This paper proposes a novel topological learning framework that integrates networks of different sizes and topology through persistent homology. Such challenging task is made possible through the introduction of a computationally efficient topological loss. The use of the proposed loss bypasses the intrinsic computational bottleneck associated with matching networks. We validate the method in extensive statistical simulations to assess its effectiveness when discriminating networks with different topology. The method is further demonstrated in a twin brain imaging study where we determine if brain networks are genetically heritable. The challenge here is due to the difficulty of overlaying the topologically different functional brain networks obtained from resting-state functional MRI onto the template structural brain network obtained through diffusion MRI.
Almost all statistical and machine learning methods in analyzing brain networks rely on distances and loss functions, which are mostly Euclidean or matrix norms. The Euclidean or matrix distances may fail to capture underlying subtle topological differences in brain networks. Further, Euclidean distances are sensitive to outliers. A few extreme edge weights may severely affect the distance. Thus it is necessary to use distances and loss functions that recognize topology of data. In this review paper, we survey various topological distance and loss functions from topological data analysis (TDA)…
The velocity of the action potential transduction along myelinated axons in the peripheral nervous system or in the white matter of brain and spinal cord reaches hundreds of meters per second to assure proper functioning of the body, which exceeds the ability of diffusive ion conduction. We propose the new model of the saltatory conduction based on a plasmon-polariton kinetics in myelinated axons, which excludes, however, the white matter form the information storage and its identification in the brain via e-m response. We propose a nonlocal topological approach to information processing in th…
Characterizing the transcriptome architecture of the human brain is fundamental in gaining an understanding of brain function and disease. A number of recent studies have investigated patterns of brain gene expression obtained from an extensive anatomical coverage across the entire human brain using experimental data generated by the Allen Human Brain Atlas (AHBA) project. In this paper, we propose a new representation of a gene's transcription activity that explicitly captures the pattern of spatial co-expression across different anatomical brain regions. For each gene, we define a Spatial Ex…
The organization in brain networks shows highly modular features with weak inter-modular interaction. The topology of the networks involves emergence of modules and sub-modules at different levels of constitution governed by fractal laws. The modular organization, in terms of modular mass, inter-modular, and intra-modular interaction, also obeys fractal nature. The parameters which characterize topological properties of brain networks follow one parameter scaling theory in all levels of network structure which reveals the self-similar rules governing the network structure. The calculated fract…
Existing brain network distances are often based on matrix norms. The element-wise differences in the existing matrix norms may fail to capture underlying topological differences. Further, matrix norms are sensitive to outliers. A major disadvantage to element-wise distance calculations is that it could be severely affected even by a small number of extreme edge weights. Thus it is necessary to develop network distances that recognize topology. In this paper, we provide a survey of bottleneck, Gromov-Hausdorff (GH) and Kolmogorov-Smirnov (KS) distances that are adapted for brain networks, and …
Community detection algorithms have been widely used to study the organization of complex systems like the brain. A principal appeal of these techniques is their ability to identify a partition of brain regions (or nodes) into communities, where nodes within a community are densely interconnected. In their simplest application, community detection algorithms are agnostic to the presence of community hierarchies, but a common characteristic of many neural systems is a nested hierarchy. To address this limitation, we exercise a multi-scale extension of a community detection technique known as mo…
Understanding brain connectivity has become one of the most important issues in neuroscience. But connectivity data can reflect either the functional relationships of the brain activities or the anatomical properties between brain areas. Although one should expect a clear relationship between both representations it is not straightforward. Here we present a formalism that allows for the comparison of structural (DTI) and functional (fMRI) networks by embedding both in a common metric space. In this metric space one can then find for which regions the two networks are significantly different. O…
In recent years, network analyses have been used to evaluate brain reorganization following stroke. However, many studies have often focused on single topological scales, leading to an incomplete model of how focal brain lesions affect multiple network properties simultaneously and how changes on smaller scales influence those on larger scales. In an EEG-based experiment on the performance of hand motor imagery (MI) in 20 patients with unilateral stroke, we observed that the anatomic lesion affects the functional brain network on multiple levels. In the beta (13-30 Hz) frequency band, the MI o…
Persistent Homological State-Space Estimation of Functional Human Brain Networks at Rest
University of Wisconsin, Madison（威斯康星大学麦迪逊分校）；Taiwan（中国台湾）；New York University Grossman School of Medicine（纽约大学格罗斯曼医学院）；Georgia State University（佐治亚州立大学）；Waisman Center, University of Wisconsin-Madison（威斯康星大学麦迪逊分校怀斯曼中心）
Group-based brain connectivity networks have great appeal for researchers interested in gaining further insight into complex brain function and how it changes across different mental states and disease conditions. Accurately constructing these networks presents a daunting challenge given the difficulties associated with accounting for inter-subject topological variability. Viable approaches to this task must engender networks that capture the constitutive topological properties of the group of subjects' networks that it is aiming to represent. The conventional approach has been to use a mean o…
