Fundamental limits on dynamic inference from single-cell snapshots

Seeing a snapshot of individuals at different stages of a dynamic process can reveal what the process would look like for a single individual over time. Biologists apply this principle to infer temporal sequences of gene expression states in cells from measurements made at a single moment in time. However, the sparsity and high dimensionality of single-cell data have made inference difficult using formal approaches. Here, we apply recent innovations in spectral graph theory to devise a simple and asymptotically exact algorithm for inferring the unique dynamic solution under defined approximations and apply it to data from bone marrow stem cells.

Single-cell expression profiling reveals the molecular states of individual cells with unprecedented detail. Because these methods destroy cells in the process of analysis, they cannot measure how gene expression changes over time. However, some information on dynamics is present in the data: the continuum of molecular states in the population can reflect the trajectory of a typical cell. Many methods for extracting single-cell dynamics from population data have been proposed. However, all such attempts face a common limitation: for any measured distribution of cell states, there are multiple dynamics that could give rise to it, and by extension, multiple possibilities for underlying mechanisms of gene regulation. Here, we describe the aspects of gene expression dynamics that cannot be inferred from a static snapshot alone and identify assumptions necessary to constrain a unique solution for cell dynamics from static snapshots. We translate these constraints into a practical algorithmic approach, population balance analysis (PBA), which makes use of a method from spectral graph theory to solve a class of high-dimensional differential equations. We use simulations to show the strengths and limitations of PBA, and then apply it to single-cell profiles of hematopoietic progenitor cells (HPCs). Cell state predictions from this analysis agree with HPC fate assays reported in several papers over the past two decades. By highlighting the fundamental limits on dynamic inference faced by any method, our framework provides a rigorous basis for dynamic interpretation of a gene expression continuum and clarifies best experimental designs for trajectory reconstruction from static snapshot measurements.

Spectral graph theory of brain oscillations

The relationship between the brain's structural wiring and the functional patterns of neural activity is of fundamental interest in computational neuroscience. We examine a hierarchical, linear graph spectral model of brain activity at mesoscopic and macroscopic scales. The model formulation yields an elegant closed-form solution for the structure-function problem, specified by the graph spectrum of the structural connectome's Laplacian, with simple, universal rules of dynamics specified by a minimal set of global parameters. The resulting parsimonious and analytical solution stands in contrast to complex numerical simulations of high dimensional coupled nonlinear neural field models. This spectral graph model accurately predicts spatial and spectral features of neural oscillatory activity across the brain and was successful in simultaneously reproducing empirically observed spatial and spectral patterns of alpha-band (8-12 Hz) and beta-band (15-30 Hz) activity estimated from source localized magnetoencephalography (MEG). This spectral graph model demonstrates that certain brain oscillations are emergent properties of the graph structure of the structural connectome and provides important insights towards understanding the fundamental relationship between network topology and macroscopic whole-brain dynamics. .

Spectral graph theory of brain oscillations - revisited and improved

Mathematical modeling of the relationship between the functional activity and the structural wiring of the brain has largely been undertaken using non-linear and biophysically detailed mathematical models with regionally varying parameters. While this approach provides us a rich repertoire of multistable dynamics that can be displayed by the brain, it is computationally demanding. Moreover, although neuronal dynamics at the microscopic level are nonlinear and chaotic, it is unclear if such detailed nonlinear models are required to capture the emergent meso-(regional population ensemble) and macro-scale (whole brain) behavior, which is largely deterministic and reproducible across individuals. Indeed, recent modeling effort based on spectral graph theory has shown that an analytical model without regionally varying parameters and without multistable dynamics can capture the empirical magnetoencephalography frequency spectra and the spatial patterns of the alpha and beta frequency bands accurately.

