed that tic severity and the frequency of premonitory urges (PU) were associated with increased connectivity between primary motor cortex (M1) and the caudate nuclei, and increased information transfer between M1 and the insula, respectively. This is to our knowledge the first study to employ both TBSS and probabilistic tractography in a sample of young patients with TS. Our results contribute to the limited existing literature demonstrating altered connectivity in TS and confirm previous results suggesting in particular, that altered insular function contributes to increased frequency of PU

Resting-State Functional Connectivity and Network Analysis of Cerebellum with Respect to Crystallized IQ and Gender

During the last years, it has been established that the prefrontal and posterior parietal brain lobes, which are mostly related to intelligence, have many connections to cerebellum. However, there is a limited research investigating cerebellum's relationship with cognitive processes. In this study, the network of cerebellum was analyzed in order to investigate its overall organization in individuals with low and high crystallized Intelligence Quotient (IQ). Functional magnetic resonance imaging (fMRI) data were selected from 136 subjects in resting-state from the Human Connectome Project (HCP) database and were further separated into two IQ groups composed of 69 low-IQ and 67 high-IQ subjects. Cerebellum was parcellated into 28 lobules/ROIs (per subject) using a standard cerebellum anatomical atlas. Thereafter, correlation matrices were constructed by computing Pearson's correlation coefficients between the average BOLD time-series for each pair of ROIs inside the cerebellum. By computing conventional graph metrics, small-world network properties were verified using the weighted clustering coefficient and the characteristic path length for estimating the trade-off between segregation and integration. In addition, a connectivity metric was computed for extracting the average cost per network. The concept of the Minimum Spanning Tree (MST) was adopted and implemented in order to avoid methodological biases in graph comparisons and retain only the strongest connections per network. Subsequently, six global and three local metrics were calculated in order to retrieve useful features concerning the characteristics of each MST. Moreover, the local metrics of degree and betweenness centrality were used to detect hubs, i.e., nodes with high importance. The computed set of metrics gave rise to extensive statistical analysis in order to examine differences between low and high-IQ groups, as well as between all possible gender-based group combinations. Our results reveal that both male and female networks have small-world properties with differences in females (especially in higher IQ females) indicative of higher neural efficiency in cerebellum. There is a trend toward the same direction in men, but without significant differences. Finally, three lobules showed maximum correlation with the median response time in low-IQ individuals, implying that there is an increased effort dedicated locally by this population in cognitive tasks

Aging brain from a network science perspective: Something to be positive about?

To better understand age differences in brain function and behavior, the current study applied network science to model functional interactions between brain regions. We observed a shift in network topology whereby for older adults subcortical and cerebellar structures overlapping with the Salience network had more connectivity to the rest of the brain, coupled with fragmentation of large-scale cortical networks such as the Default and Fronto-Parietal networks. Additionally, greater integration of the dorsal medial thalamus and red nucleus in the Salience network was associated with greater satisfaction with life for older adults, which is consistent with theoretical predictions of age-related increases in emotion regulation that are thought to help maintain well-being and life satisfaction in late adulthood. In regard to cognitive abilities, greater ventral medial prefrontal cortex coherence with its topological neighbors in the Default Network was associated with faster processing speed. Results suggest that large-scale organizing properties of the brain differ with normal aging, and this perspective may offer novel insight into understanding age-related differences in cognitive function and well-being. © 2013 Voss et al