The topology of higher-order complexes associated with brain hubs in human connectomes

A topological deep learning framework for neural spike decoding

The brain's spatial orientation system uses different neuron ensembles to aid in environment-based navigation. Two of the ways brains encode spatial information are through head direction cells and grid cells. Brains use head direction cells to determine orientation, whereas grid cells consist of layers of decked neurons that overlay to provide environment-based navigation. These neurons fire in ensembles where several neurons fire at once to activate a single head direction or grid. We want to capture this firing structure and use it to decode head direction and animal location from head direction and grid cell activity. Understanding, representing, and decoding these neural structures require models that encompass higher-order connectivity, more than the one-dimensional connectivity that traditional graph-based models provide. To that end, in this work, we develop a topological deep learning framework for neural spike train decoding. Our framework combines unsupervised simplicial complex discovery with the power of deep learning via a new architecture we develop herein called a simplicial convolutional recurrent neural network. Simplicial complexes, topological spaces that use not only vertices and edges but also higher-dimensional objects, naturally generalize graphs and capture more than just pairwise relationships. Additionally, this approach does not require prior knowledge of the neural activity beyond spike counts, which removes the need for similarity measurements. The effectiveness and versatility of the simplicial convolutional neural network is demonstrated on head direction and trajectory prediction via head direction and grid cell datasets.

Simplicial complex entropy for time series analysis

The complex behavior of many systems in nature requires the application of robust methodologies capable of identifying changes in their dynamics. In the case of time series (which are sensed values of a system during a time interval), several methods have been proposed to evaluate their irregularity. However, for some types of dynamics such as stochastic and chaotic, new approaches are required that can provide a better characterization of them. In this paper we present the simplicial complex approximate entropy, which is based on the conditional probability of the occurrence of elements of a simplicial complex. Our results show that this entropy measure provides a wide range of values with details not easily identifiable with standard methods. In particular, we show that our method is able to quantify the irregularity in simulated random sequences and those from low-dimensional chaotic dynamics. Furthermore, it is possible to consistently differentiate cardiac interbeat sequences from healthy subjects and from patients with heart failure, as well as to identify changes between dynamical states of coupled chaotic maps. Our results highlight the importance of the structures revealed by the simplicial complexes, which holds promise for applications of this approach in various contexts.

Rehabilitation Modulates High-Order Interactions Among Large-Scale Brain Networks in Subacute Stroke

The recovery of motor functions after stroke is fostered by the functional integration of large-scale brain networks, including the motor network (MN) and high-order cognitive controls networks, such as the default mode (DMN) and executive control (ECN) networks. In this paper, electroencephalography signals are used to investigate interactions among these three resting state networks (RSNs) in subacute stroke patients after motor rehabilitation. A novel metric, the O-information rate (OIR), is used to quantify the balance between redundancy and synergy in the complex high-order interactions among RSNs, as well as its causal decomposition to identify the direction of information flow. The paper also employs conditional spectral Granger causality to assess pairwise directed functional connectivity between RSNs. After rehabilitation, a synergy increase among these RSNs is found, especially driven by MN. From the pairwise description, a reduced directed functional connectivity towards MN is enhanced after treatment. Besides, inter-network connectivity changes are associated with motor recovery, for which the mediation role of ECN seems to play a relevant role, both from pairwise and high-order interactions perspective.

Topology switching during window thresholding fMRI-based functional networks of patients with major depressive disorder: Consensus network approach

We present a novel method for analyzing brain functional networks using functional magnetic resonance imaging data, which involves utilizing consensus networks. In this study, we compare our approach to a standard group-based method for patients diagnosed with major depressive disorder (MDD) and a healthy control group, taking into account different levels of connectivity. Our findings demonstrate that the consensus network approach uncovers distinct characteristics in network measures and degree distributions when considering connection strengths. In the healthy control group, as connection strengths increase, we observe a transition in the network topology from a combination of scale-free and random topologies to a small-world topology. Conversely, the MDD group exhibits uncertainty in weak connections, while strong connections display small-world properties. In contrast, the group-based approach does not exhibit significant differences in behavior between the two groups. However, it does indicate a transition in topology from a scale-free-like structure to a combination of small-world and scale-free topologies. The use of the consensus network approach also holds immense potential for the classification of MDD patients, as it unveils substantial distinctions between the two groups.