In this work, we demonstrate an improved hierarchical, linearized, and analytic spectral graph theory-based model that can capture the frequency spectra obtained from magnetoencephalography recordings of resting healthy subjects. We reformulated the spectral graph theory model in line with classical neural mass models, therefore providing more biologically interpretable parameters, especially at the local scale. We demonstrated that this model performs better than the original model when comparing the spectral correlation of modeled frequency spectra and that obtained from the magnetoencephalography recordings. This model also performs equally well in predicting the spatial patterns of the empirical alpha and beta frequency bands.

Transcranial photobiomodulation changes topology, synchronizability, and complexity of resting state brain networks

Objective. Transcranial photobiomodulation (tPBM) is a recently proposed non-invasive brain stimulation approach with various effects on the nervous system from the cells to the whole brain networks. Specially in the neural network level, tPBM can alter the topology and synchronizability of functional brain networks. However, the functional properties of the neural networks after tPBM are still poorly clarified.Approach. Here, we employed electroencephalography and different methods (conventional and spectral) in the graph theory analysis to track the significant effects of tPBM on the resting state brain networks. The non-parametric statistical analysis showed that just one short-term tPBM session over right medial frontal pole can significantly change both topological (i.e. clustering coefficient, global efficiency, local efficiency, eigenvector centrality) and dynamical (i.e. energy, largest eigenvalue, and entropy) features of resting state brain networks.Main results. The topological results revealed that tPBM can reduce local processing, centrality, and laterality. Furthermore, the increased centrality of central electrode was observed.Significance. These results suggested that tPBM can alter topology of resting state brain network to facilitate the neural information processing. On the other hand, the dynamical results showed that tPBM reduced stability of synchronizability and increased complexity in the resting state brain networks. These effects can be considered in association with the increased complexity of connectivity patterns among brain regions and the enhanced information propagation in the resting state brain networks. Overall, both topological and dynamical features of brain networks suggest that although tPBM decreases local processing (especially in the right hemisphere) and disrupts synchronizability of network, but it can increase the level of information transferring and processing in the brain network.

HERMES: PERSISTENT SPECTRAL GRAPH SOFTWARE

Persistent homology (PH) is one of the most popular tools in topological data analysis (TDA), while graph theory has had a significant impact on data science. Our earlier work introduced the persistent spectral graph (PSG) theory as a unified multiscale paradigm to encompass TDA and geometric analysis. In PSG theory, families of persistent Laplacian matrices (PLMs) corresponding to various topological dimensions are constructed via a filtration to sample a given dataset at multiple scales. The harmonic spectra from the null spaces of PLMs offer the same topological invariants, namely persistent Betti numbers, at various dimensions as those provided by PH, while the non-harmonic spectra of PLMs give rise to additional geometric analysis of the shape of the data. In this work, we develop an open-source software package, called highly efficient robust multidimensional evolutionary spectra (HERMES), to enable broad applications of PSGs in science, engineering, and technology. To ensure the reliability and robustness of HERMES, we have validated the software with simple geometric shapes and complex datasets from three-dimensional (3D) protein structures. We found that the smallest non-zero eigenvalues are very sensitive to data abnormality.

Sensor Network Localization by Eigenvector Synchronization Over the Euclidean Group

We present a new approach to localization of sensors from noisy measurements of a subset of their Euclidean distances. Our algorithm starts by finding, embedding, and aligning uniquely realizable subsets of neighboring sensors called patches. In the noise-free case, each patch agrees with its global positioning up to an unknown rigid motion of translation, rotation, and possibly reflection. The reflections and rotations are estimated using the recently developed eigenvector synchronization algorithm, while the translations are estimated by solving an overdetermined linear system. The algorithm is scalable as the number of nodes increases and can be implemented in a distributed fashion. Extensive numerical experiments show that it compares favorably to other existing algorithms in terms of robustness to noise, sparse connectivity, and running time. While our approach is applicable to higher dimensions, in the current article, we focus on the two-dimensional case.

