How to Direct the Edges of the Connectomes: Dynamics of the Consensus Connectomes and the Development of the Connections in the Human Brain

The length and the width of the human brain circuit connections are strongly correlated

The correlations of several fundamental properties of human brain connections are investigated in a consensus connectome, constructed from 1064 braingraphs, each on 1015 vertices, corresponding to 1015 anatomical brain areas. The properties examined include the edge length, the fiber count, or edge width, meaning the number of discovered axon bundles forming the edge and the occurrence number of the edge, meaning the number of individual braingraphs where the edge exists. By using our previously published robust braingraphs at https://braingraph.org, we have prepared a single consensus graph from the data and compared the statistical similarity of the edge occurrence numbers, edge lengths, and fiber counts of the edges. We have found a strong positive Spearman correlation between the edge occurrence numbers and the fiber count numbers, showing that statistically, the most frequent cerebral connections have the largest widths, i.e., the fiber count. We have found a negative Spearman correlation between the fiber lengths and fiber counts, showing that, typically, the shortest edges are the widest or strongest by their fiber counts. We have also found a negative Spearman correlation between the occurrence numbers and the edge lengths: it shows that typically, the long edges are infrequent, and the frequent edges are short.

New Graphs at the braingraph.org Website for Studying the Aging Brain Circuitry

Human braingraphs or connectomes are widely studied in the last decade to understand the structural and functional properties of our brain. In the last several years our research group has computed and deposited thousands of human braingraphs to the braingraph.org site, by applying public structural (diffusion) MRI data from young and healthy subjects. Here we describe a recent addition to the braingraph.org site, which contains connectomes from healthy and demented subjects between 42 and 95 years of age, based on the public release of the OASIS-3 dataset. The diffusion MRI data was processed with the Connectome Mapper Toolkit v.3.1. We believe that the new addition to the braingraph.org site will become a useful resource for enlightening the aging circuitry of the human brain in healthy and diseased subjects, including those with Alzheimer's disease in several stages.

Robust circuitry-based scores of structural importance of human brain areas

We consider the 1015-vertex human consensus connectome computed from the diffusion MRI data of 1064 subjects. We define seven different orders on these 1015 graph vertices, where the orders depend on parameters derived from the brain circuitry, that is, from the properties of the edges (or connections) incident to the vertices ordered. We order the vertices according to their degree, the sum, the maximum, and the average of the fiber counts on the incident edges, and the sum, the maximum and the average length of the fibers in the incident edges. We analyze the similarities of these seven orders by the Spearman correlation coefficient and by their inversion numbers and have found that all of these seven orders have great similarities. In other words, if we interpret the orders as scoring of the importance of the vertices in the consensus connectome, then the scores of the vertices will be similar in all seven orderings. That is, important vertices of the human connectome typically have many neighbors connected with long and thick axonal fibers (where thickness is measured by fiber numbers), and their incident edges have high maximum and average values of length and fiber-number parameters, too. Therefore, these parameters may yield robust ways of deciding which vertices are more important in the anatomy of our brain circuitry than the others.

Analysis of Connectome Graphs Based on Boundary Scale

The purpose of this work is to advance in the computational study of connectome graphs from a topological point of view. Specifically, starting from a sequence of hypergraphs associated to a brain graph (obtained using the Boundary Scale model, BS2), we analyze the resulting scale-space representation using classical topological features, such as Betti numbers and average node and edge degrees. In this way, the topological information that can be extracted from the original graph is substantially enriched, thus providing an insightful description of the graph from a clinical perspective. To assess the qualitative and quantitative topological information gain of the BS2 model, we carried out an empirical analysis of neuroimaging data using a dataset that contains the connectomes of 96 healthy subjects, 52 women and 44 men, generated from MRI scans in the Human Connectome Project. The results obtained shed light on the differences between these two classes of subjects in terms of neural connectivity.