Effect of hidden geometry and higher-order interactions on the synchronization and hysteresis behavior of phase oscillators on 5-clique simplicial assemblies

The hidden geometry of simplicial complexes can influence the collective dynamics of nodes in different ways depending on the simplex-based interactions of various orders and competition between local and global structural features. We study a system of phase oscillators attached to nodes of four-dimensional simplicial complexes and interacting via positive/negative edges-based pairwise K_{1} and triangle-based triple K_{2}≥0 couplings. Three prototypal simplicial complexes are grown by aggregation of 5-cliques, controlled by the chemical affinity parameter ν, resulting in sparse, mixed, and compact architecture, all of which have 1-hyperbolic graphs but different spectral dimensions. By changing the interaction strength K_{1}∈[-4,2] along the forward and backward sweeps, we numerically determine individual phases of each oscillator and a global order parameter to measure the level of synchronization. Our results reveal how different architectures of simplicial complexes, in conjunction with the interactions and internal-frequency distributions, impact the shape of the hysteresis loop and lead to patterns of locally synchronized groups that hinder global network synchronization. Remarkably, these groups are differently affected by the size of the shared faces between neighboring 5-cliques and the presence of higher-order interactions. At K_{1}<0, partial synchronization is much higher in the compact community than in the assemblies of cliques sharing single nodes, at least occasionally. These structures also partially desynchronize at a lower triangle-based coupling K_{2} than the compact assembly. Broadening of the internal frequency distribution gradually reduces the synchronization level in the mixed and sparse communities, even at positive pairwise couplings. The order-parameter fluctuations in these partially synchronized states are quasicyclical with higher harmonics, described by multifractal analysis and broad singularity spectra.

Topological data analysis for revealing dynamic brain reconfiguration in MEG data

In recent years, the focus of the functional connectivity community has shifted from stationary approaches to the ones that include temporal dynamics. Especially, non-invasive electrophysiological data (magnetoencephalography/electroencephalography (MEG/EEG)) with high temporal resolution and good spatial coverage have made it possible to measure the fast alterations in the neural activity in the brain during ongoing cognition. In this article, we analyze dynamic brain reconfiguration using MEG images collected from subjects during the rest and the cognitive tasks. Our proposed topological data analysis method, called Mapper, produces biomarkers that differentiate cognitive tasks without prior spatial and temporal collapse of the data. The suggested method provides an interactive visualization of the rapid fluctuations in electrophysiological data during motor and cognitive tasks; hence, it has the potential to extract clinically relevant information at an individual level without temporal and spatial collapse.

The expanding horizons of network neuroscience: From description to prediction and control

The field of network neuroscience has emerged as a natural framework for the study of the brain and has been increasingly applied across divergent problems in neuroscience. From a disciplinary perspective, network neuroscience originally emerged as a formal integration of graph theory (from mathematics) and neuroscience (from biology). This early integration afforded marked utility in describing the interconnected nature of neural units, both structurally and functionally, and underscored the relevance of that interconnection for cognition and behavior. But since its inception, the field has not remained static in its methodological composition. Instead, it has grown to use increasingly advanced graph-theoretic tools and to bring in several other disciplinary perspectives-including machine learning and systems engineering-that have proven complementary. In doing so, the problem space amenable to the discipline has expanded markedly. In this review, we discuss three distinct flavors of investigation in state-of-the-art network neuroscience: (i) descriptive network neuroscience, (ii) predictive network neuroscience, and (iii) a perturbative network neuroscience that draws on recent advances in network control theory. In considering each area, we provide a brief summary of the approaches, discuss the nature of the insights obtained, and highlight future directions.

Automatic segmentation of Deep Grey Nuclei using a high-resolution 7T MRI Atlas - quantification of T1 values in healthy volunteers

We present a new consensus atlas of deep grey nuclei obtained by shape-based averaging of manual segmentation of two experienced neuroradiologists and optimized from 7T MP2RAGE images acquired at (0.6mm)³ in 60 healthy subjects. A group-wise normalization method was used to build a high-contrast and high-resolution T₁ -weighted brain template (0.5mm)³ using data from 30 out of the 60 controls. Delineation of 24 deep grey nuclei per hemisphere, including the claustrum and twelve thalamic nuclei, was then performed by two expert neuroradiologists and reviewed by a third neuroradiologist according to tissue contrast and external references based on the Morel atlas. Corresponding deep grey matter structures were also extracted from the Morel and CIT168 atlases. The data-derived, Morel and CIT168 atlases were all applied at the individual level using non-linear registration to fit the subject reference and to extract absolute mean quantitative T₁ values derived from the 3D-MP2RAGE volumes, after correction for residual B₁ ⁺ biases. Three metrics (The Dice and the volumetric similarity coefficients, and a novel Hausdorff distance) were used to estimate the inter-rater agreement of manual MRI segmentation and inter-atlas variability, and these metrics were measured to quantify biases due to image registration and their impact on the measurements of the quantitative T₁ values was highlighted. This represents a fully-automated segmentation process permitting the extraction of unbiased normative T₁ values in a population of young healthy controls as a reference for characterizing subtle structural alterations of deep grey nuclei relevant to a range of neurological diseases.