Eigenvector synchronization, graph rigidity and the molecule problem

The graph realization problem has received a great deal of attention in recent years, due to its importance in applications such as wireless sensor networks and structural biology. In this paper, we extend the previous work and propose the 3D-As-Synchronized-As-Possible (3D-ASAP) algorithm, for the graph realization problem in ℝ3, given a sparse and noisy set of distance measurements. 3D-ASAP is a divide and conquer, non-incremental and non-iterative algorithm, which integrates local distance information into a global structure determination. Our approach starts with identifying, for every node, a subgraph of its 1-hop neighborhood graph, which can be accurately embedded in its own coordinate system. In the noise-free case, the computed coordinates of the sensors in each patch must agree with their global positioning up to some unknown rigid motion, that is, up to translation, rotation and possibly reflection. In other words, to every patch, there corresponds an element of the Euclidean group, Euc(3), of rigid transformations in ℝ3, and the goal was to estimate the group elements that will properly align all the patches in a globally consistent way. Furthermore, 3D-ASAP successfully incorporates information specific to the molecule problem in structural biology, in particular information on known substructures and their orientation. In addition, we also propose 3D-spectral-partitioning (SP)-ASAP, a faster version of 3D-ASAP, which uses a spectral partitioning algorithm as a pre-processing step for dividing the initial graph into smaller subgraphs. Our extensive numerical simulations show that 3D-ASAP and 3D-SP-ASAP are very robust to high levels of noise in the measured distances and to sparse connectivity in the measurement graph, and compare favorably with similar state-of-the-art localization algorithms.

PERSISTENT PATH LAPLACIAN

Path homology proposed by S.-T.Yau and his co-workers provides a new mathematical model for directed graphs and networks. Persistent path homology (PPH) extends the path homology with filtration to deal with asymmetry structures. However, PPH is constrained to purely topological persistence and cannot track the homotopic shape evolution of data during filtration. To overcome the limitation of PPH, persistent path Laplacian (PPL) is introduced to capture the shape evolution of data. PPL's harmonic spectra fully recover PPH's topological persistence and its non-harmonic spectra reveal the homotopic shape evolution of data during filtration.

Entropy and optimality in river deltas

River deltas are critically important Earthscapes at the land–water interface, supporting dense populations and diverse ecosystems while also providing disproportionately large food and energy resources. Deltas exhibit complex channel networks that dictate how water, sediment, and nutrients are spread over the delta surface. By adapting concepts from information theory, we show that a range of field and numerically generated deltas obey an optimality principle that suggests that deltas self-organize to increase the diversity of sediment transport pathways across the delta channels to the shoreline. We suggest that optimal delta configurations are also more resilient because the same mechanism that diversifies the delivery of fluxes to the shoreline also enhances the dampening of possible perturbations.

The form and function of river deltas is intricately linked to the evolving structure of their channel networks, which controls how effectively deltas are nourished with sediments and nutrients. Understanding the coevolution of deltaic channels and their flux organization is crucial for guiding maintenance strategies of these highly stressed systems from a range of anthropogenic activities. To date, however, a unified theory explaining how deltas self-organize to distribute water and sediment up to the shoreline remains elusive. Here, we provide evidence for an optimality principle underlying the self-organized partition of fluxes in delta channel networks. By introducing a suitable nonlocal entropy rate (nER) and by analyzing field and simulated deltas, we suggest that delta networks achieve configurations that maximize the diversity of water and sediment flux delivery to the shoreline. We thus suggest that prograding deltas attain dynamically accessible optima of flux distributions on their channel network topologies, thus effectively decoupling evolutionary time scales of geomorphology and hydrology. When interpreted in terms of delta resilience, high nER configurations reflect an increased ability to withstand perturbations. However, the distributive mechanism responsible for both diversifying flux delivery to the shoreline and dampening possible perturbations might lead to catastrophic events when those perturbations exceed certain intensity thresholds.