Analyzing bursting synchronization in structural connectivity matrix of a human brain under external pulsed currents

Cognitive tasks in the human brain are performed by various cortical areas located in the cerebral cortex. The cerebral cortex is separated into different areas in the right and left hemispheres. We consider one human cerebral cortex according to a network composed of coupled subnetworks with small-world properties. We study the burst synchronization and desynchronization in a human neuronal network under external periodic and random pulsed currents. With and without external perturbations, the emergence of bursting synchronization is observed. Synchronization can contribute to the processing of information, however, there are evidences that it can be related to some neurological disorders. Our results show that synchronous behavior can be suppressed by means of external pulsed currents.

Nonlocal models in the analysis of brain neurodegenerative protein dynamics with application to Alzheimer's disease

It is well known that today nearly one in six of the world’s population has to deal with neurodegenerative disorders. While a number of medical devices have been developed for the detection, prevention, and treatments of such disorders, some fundamentals of the progression of associated diseases are in urgent need of further clarification. In this paper, we focus on Alzheimer’s disease, where it is believed that the concentration changes in amyloid-beta and tau proteins play a central role in its onset and development. A multiscale model is proposed to analyze the propagation of these concentrations in the brain connectome. In particular, we consider a modified heterodimer model for the protein–protein interactions. Higher toxic concentrations of amyloid-beta and tau proteins destroy the brain cell. We have studied these propagations for the primary and secondary and their mixed tauopathies. We model the damage of a brain cell by the nonlocal contributions of these toxic loads present in the brain cells. With the help of rigorous analysis, we check the stability behaviour of the stationary points corresponding to the homogeneous system. After integrating the brain connectome data into the developed model, we see that the spreading patterns of the toxic concentrations for the whole brain are the same, but their concentrations are different in different regions. Also, the time to propagate the damage in each region of the brain connectome is different.

Introducing and applying Newtonian blurring: an augmented dataset of 126,000 human connectomes at braingraph.org

Gaussian blurring is a well-established method for image data augmentation: it may generate a large set of images from a small set of pictures for training and testing purposes for Artificial Intelligence (AI) applications. When we apply AI for non-imagelike biological data, hardly any related method exists. Here we introduce the "Newtonian blurring" in human braingraph (or connectome) augmentation: Started from a dataset of 1053 subjects from the public release of the Human Connectome Project, we first repeat a probabilistic weighted braingraph construction algorithm 10 times for describing the connections of distinct cerebral areas, then for every possible set of 7 of these graphs, delete the lower and upper extremes, and average the remaining 7 - 2 = 5 edge-weights for the data of each subject. This way we augment the 1053 graph-set to 120 [Formula: see text] 1053 = 126,360 graphs. In augmentation techniques, it is an important requirement that no artificial additions should be introduced into the dataset. Gaussian blurring and also this Newtonian blurring satisfy this goal. The resulting dataset of 126,360 graphs, each in 5 resolutions (i.e., 631,800 graphs in total), is freely available at the site https://braingraph.org/cms/download-pit-group-connectomes/ . Augmenting with Newtonian blurring may also be applicable in other non-image-related fields, where probabilistic processing and data averaging are implemented.

Identifying super-feminine, super-masculine and sex-defining connections in the human braingraph

