Homological analyses of network dynamics

Abstract: Dynamical processes over networks have widespread applications including the study of social activity, brain dynamics, and heat diffusion, the latter two of which also help provide mathematical foundations for algorithms in machine learning and AI. In this talk, I will present my work that approaches this field from the perspective from homology, the mathematical subfield that formally studies ``holes'' in structures. In particular, I will present theory and experiments that use the TDA framework of persistent homology to study wavefront propagation for social contagions [Nat. Comms. 6, 7723 (2015)] and convection cycles for irreversible Markov chains [arXiv:2109.08746 (2021)]. I will also explore how these dynamical phenomena change as one considers "interconnected networks," such as a transportation system with multiple modes of transportation or a social network with different types of social relations.

Dynamics embedded in the brain network

Abstract: One of the important questions about the brain is how dynamic functionalities arise from a stable anatomy. This question was recently formulated in terms of network architecture of the whole brain. When we look at the brain as a whole, the organization of brain networks appears to have properties of complex network systems from which diverse functions emerge. The importance of this systems approach is in its ability to explain intact (or less deficient) information exchange (functional) among brain regions even after anatomical malformation or damage has taken place. Network brain science has disclosed dynamics in the functional networks but also revealed configuration of human brains suitable for transitions among multistable brain states. According to the energy landscape analysis, the brain appears to exhibit multiple stable patterns of the brain activity (multistability) and well-organized transitions among those stable patterns. These dynamic properties of the human brain system may be embedded in the brain network, which can be inferred from neuroimaging data. In the current presentation, dynamic properties of brain networks and brain states estimated from neuroimaging data will be presented. Furthermore, the utility of systems approach and computational theory to treat dynamic brain networks will be discussed.

Learning phylogenetic networks using invariants

Abstract: Phylogenetic networks provide a means of describing the evolutionary history of sets of species believed to have undergone hybridization or gene flow during the course of their evolution. The mutation process for a set of such species can be modeled as a Markov process on a phylogenetic network. Previous work has shown that a site-pattern probability distributions from a Jukes-Cantor phylogenetic network model must satisfy certain algebraic invariants. As a corollary, aspects of the phylogenetic network are theoretically identifiable from site-pattern frequencies. In practice, because of the probabilistic nature of sequence evolution, the phylogenetic network invariants will rarely be satisfied, even for data generated under the model. Thus, using network invariants for inferring phylogenetic networks requires some means of interpreting the residuals, or deviations from zero, when observed site-pattern frequencies are substituted into the invariants. In this work, we propose a machine learning algorithm utilizing invariants to infer small, level-one phylogenetic networks. Given a data set, the algorithm is trained on model data to learn the patterns of residuals corresponding to different network structures to classify the network that produced the data. This is joint work with Travis Barton, Colby Long, and Joseph Rusinko.

An Introduction to Multiparameter Persistent Homology

Abstract: In topological data analysis (TDA), one often studies the shape of data by constructing a filtered topological space, whose structure is then examined using persistent homology. However, a single filtered space often does not adequately capture the structure of interest in the data, and one is led to consider multiparameter persistent homology, which associates to the data a space equipped with two or more filtrations. This has become one of the most active areas of research within TDA, with exciting progress on several fronts. My talk will introduce multiparameter persistent homology and survey some of this recent progress.

The talk will cover some of the same territory as a forthcoming expository article with the same title, jointly written with Magnus Botnan.

On the Achievability of Blind Source Separation for High-Dimensional Nonlinear Source Mixtures

Abstract: A combination of principal component analysis (PCA) and independent component analysis (ICA) has been frequently used for blind source separation (BSS). We have previously developed a biologically plausible unsupervised learning algorithm that performs PCA and ICA in a single-layer feedforward neural network with local information available at synapses. However, it remains unclear why these linear methods work well with real-world data that involve nonlinear source mixtures. We theoretically validate that a cascade of linear PCA and ICA can solve a nonlinear BSS problem accurately—when the sensory inputs are generated from hidden sources via nonlinear mappings with sufficient dimensionality. Applying linear PCA to the inputs can reliably extract a subspace spanned by the linear projections from every hidden source as the major components—and thus projecting the inputs onto their major eigenspace can effectively recover a linear transformation of the hidden sources. Then the subsequent application of linear ICA can separate all the true independent hidden sources accurately. Zero-element-wise-error nonlinear BSS is asymptotically attained when the source dimensionality is large and the input dimensionality is sufficiently larger than the source dimensionality. Our results highlight the utility of linear PCA and ICA for accurately and reliably recovering nonlinearly mixed sources and suggest the importance of employing sensors with sufficient dimensionality to identify true hidden sources of real-world data.

Quantifying neural interactions based on information geometry

Abstract: Assessment of interactions between neurons is an important subject to understand the nature of information processing in the brain. We have previously proposed a unified framework for quantifying neural interactions based on information geometry. In the proposed framework, the degree of interactions is quantified by the divergence between the actual probability distribution of a system and a constrained probability distribution where the interactions of interest are disconnected. The degree of interactions quantified in this way can be interpreted as the amount of information loss associated with certain disconnections. Based on the framework, I will cover two topics in this talk; higher-order interactions and integrated information (causal interactions in other words). I will explain how to utilize information geometry for assessing neural interactions and then, introduce some applications to neural data.

Finding curvature threshold in the ephaptic propagation for white matter tractography

Abstract: If neural fiber were fully wrapped with myeline, the neural network would be random and arbitrary. The shape of the neural fiber bundle does not matter because the propagation along a fiber does not affect the propagation in the neighborhood fibers. However, the Ranvier nodes of 1/1000 scale to the length of myelin, which compensates the energy loss by myeline leakage, place critical geometric constraints or facilitators on its propagation. Though it is not completely multidimensional propagation as happened in the heart, the neural spike propagation retains some properties in the ephaptic propagation in the neural fiber bundle. In this talk, we explain the motivation, inferred principle, and numerical schemes of neural spike propagation. Also, direct application to white matter tractography is explained from the pros and cons of state-of-the-art tractography.