Automatic detection of regional heart rejection in USPIO-enhanced MRI

Contrast-enhanced magnetic resonance imaging (MRI) is useful to study the infiltration of cells in vivo. This research adopts ultrasmall superparamagnetic iron oxide (USPIO) particles as contrast agents. USPIO particles administered intravenously can be endocytosed by circulating immune cells, in particular, macrophages. Hence, macrophages are labeled with USPIO particles. When a transplanted heart undergoes rejection, immune cells will infiltrate the allograft. Imaged by T(2)(*)-weighted MRI, USPIO-labeled macrophages display dark pixel intensities. Detecting these labeled cells in the image facilitates the identification of acute heart rejection. This paper develops a classifier to detect the presence of USPIO-labeled macrophages in the myocardium in the framework of spectral graph theory. First, we describe a USPIO-enhanced heart image with a graph. Classification becomes equivalent to partitioning the graph into two disjoint subgraphs. We use the Cheeger constant of the graph as an objective functional to derive the classifier. We represent the classifier as a linear combination of basis functions given from the spectral analysis of the graph Laplacian. Minimization of the Cheeger constant based functional leads to the optimal classifier. Experimental results and comparisons with other methods suggest the feasibility of our approach to study the rejection of hearts imaged by USPIO-enhanced MRI.

Structure-function models of temporal, spatial, and spectral characteristics of non-invasive whole brain functional imaging

We review recent advances in using mathematical models of the relationship between the brain structure and function that capture features of brain dynamics. We argue the need for models that can jointly capture temporal, spatial, and spectral features of brain functional activity. We present recent work on spectral graph theory based models that can accurately capture spectral as well as spatial patterns across multiple frequencies in MEG reconstructions.

On the subgroup structure of the hyperoctahedral group in six dimensions

The subgroup structure of the hyperoctahedral group in six dimensions is studied, with particular attention to the subgroups isomorphic to the icosahedral group. The orthogonal crystallographic representations of the icosahedral group are classified, and their intersections are studied in some detail, using a combinatorial approach which involves results from graph theory and their spectra.

The subgroup structure of the hyperoctahedral group in six dimensions is investigated. In particular, the subgroups isomorphic to the icosahedral group are studied. The orthogonal crystallographic representations of the icosahedral group are classified and their intersections and subgroups analysed, using results from graph theory and their spectra.

Field-theoretic density estimation for biological sequence space with applications to 5' splice site diversity and aneuploidy in cancer

Density estimation in sequence space is a fundamental problem in machine learning that is also of great importance in computational biology. Due to the discrete nature and large dimensionality of sequence space, how best to estimate such probability distributions from a sample of observed sequences remains unclear. One common strategy for addressing this problem is to estimate the probability distribution using maximum entropy (i.e., calculating point estimates for some set of correlations based on the observed sequences and predicting the probability distribution that is as uniform as possible while still matching these point estimates). Building on recent advances in Bayesian field-theoretic density estimation, we present a generalization of this maximum entropy approach that provides greater expressivity in regions of sequence space where data are plentiful while still maintaining a conservative maximum entropy character in regions of sequence space where data are sparse or absent. In particular, we define a family of priors for probability distributions over sequence space with a single hyperparameter that controls the expected magnitude of higher-order correlations. This family of priors then results in a corresponding one-dimensional family of maximum a posteriori estimates that interpolate smoothly between the maximum entropy estimate and the observed sample frequencies. To demonstrate the power of this method, we use it to explore the high-dimensional geometry of the distribution of 5' splice sites found in the human genome and to understand patterns of chromosomal abnormalities across human cancers.

Further Exploration of an Upper Bound for Kemeny's Constant

Even though Kemeny's constant was first discovered in Markov chains and expressed by Kemeny in terms of mean first passage times on a graph, it can also be expressed using the pseudo-inverse of the Laplacian matrix representing the graph, which facilitates the calculation of a sharp upper bound of Kemeny's constant. We show that for certain classes of graphs, a previously found bound is tight, which generalises previous results for bipartite and (generalised) windmill graphs. Moreover, we show numerically that for real-world networks, this bound can be used to find good numerical approximations for Kemeny's constant. For certain graphs consisting of up to 100 K nodes, we find a speedup of a factor 30, depending on the accuracy of the approximation that can be achieved. For networks consisting of over 500 K nodes, the approximation can be used to estimate values for the Kemeny constant, where exact calculation is no longer feasible within reasonable computation time.