Hysteresis and synchronization processes of Kuramoto oscillators on high-dimensional simplicial complexes with competing simplex-encoded couplings

Recent studies of dynamic properties in complex systems point out the profound impact of hidden geometry features known as simplicial complexes, which enable geometrically conditioned many-body interactions. Studies of collective behaviors on the controlled-structure complexes can reveal the subtle interplay of geometry and dynamics. Here we investigate the phase synchronization (Kuramoto) dynamics under the competing interactions embedded on 1-simplex (edges) and 2-simplex (triangles) faces of a homogeneous four-dimensional simplicial complex. Its underlying network is a 1-hyperbolic graph with the assortative correlations among the node's degrees and the spectral dimension that exceeds d_{s}=4. By numerically solving the set of coupled equations for the phase oscillators associated with the network nodes, we determine the time-averaged system's order parameter to characterize the synchronization level. Our results reveal a variety of synchronization and desynchronization scenarios, including partially synchronized states and nonsymmetrical hysteresis loops, depending on the sign and strength of the pairwise interactions and the geometric frustrations promoted by couplings on triangle faces. For substantial triangle-based interactions, the frustration effects prevail, preventing the complete synchronization and the abrupt desynchronization transition disappears. These findings shed new light on the mechanisms by which the high-dimensional simplicial complexes in natural systems, such as human connectomes, can modulate their native synchronization processes.

Sex Differences in Cortical Morphometry and White Matter Microstructure During Brain Aging and Their Relationships to Cognition

Changes in brain structure are associated with aging, and accompanied by the gradual deterioration of cognitive functions, which manifests differently in males and females. Here, we quantify the age-related spatial aging patterns of brain gray and white matter structures, their volume reduction rate, their relationships with specific cognitive functions, as well as differences between males and females in a cross-sectional nondementia dataset. We found that both males and females showed extensive age-related decreases in the volumes of most gray matter and white matter regions. Females have larger regions where the volume decreases with age and a greater slope (females: 0.199%, males: 0.183%) of volume decrease in gray matter. For white matter, no significant sex differences were found in age-related regions, and the slope of volume decrease. More significant associations were identified between brain structures and cognition in males during aging than females. This study explored the age-related regional variations in gray matter and white matter, as well as the sex differences in a nondemented elderly population. This study helps to further understand the aging of the brain structure and sex differences in the aging of brain structures and provides new evidence for the aging of nondemented individuals.

Topological analysis of interaction patterns in cancer-specific gene regulatory network: persistent homology approach

In this study, we investigated cancer cellular networks in the context of gene interactions and their associated patterns in order to recognize the structural features underlying this disease. We aim to propose that the quest of understanding cancer takes us beyond pairwise interactions between genes to a higher-order construction. We characterize the most prominent network deviations in the gene interaction patterns between cancer and normal samples that contribute to the complexity of this disease. What we hope is that through understanding these interaction patterns we will notice a deeper structure in the cancer network. This study uncovers the significant deviations that topological features in cancerous cells show from the healthy one, where the last stage of filtration confirms the importance of one-dimensional holes (topological loops) in cancerous cells and two-dimensional holes (topological voids) in healthy cells. In the small threshold region, the drop in the number of connected components of the cancer network, along with the rise in the number of loops and voids, all occurring at some smaller weight values compared to the normal case, reveals the cancerous network tendency to certain pathways.

Joint hub identification for brain networks by multivariate graph inference

Recent developments in neuroimaging allow us to investigate the structural and functional connectivity between brain regions in vivo. Mounting evidence suggests that hub nodes play a central role in brain communication and neural integration. Such high centrality, however, makes hub nodes particularly susceptible to pathological network alterations and the identification of hub nodes from brain networks has attracted much attention in neuroimaging. Current popular hub identification methods often work in a univariate manner, i.e., selecting the hub nodes one after another based on either heuristic of the connectivity profile at each node or predefined settings of network modules. Since the topological information of the entire network (such as network modules) is not fully utilized, current methods have limited power to identify hubs that link multiple modules (connector hubs) and are biased toward identifying hubs having many connections within the same module (provincial hubs). To address this challenge, we propose a novel multivariate hub identification method. Our method identifies connector hubs as those that partition the network into disconnected components when they are removed from the network. Furthermore, we extend our hub identification method to find the population-based hub nodes from a group of network data. We have compared our hub identification method with existing methods on both simulated and human brain network data. Our proposed method achieves more accurate and replicable discovery of hub nodes and exhibits enhanced statistical power in identifying network alterations related to neurological disorders such as Alzheimer's disease and obsessive-compulsive disorder.

Homological scaffold via minimal homology bases

The homological scaffold leverages persistent homology to construct a topologically sound summary of a weighted network. However, its crucial dependency on the choice of representative cycles hinders the ability to trace back global features onto individual network components, unless one provides a principled way to make such a choice. In this paper, we apply recent advances in the computation of minimal homology bases to introduce a quasi-canonical version of the scaffold, called minimal, and employ it to analyze data both real and in silico. At the same time, we verify that, statistically, the standard scaffold is a good proxy of the minimal one for sufficiently complex networks.