High-throughput methods for yielding the set of connections in a neural system, the connectome, are now being developed. This tutorial describes ways to analyze the topological and spatial organization of the connectome at the macroscopic level of connectivity between brain regions as well as the microscopic level of connectivity between neurons. We will describe topological features at three different levels: the local scale of individual nodes, the regional scale of sets of nodes, and the global scale of the complete set of nodes in a network. Such features can be used to characterize compon…
In many biological systems, the network of interactions between the elements can only be inferred from experimental measurements. In neuroscience, non-invasive imaging tools are extensively used to derive either structural or functional brain networks in-vivo. As a result of the inference process, we obtain a matrix of values corresponding to an unrealistic fully connected and weighted network. To turn this into a useful sparse network, thresholding is typically adopted to cancel a percentage of the weakest connections. The structural properties of the resulting network depend on how much of t…
The connectional brain template (CBT) captures the shared traits across all individuals of a given population of brain connectomes, thereby acting as a fingerprint. Estimating a CBT from a population where brain graphs are derived from diverse neuroimaging modalities (e.g., functional and structural) and at different resolutions (i.e., number of nodes) remains a formidable challenge to solve. Such network integration task allows for learning a rich and universal representation of the brain connectivity across varying modalities and resolutions. The resulting CBT can be substantially used to ge…
Graph theoretical approach has proved an effective tool to understand, characterize and quantify the complex brain network. However, much less attention has been paid to methods that quantitatively compare two graphs, a crucial issue in the context of brain networks. Comparing brain networks is indeed mandatory in several network neuroscience applications. Here, we discuss the current state of the art, challenges, and a collection of analysis tools that have been developed in recent years to compare brain networks. We first introduce the graph similarity problem in brain network application. W…
Topological Cycle Graph Attention Network for Brain Functional Connectivity
National University of Singapore（新加坡国立大学）；The Hong Kong Polytechnic University（香港理工大学）；NUS (Suzhou) Research Institute, National University of Singapore（新加坡国立大学（苏州）研究院）；The N.1 Institute for Health, National University of Singapore（新加坡国立大学N.1健康研究院）；The Johns Hopkins University（约翰霍普金斯大学）
Spectral Topological Data Analysis of Brain Signals
King Abdullah University of Science Technology（阿卜杜拉国王科技大学）；Department of Biostatistics, City University of Hong Kong（香港城市大学生物统计系）；Department of Biostatistics & Medical Informatics, University of Wisconsin–Madison（威斯康星大学麦迪逊分校生物统计与医学信息学系）；Statistics Program, King Abdullah University of Science and Technology（阿卜杜拉国王科技大学统计学项目）
What is the relationship between brain and behavior? The answer to this question necessitates characterizing the mapping between structure and function. The aim of this paper is to discuss broad issues surrounding the link between structure and function in the brain that will motivate a network perspective to understanding this question. As others in the past, I argue that a network perspective should supplant the common strategy of understanding the brain in terms of individual regions. Whereas this perspective is needed for a fuller characterization of the mind-brain, it should not be viewed…
We explore the main characteristics of big brain network data that offer unique statistical challenges. The brain networks are biologically expected to be both sparse and hierarchical. Such unique characterizations put specific topological constraints onto statistical approaches and models we can use effectively. We explore the limitations of the current models used in the field and offer alternative approaches and explain new challenges.
The brain can be regarded as a network: a connected system where nodes, or units, represent different specialized regions and links, or connections, represent communication pathways. From a functional perspective communication is coded by temporal dependence between the activities of different brain areas. In the last decade, the abstract representation of the brain as a graph has allowed to visualize functional brain networks and describe their non-trivial topological properties in a compact and objective way. Nowadays, the use of graph analysis in translational neuroscience has become essent…
Exploring the human brain networks during rest is a topic of great interest. Several structural and functional studies have previously been conducted to study the intrinsic brain networks. In this paper, we focus on investigating the human brain network topology using dense Electroencephalography (EEG) source connectivity approach. We applied graph theoretical methods on functional networks reconstructed from resting state data acquired using EEG in 14 healthy subjects. Our findings confirmed the existence of sets of brain regions considered as functional hubs. In particular, the isthmus cingu…
Topological data analyses are rapidly turning into key tools for quantifying large volumes of neurobiological data, e.g., for organizing the spiking outputs of large neuronal ensembles and thus gaining insights into the information produced by various networks. Below we discuss a case in which several convergent topological analyses not only provide a description of the data structure, but also produce insights into how these data may be processed in the hippocampus---a brain part that plays a key role in learning and memory. The resulting functional model provides a unifying framework for int…
Higher brain function relies upon the ability to flexibly integrate information across specialized communities of brain regions, however it is unclear how this mechanism manifests over time. In this study, we use time-resolved network analysis of functional magnetic resonance imaging data to demonstrate that the human brain traverses between two functional states that maximize either segregation into tight-knit communities or integration across otherwise disparate neural regions. The integrated state enables faster and more accurate performance on a cognitive task, and is associated with dilat…
We investigated the degree distribution of brain networks extracted from functional magnetic resonance imaging of the human brain. In particular, the distributions are compared between macroscopic brain networks using region-based nodes and mesoscopic brain networks using voxel-based nodes. We found that the distribution from these networks follow the same family of distributions and represent a continuum of exponentially truncated power law distributions.