Impaired long-range excitatory time scale predicts abnormal neural oscillations and cognitive deficits in Alzheimer's disease

Alzheimer's disease (AD) is the most common form of dementia, progressively impairing memory and cognition. While neuroimaging studies have revealed functional abnormalities in AD, how these relate to aberrant neuronal circuit mechanisms remains unclear. Using magnetoencephalography imaging we documented abnormal local neural synchrony patterns in patients with AD. To identify abnormal biophysical mechanisms underlying these abnormal electrophysiological patterns, we estimated the parameters of a spectral graph-theory model (SGM). SGM is an analytic model that describes how long-range fiber projections in the brain mediate the excitatory and inhibitory activity of local neuronal subpopulations. The long-range excitatory time scale was associated with greater deficits in global cognition and was able to distinguish AD patients from controls with high accuracy. These results demonstrate that long-range excitatory time scale of neuronal activity, despite being a global measure, is a key determinant in the spatiospectral signatures and cognition in AD.

Voxel-Wise Brain Graphs From Diffusion MRI: Intrinsic Eigenspace Dimensionality and Application to Functional MRI

Goal: Structural brain graphs are conventionally limited to defining nodes as gray matter regions from an atlas, with edges reflecting the density of axonal projections between pairs of nodes. Here we explicitly model the entire set of voxels within a brain mask as nodes of high-resolution, subject-specific graphs. Methods: We define the strength of local voxel-to-voxel connections using diffusion tensors and orientation distribution functions derived from diffusion MRI data. We study the graphs' Laplacian spectral properties on data from the Human Connectome Project. We then assess the extent of inter-subject variability of the Laplacian eigenmodes via a procrustes validation scheme. Finally, we demonstrate the extent to which functional MRI data are shaped by the underlying anatomical structure via graph signal processing. Results: The graph Laplacian eigenmodes manifest highly resolved spatial profiles, reflecting distributed patterns that correspond to major white matter pathways. We show that the intrinsic dimensionality of the eigenspace of such high-resolution graphs is only a mere fraction of the graph dimensions. By projecting task and resting-state data on low-frequency graph Laplacian eigenmodes, we show that brain activity can be well approximated by a small subset of low-frequency components. Conclusions: The proposed graphs open new avenues in studying the brain, be it, by exploring their organisational properties via graph or spectral graph theory, or by treating them as the scaffold on which brain function is observed at the individual level.

Impaired long-range excitatory time scale predicts abnormal neural oscillations and cognitive deficits in Alzheimer's disease

Alzheimer's disease (AD) is the most common form of dementia, progressively impairing memory and cognition. While neuroimaging studies have revealed functional abnormalities in AD, how these relate to aberrant neuronal circuit mechanisms remains unclear. Using magnetoencephalography imaging we documented abnormal local neural synchrony patterns in patients with AD. To identify abnormal biophysical mechanisms underlying these abnormal electrophysiological patterns, we estimated the parameters of a spectral graph-theory model (SGM). SGM is an analytic model that describes how long-range fiber projections in the brain mediate the excitatory and inhibitory activity of local neuronal subpopulations. The long-range excitatory time scale was associated with greater deficits in global cognition and was able to distinguish AD patients from controls with high accuracy. These results demonstrate that long-range excitatory time scale of neuronal activity, despite being a global measure, is a key determinant in the spatiospectral signatures and cognition in AD.