For more than a decade now, we can discover and study thousands of cerebral connections with the application of diffusion magnetic resonance imaging (dMRI) techniques and the accompanying algorithmic workflow. While numerous connectomical results were published enlightening the relation between the braingraph and certain biological, medical, and psychological properties, it is still a great challenge to identify a small number of brain connections closely related to those conditions. In the present contribution, by applying the 1200 Subjects Release of the Human Connectome Project (HCP) and Support Vector Machines, we identify just 102 connections out of the total number of 1950 connections in the 83-vertex graphs of 1064 subjects, which-by a simple linear test-precisely, without any error determine the sex of the subject. Next, we re-scaled the weights of the edges-corresponding to the discovered fibers-to be between 0 and 1, and, very surprisingly, we were able to identify two graph edges out of these 102, such that, if their weights are both 1, then the connectome always belongs to a female subject, independently of the other edges. Similarly, we have identified 3 edges from these 102, whose weights, if two of them are 1 and one is 0, imply that the graph belongs to a male subject-again, independently of the other edges. We call the former 2 edges superfeminine and the first two of the 3 edges supermasculine edges of the human connectome. Even more interestingly, the edge, connecting the right Pars Triangularis and the right Superior Parietal areas, is one of the 2 superfeminine edges, and it is also the third edge, accompanying the two supermasculine connections if its weight is 0; therefore, it is also a "switching" edge. Identifying such edge-sets of distinction is the unprecedented result of this work.

SUPPLEMENTARY INFORMATION

The online version contains supplementary material available at 10.1007/s11571-021-09687-w.

The braingraph.org database with more than 1000 robust human connectomes in five resolutions

The human brain is the most complex object of study we encounter today. Mapping the neuronal-level connections between the more than 80 billion neurons in the brain is a hopeless task for science. By the recent advancement of magnetic resonance imaging (MRI), we are able to map the macroscopic connections between about 1000 brain areas. The MRI data acquisition and the subsequent algorithmic workflow contain several complex steps, where errors can occur. In the present contribution we describe and publish 1064 human connectomes, computed from the public release of the Human Connectome Project. Each connectome is available in 5 resolutions, with 83, 129, 234, 463 and 1015 anatomically labeled nodes. For error correction we follow an averaging and extreme value deleting strategy for each edge and for each connectome. The resulting 5320 braingraphs can be downloaded from the https://braingraph.org site. This dataset makes possible the access to this graphs for scientists unfamiliar with neuroimaging- and connectome-related tools: mathematicians, physicists and engineers can use their expertize and ideas in the analysis of the connections of the human brain. Brain scientists and computational neuroscientists also have a robust and large, multi-resolution set for connectomical studies.

The Graph of Our Mind

Graph theory in the last two decades penetrated sociology, molecular biology, genetics, chemistry, computer engineering, and numerous other fields of science. One of the more recent areas of its applications is the study of the connections of the human brain. By the development of diffusion magnetic resonance imaging (diffusion MRI), it is possible today to map the connections between the 1-1.5 cm2 regions of the gray matter of the human brain. These connections can be viewed as a graph. We have computed 1015-vertex graphs with thousands of edges for hundreds of human brains from one of the highest quality data sources: the Human Connectome Project. Here we analyze the male and female braingraphs graph-theoretically and show statistically significant differences in numerous parameters between the sexes: the female braingraphs are better expanders, have more edges, larger bipartition widths, and larger vertex cover than the braingraphs of the male subjects. These parameters are closely related to the quality measures of highly parallel computer interconnection networks: the better expanding property, the large bipartition width, and the large vertex cover characterize high-quality interconnection networks. We apply the data of 426 subjects and demonstrate the statistically significant (corrected) differences in 116 graph parameters between the sexes.

The frequent complete subgraphs in the human connectome

While it is still not possible to describe the neuronal-level connections of the human brain, we can map the human connectome with several hundred vertices, by the application of diffusion-MRI based techniques. In these graphs, the nodes correspond to anatomically identified gray matter areas of the brain, while the edges correspond to the axonal fibers, connecting these areas. In our previous contributions, we have described numerous graph-theoretical phenomena of the human connectomes. Here we map the frequent complete subgraphs of the human brain networks: in these subgraphs, every pair of vertices is connected by an edge. We also examine sex differences in the results. The mapping of the frequent subgraphs gives robust substructures in the graph: if a subgraph is present in the 80% of the graphs, then, most probably, it could not be an artifact of the measurement or the data processing workflow. We list here the frequent complete subgraphs of the human braingraphs of 413 subjects (238 women and 175 men), each with 463 nodes, with a frequency threshold of 80%, and identify 812 complete subgraphs, which are more frequent in male and 224 complete subgraphs, which are more frequent in female connectomes.