Network models describe the brain as sets of nodes and edges that represent its distributed organization. So far, most discoveries in network neuroscience have prioritized insights that highlight distinct groupings and specialized functional contributions of network nodes. Importantly, these functional contributions are determined and expressed by the web of their interrelationships, formed by network edges. Here, we underscore the important contributions made by brain network edges for understanding distributed brain organization. Different types of edges represent different types of relation…
The closed loops or cycles in a brain network embeds higher order signal transmission paths, which provide fundamental insights into the functioning of the brain. In this work, we propose an efficient algorithm for systematic identification and modeling of cycles using persistent homology and the Hodge Laplacian. Various statistical inference procedures on cycles are developed. We validate the our methods on simulations and apply to brain networks obtained through the resting state functional magnetic resonance imaging. The computer codes for the Hodge Laplacian are given in https://github.com…
It is widely believed that the formation of brain network structure is under the pressure of optimal trade-off between reducing wiring cost and promoting communication efficiency. However, the question of whether this trade-off exists in empirical human brain networks and, if so, how it takes effect is still not well understood. Here, we employed a multiobjective evolutionary algorithm to directly and quantitatively explore the cost-efficiency trade-off in human brain networks. Using this algorithm, we generated a population of synthetic networks with optimal but diverse cost-efficiency trade-…
Many recent efforts in computational modeling of macro-scale brain dynamics have begun to take a data-driven approach by incorporating structural and/or functional information derived from subject data. Here, we discuss recent work using personalized brain network models to study structure-function relationships in human brains. We describe the steps necessary to build such models and show how this computational approach can provide previously unobtainable information through the ability to perform virtual experiments. Finally, we present examples of how personalized brain network models can b…
In Indian history of arts, Rasas are the aesthetics associated with any auditory, visual, literary or musical piece of art that evokes highly orchestrated emotional states. In this work, we study the functional response of the brain to movie clippings meant to evoke the Rasas through network analysis. We extract functional brain networks using coherence measures on EEG recordings of film clips from popular Indian Bollywood movies representing nine Rasas in the Indian Natyasastra. Structural and functional network measures were computed for these brain networks, averaging over a range of signif…
In this research, a number of popular network measurement algorithms have been applied to several brain networks (based on applicability of algorithms) for finding out statistical correlation among these popular network measurements which will help scientists to understand these popular network measurement algorithms and their applicability to brain networks. By analysing the results of correlations among these network measurement algorithms, statistical comparison among selected brain networks has also been summarized. Besides that, to understand each brain network, the visualization of each …
Rich, spontaneous brain activity has been observed across a range of different temporal and spatial scales. These dynamics are thought to be important t for efficient neural functioning. Experimental evidence suggests that these neural dynamics are maintained across a variety of different cognitive states, in response to alterations of the environment and to changes in brain configuration (e.g., across individuals, development and in many neurological disorders). This suggests that the brain has evolved mechanisms to stabilize dynamics and maintain them across a range of situations. Here, we e…
Converging evidence suggests that the mammalian ventral visual pathway encodes increasingly complex stimulus features in downstream areas. Using deep convolutional neural networks, we can now quantitatively demonstrate that there is indeed an explicit gradient for feature complexity in the ventral pathway of the human brain. Our approach also allows stimulus features of increasing complexity to be mapped across the human brain, providing an automated approach to probing how representations are mapped across the cortical sheet. Finally, it is shown that deep convolutional neural networks allow …
Recent developments in graph theoretic analysis of complex networks have led to deeper understanding of brain networks. Many complex networks show similar macroscopic behaviors despite differences in the microscopic details. Probably two most often observed characteristics of complex networks are scale-free and small-world properties. In this paper, we will explore whether brain networks follow scale-free and small-worldness among other graph theory properties.
Higher-order connectivity in complex systems described by simplexes of different orders provides a geometry for simplex-based dynamical variables and interactions. Simplicial complexes that constitute a functional geometry of the human connectome can be crucial for the brain complex dynamics. In this context, the best-connected brain areas, designated as hub nodes, play a central role in supporting integrated brain function. Here, we study the structure of simplicial complexes attached to eight global hubs in the female and male connectomes and identify the core networks among the affected bra…
Emory University（埃默里大学）；University of California, Los Angeles（加州大学洛杉矶分校）；University of Pittsburgh（匹兹堡大学）；Lehigh University（理海大学）
Mapping the connectome of the human brain using structural or functional connectivity has become one of the most pervasive paradigms for neuroimaging analysis. Recently, Graph Neural Networks (GNNs) motivated from geometric deep learning have attracted broad interest due to their established power for modeling complex networked data. Despite their superior performance in many fields, there has not yet been a systematic study of how to design effective GNNs for brain network analysis. To bridge this gap, we present BrainGB, a benchmark for brain network analysis with GNNs. BrainGB standardizes …