Bayesian Inference of a Spectral Graph Model for Brain Oscillations

The relationship between brain functional connectivity and structural connectivity has caught extensive attention of the neuroscience community, commonly inferred using mathematical modeling. Among many modeling approaches, spectral graph model (SGM) is distinctive as it has a closed-form solution of the wide-band frequency spectra of brain oscillations, requiring only global biophysically interpretable parameters. While SGM is parsimonious in parameters, the determination of SGM parameters is non-trivial. Prior works on SGM determine the parameters through a computational intensive annealing algorithm, which only provides a point estimate with no confidence intervals for parameter estimates. To fill this gap, we incorporate the simulation-based inference (SBI) algorithm and develop a Bayesian procedure for inferring the posterior distribution of the SGM parameters. Furthermore, using SBI dramatically reduces the computational burden for inferring the SGM parameters. We evaluate the proposed SBI-SGM framework on the resting-state magnetoencephalography recordings from healthy subjects and show that the proposed procedure has similar performance to the annealing algorithm in recovering power spectra and the spatial distribution of the alpha frequency band. In addition, we also analyze the correlations among the parameters and their uncertainty with the posterior distribution which can not be done with annealing inference. These analyses provide a richer understanding of the interactions among biophysical parameters of the SGM. In general, the use of simulation-based Bayesian inference enables robust and efficient computations of generative model parameter uncertainties and may pave the way for the use of generative models in clinical translation applications.

A graph convolutional neural network for gene expression data analysis with multiple gene networks

Spectral graph convolutional neural networks (GCN) are proposed to incorporate important information contained in graphs such as gene networks. In a standard spectral GCN, there is only one gene network to describe the relationships among genes. However, for genomic applications, due to condition- or tissue-specific gene function and regulation, multiple gene networks may be available; it is unclear how to apply GCNs to disease classification with multiple networks. Besides, which gene networks may provide more effective prior information for a given learning task is unknown a priori and is not straightforward to discover in many cases. A deep multiple graph convolutional neural network is therefore developed here to meet the challenge. The new approach not only computes a feature of a gene as the weighted average of those of itself and its neighbors through spectral GCNs, but also extracts features from gene-specific expression (or other feature) profiles via a feed-forward neural networks (FNN). We also provide two measures, the importance of a given gene and the relative importance score of each gene network, for the genes' and gene networks' contributions, respectively, to the learning task. To evaluate the new method, we conduct real data analyses using several breast cancer and diffuse large B-cell lymphoma datasets and incorporating multiple gene networks obtained from "GIANT 2.0" Compared with the standard FNN, GCN, and random forest, the new method not only yields high classification accuracy but also prioritizes the most important genes confirmed to be highly associated with cancer, strongly suggesting the usefulness of the new method in incorporating multiple gene networks.

Graph metric learning quantifies morphological differences between two genotypes of shoot apical meristem cells in Arabidopsis

We present a method for learning 'spectrally descriptive' edge weights for graphs. We generalize a previously known distance measure on graphs (graph diffusion distance [GDD]), thereby allowing it to be tuned to minimize an arbitrary loss function. Because all steps involved in calculating this modified GDD are differentiable, we demonstrate that it is possible for a small neural network model to learn edge weights which minimize loss. We apply this method to discriminate between graphs constructed from shoot apical meristem images of two genotypes of Arabidopsis thaliana specimens: wild-type and trm678 triple mutants with cell division phenotype. Training edge weights and kernel parameters with contrastive loss produce a learned distance metric with large margins between these graph categories. We demonstrate this by showing improved performance of a simple k -nearest-neighbour classifier on the learned distance matrix. We also demonstrate a further application of this method to biological image analysis. Once trained, we use our model to compute the distance between the biological graphs and a set of graphs output by a cell division simulator. Comparing simulated cell division graphs to biological ones allows us to identify simulation parameter regimes which characterize mutant versus wild-type Arabidopsis cells. We find that trm678 mutant cells are characterized by increased randomness of division planes and decreased ability to avoid previous vertices between cell walls.