Good neighbors, bad neighbors: the frequent network neighborhood mapping of the hippocampus enlightens several structural factors of the human intelligence on a 414-subject cohort

The human connectome has become the very frequent subject of study of brain-scientists, psychologists and imaging experts in the last decade. With diffusion magnetic resonance imaging techniques, united with advanced data processing algorithms, today we are able to compute braingraphs with several hundred, anatomically identified nodes and thousands of edges, corresponding to the anatomical connections of the brain. The analysis of these graphs without refined mathematical tools is hopeless. These tools need to address the high error rate of the MRI processing workflow, and need to find structural causes or at least correlations of psychological properties and cerebral connections. Until now, structural connectomics was only rarely able of identifying such causes or correlations. In the present work we study the frequent neighbor sets of the most deeply investigated brain area, the hippocampus. By applying the Frequent Network Neighborhood mapping method, we identified frequent neighbor-sets of the hippocampus, which may influence numerous psychological parameters, including intelligence-related ones. We have found "Good Neighbor" sets, which correlate with better test results and also "Bad Neighbor" sets, which correlate with worse test results. Our study utilizes the braingraphs, computed from the imaging data of the Human Connectome Project's 414 subjects, each with 463 anatomically identified nodes.

Scaling in Colloidal and Biological Networks

Scaling and dimensional analysis is applied to networks that describe various physical systems. Some of these networks possess fractal, scale-free, and small-world properties. The amount of information contained in a network is found by calculating its Shannon entropy. First, we consider networks arising from granular and colloidal systems (small colloidal and droplet clusters) due to pairwise interaction between the particles. Many networks found in colloidal science possess self-organizing properties due to the effect of percolation and/or self-organized criticality. Then, we discuss the allometric laws in branching vascular networks, artificial neural networks, cortical neural networks, as well as immune networks, which serve as a source of inspiration for both surface engineering and information technology. Scaling relationships in complex networks of neurons, which are organized in the neocortex in a hierarchical manner, suggest that the characteristic time constant is independent of brain size when interspecies comparison is conducted. The information content, scaling, dimensional, and topological properties of these networks are discussed.

The Frequent Network Neighborhood Mapping of the human hippocampus shows much more frequent neighbor sets in males than in females

In the study of the human connectome, the vertices and the edges of the network of the human brain are analyzed: the vertices of the graphs are the anatomically identified gray matter areas of the subjects; this set is exactly the same for all the subjects. The edges of the graphs correspond to the axonal fibers, connecting these areas. In the biological applications of graph theory, it happens very rarely that scientists examine numerous large graphs on the very same, labeled vertex set. Exactly this is the case in the study of the connectomes. Because of the particularity of these sets of graphs, novel, robust methods need to be developed for their analysis. Here we introduce the new method of the Frequent Network Neighborhood Mapping for the connectome, which serves as a robust identification of the neighborhoods of given vertices of special interest in the graph. We apply the novel method for mapping the neighborhoods of the human hippocampus and discover strong statistical asymmetries between the connectomes of the sexes, computed from the Human Connectome Project. We analyze 413 braingraphs, each with 463 nodes. We show that the hippocampi of men have much more significantly frequent neighbor sets than women; therefore, in a sense, the connections of the hippocampi are more regularly distributed in men and more varied in women. Our results are in contrast to the volumetric studies of the human hippocampus, where it was shown that the relative volume of the hippocampus is the same in men and women.

The frequent subgraphs of the connectome of the human brain

In mapping the human structural connectome, we are in a very fortunate situation: one can compute and compare graphs, describing the cerebral connections between the very same, anatomically identified small regions of the gray matter among hundreds of human subjects. The comparison of these graphs has led to numerous recent results, as the (1) discovery that women's connectomes have deeper and richer connectivity-related graph parameters like those of men, or (2) the description of more and less conservatively connected lobes and cerebral regions, and (3) the discovery of the phenomenon of the consensus connectome dynamics. Today one of the greatest challenges of brain science is the description and modeling of the circuitry of the human brain. For this goal, we need to identify sub-circuits that are present in almost all human subjects and those, which are much less frequent: the former sub-circuits most probably have functions with general importance, the latter sub-circuits are probably related to the individual variability of the brain structure and function. The present contribution describes the frequent connected subgraphs of at most six edges in the human brain. We analyze these frequent graphs and also examine sex differences in these graphs: we demonstrate numerous connected subgraphs that are more frequent in female or male connectomes. While there is no difference in the number of k edge connected subgraphs in males or females for k = 1 , and for k = 2 males have slightly more frequent subgraphs, for k = 6 there is a very strong advantage in the case of female braingraphs. Our data source is the public release of the Human Connectome Project, and we are applying the data of 426 human subjects in this study.

High-resolution directed human connectomes and the Consensus Connectome Dynamics

Here we show a method of directing the edges of the connectomes, prepared from HARDI datasets from the human brain. Before the present work, no high-definition directed braingraphs were published, because the tractography methods in use are not capable of assigning directions to the neural tracts discovered. Previous work on the functional connectomes applied low-resolution functional MRI-detected statistical causality for the assignment of directions of connectomes of typically several dozens of vertices. Our method is based on the phenomenon of the “Consensus Connectome Dynamics”, described earlier by our research group. In this contribution, we apply the method to the 423 braingraphs, each with 1015 vertices, computed from the public release of the Human Connectome Project, and we also made the directed connectomes publicly available at the site http://braingraph.org. We also show the robustness of our edge directing method in four independently chosen connectome datasets: we have found that 86% of the edges, which were present in all four datasets, get the same directions in all datasets; therefore the direction method is robust. While our new edge-directing method still needs more empirical validation, we think that our present contribution opens up new possibilities in the analysis of the high-definition human connectome.

The dorsal striatum and the dynamics of the consensus connectomes in the frontal lobe of the human brain

In the applications of the graph theory, it is unusual that one considers numerous, pairwise different graphs on the very same set of vertices. In the case of human braingraphs or connectomes, however, this is the standard situation: the nodes correspond to anatomically identified cerebral regions, and two vertices are connected by an edge if a diffusion MRI-based workflow identifies a fiber of axons, running between the two regions, corresponding to the two vertices. Therefore, if we examine the braingraphs of n subjects, then we have n graphs on the very same, anatomically identified vertex set. It is a natural idea to describe the k-frequently appearing edges in these graphs: the edges that are present between the same two vertices in at least k out of the n graphs. Based on the NIH-funded large Human Connectome Project's public data release, we have reported the construction of the Budapest Reference Connectome Server http://www.connectome.pitgroup.org that generates and visualizes these k-frequently appearing edges. We call the graphs of the k-frequently appearing edges "k-consensus connectomes" since an edge could be included only if it is present in at least k graphs out of n. Considering the whole human brain, we have reported a surprising property of these consensus connectomes earlier. In the present work we are focusing on the frontal lobe of the brain, and we report here a similarly surprising dynamical property of the consensus connectomes when k is gradually changed from k = n to k = 1: the connections between the nodes of the frontal lobe are seemingly emanating from those nodes that were connected to sub-cortical structures of the dorsal striatum: the caudate nucleus, and the putamen. We hypothesize that this dynamic behavior copies the axonal fiber development of the frontal lobe. An animation of the phenomenon is presented at https://youtu.be/wBciB2eW6_8.

The Robustness and the Doubly-Preferential Attachment Simulation of the Consensus Connectome Dynamics of the Human Brain

Consensus Connectome Dynamics (CCD) is a remarkable phenomenon of the human connectomes (braingraphs) that was discovered by continuously decreasing the minimum confidence-parameter at the graphical interface of the Budapest Reference Connectome Server, which depicts the cerebral connections of n = 418 subjects with a frequency-parameter k: For any k = 1, 2, …, n one can view the graph of the edges that are present in at least k connectomes. If parameter k is decreased one-by-one from k = n through k = 1 then more and more edges appear in the graph, since the inclusion condition is relaxed. The surprising observation is that the appearance of the edges is far from random: it resembles a growing, complex structure. We hypothesize that this growing structure copies the axonal development of the human brain. Here we show the robustness of the CCD phenomenon: it is almost independent of the particular choice of the set of underlying connectomes. This result shows that the CCD phenomenon is most likely a biological property of the human brain and not just a property of the data sets examined. We also present a simulation that well-describes the growth of the CCD structure: in our random graph model a doubly-preferential attachment distribution is found to mimic the CCD.

Comparative connectomics: Mapping the inter-individual variability of connections within the regions of the human brain

The human braingraph, or connectome is a description of the connections of the brain: the nodes of the graph correspond to small areas of the gray matter, and two nodes are connected by an edge if a diffusion MRI-based workflow finds fibers between those brain areas. We have constructed 1015-vertex graphs from the diffusion MRI brain images of 392 human subjects and compared the individual graphs with respect to several different areas of the brain. The inter-individual variability of the graphs within different brain regions was discovered and described. We have found that the frontal and the limbic lobes are more conservative, while the edges in the temporal and occipital lobes are more diverse. Interestingly, a "hybrid" conservative and diverse distribution was found in the paracentral lobule and the fusiform gyrus. Smaller cortical areas were also evaluated: precentral gyri were found to be more conservative, and the postcentral and the superior temporal gyri to be very diverse. Similar studies concerning the human genome discovered more and less conservative sections of the DNA, opening up entirely new fields in genomics. We think that the present study is the first step in this direction in human connectomics. The clinical significance of the conservativity of a given cerebral area could be the higher sensitivity for traumas and developmental or neuro-degenerative events than the less conservative areas.

The braingraph.org database of high resolution structural connectomes and the brain graph tools

Based on the data of the NIH-funded Human Connectome Project, we have computed structural connectomes of 426 human subjects in five different resolutions of 83, 129, 234, 463 and 1015 nodes and several edge weights. The graphs are given in anatomically annotated GraphML format that facilitates better further processing and visualization. For 96 subjects, the anatomically classified sub-graphs can also be accessed, formed from the vertices corresponding to distinct lobes or even smaller regions of interests of the brain. For example, one can easily download and study the connectomes, restricted to the frontal lobes or just to the left precuneus of 96 subjects using the data. Partially directed connectomes of 423 subjects are also available for download. We also present a GitHub-deposited set of tools, called the Brain Graph Tools, for several processing tasks of the connectomes on the site http://braingraph.org.

Parameterizable consensus connectomes from the Human Connectome Project: the Budapest Reference Connectome Server v3.0

Connections of the living human brain, on a macroscopic scale, can be mapped by a diffusion MR imaging based workflow. Since the same anatomic regions can be corresponded between distinct brains, one can compare the presence or the absence of the edges, connecting the very same two anatomic regions, among multiple cortices. Previously, we have constructed the consensus braingraphs on 1015 vertices first in five, then in 96 subjects in the Budapest Reference Connectome Server v1.0 and v2.0, respectively. Here we report the construction of the version 3.0 of the server, generating the common edges of the connectomes of variously parameterizable subsets of the 1015-vertex connectomes of 477 subjects of the Human Connectome Project's 500-subject release. The consensus connectomes are downloadable in CSV and GraphML formats, and they are also visualized on the server's page. The consensus connectomes of the server can be considered as the "average, healthy" human connectome since all of their connections are present in at least k subjects, where the default value of [Formula: see text], but it can also be modified freely at the web server. The webserver is available at http://connectome.pitgroup.org.