Loading...
Organizing Committee
The Pennsylvania State University
University of Pennsylvania
EPFL
California Institute of Technology
Weizmann Institute of Science
Salk Institute
Abstract
In the last decade or so, applied topology and algebraic geometry have come into their own as vibrant areas of applied mathematics. At the same time, ideas and tools from topology and geometry have infiltrated theoretical and computational neuroscience. This kind of mathematics has shown itself to be a natural and useful language not only for analyzing neural data sets but also as a means of understanding principles of neural coding and computation. This workshop will bring together leading researchers at the interfaces of topology, geometry, and neuroscience to take stock of recent work and outline future directions. This includes a focus on topological data analysis (persistent homology and related methods), topological analysis of neural networks and their dynamics, topological decoding of neural activity, evolving topology of dynamic networks (e.g., networks that are changing as a result of learning), and analysis of connectome data. Related topics may include the geometry and topology of deep learning, as well as low-dimensional projections of trained networks.
Loading...
Loading participant list in background...
Workshop Schedule
Monday, October 16, 2023
8:50 - 9:00 AM EDT
Welcome
11th Floor Lecture Hall
Session Chair
Caroline Klivans, Brown University
9:00 - 9:45 AM EDT
The geometry of perceptual spaces of textures and objects
11th Floor Lecture Hall
Speaker
Jonathan Victor, Weill Cornell Medical College
Session Chair
Carina Curto, The Pennsylvania State University
Abstract
Recent technological advances allow for massive population-level recordings of neural activity, raising the hope of achieving a detailed understanding of the linkage of neurophysiology and behavior. Achieving this linkage relies on the tenet that, viewed in the right way, the mapping between neural activity and behavior preserves similarities. At the behavioral level, these similarities are captured by the topology and geometry of perceptual spaces.
With this motivation, I describe some recent studies of the geometry of several perceptual spaces, including “low-level” spaces of visual features, and “higher-level” spaces dominated by semantic content. The experiments use a new, efficient psychophysical paradigm for collecting similarity judgments, and the analysis methods range from seeking Euclidean embeddings via non-metric multidimensional scaling to strategies that make minimal assumptions about the underlying geometry. With these tools, we characterize how the geometry of the spaces vary with semantic content, and the aspects of these geometries that are task-dependent.
10:00 - 10:15 AM EDT
Coffee Break
11th Floor Collaborative Space
10:15 - 11:00 AM EDT
Topology protects emergent dynamics and long timescales in biological networks
11th Floor Lecture Hall
Speaker
Evelyn Tang, Rice University
Session Chair
Carina Curto, The Pennsylvania State University
Abstract
Long and stable timescales are often observed in complex biochemical networks, such as in emergent oscillations or memory. How these robust dynamics persist remains unclear, given the many stochastic reactions and shorter time scales of the underlying components. We propose a topological model with parsimonious parameters that produces long oscillations around the network boundary, effectively reducing the system dynamics to a lower-dimensional current. I will demonstrate how this can model the circadian clock of cyanobacteria, with efficient properties such as simultaneously increased precision and decreased cost. Our work presents a new mechanism for emergent dynamics that could be useful for various cognitive and biological functions.
11:15 - 11:45 AM EDT
Open Problems Session
Problem Session - 11th Floor Lecture Hall
Session Chair
Carina Curto, The Pennsylvania State University
11:45 AM - 1:30 PM EDT
Lunch/Free Time
1:30 - 2:15 PM EDT
Discovering the geometry of neural representations via topological tools.
11th Floor Lecture Hall
Speaker
Vladimir Itskov, The Pennsylvania State University
Session Chair
Katie Morrison, University of Northern Colorado
Abstract
Neural representations of stimulus spaces often comes with a natural geometry. Perhaps the most salient examples of such neural populations are those with convex receptive fields (or tuning curves), such as place cells in hippocampus or neurons in V1. Geometry of neural representations is understood in a very limited number of well-studied neural circuits. It is rather poorly understood in most other parts of the brain. This raises a natural question: can one infer such a geometry, based on the statistics of the neural responses alone?
A crucial tool for inferring a geometry is a basis of coordinate functions that "respects" the underlying geometry, while providing meaningful low-dimensional approximations. Eigenfunctions of a Laplacian, derived from the underlying metric, serve as such basis in many scientific fields. However, spike trains, and other derived features of neural activity do not come with a natural metric, while they do come with an "intrinsic" probability distribution of neural activity patterns.
Building on the tools from combinatorial topology, we introduce Hodge Laplacians associated with probability distributions on sequential data, such as spike trains. We demonstrate that these Laplacians have desirable properties with respect to the natural null-models, where the underlying neurons are independent. Our results establish a foundation for dimensionality reduction and Fourier analyses of probabilistic models, that are common in theoretical neuroscience and machine-learning.
2:30 - 2:40 PM EDT
Connections between the topology of tasks, classifying spaces, and learned representations
Lightning Talks - 11th Floor Lecture Hall
Speaker
Thomas Burns, ICERM
Session Chair
Katie Morrison, University of Northern Colorado
Abstract
Modified state complexes (Burns & Tang, 2022) extend the mathematical framework of reconfigurable systems and state complexes due to Abrams, Ghrist & Peterson to study gridworlds -- simple 2D environments inhabited of agents, objects, etc.. Such state complexes represent all possible configurations of a system as a single geometric space, thus making them conducive to study using geometric, topological, or combinatorial methods. Modified state complexes exhibit geometric defects (failure of Gromov's Link Condition) exactly where undesirable or dangerous states appear in the gridworld. We hypothesize that the modified state complex should be a classifying space for the n–strand braid group and that social place cell circuits in mammalian hippocampus use similar principles to represent and avoid danger.
2:40 - 2:50 PM EDT
Emergence of high-order functional hubs in the human brain
Lightning Talks - 11th Floor Lecture Hall
Speaker
Fernando Nobrega Santos, University of Amsterdam
Session Chair
Katie Morrison, University of Northern Colorado
Abstract
Network theory is often based on pairwise relationships between nodes, which is not necessarily realistic for modeling complex systems. Importantly, it does not accurately capture non-pairwise interactions in the human brain, often considered one of the most complex systems. In this work, we develop a multivariate signal processing pipeline to build high-order networks from time series and apply it to resting-state functional magnetic resonance imaging (fMRI) signals to characterize high-order communication between brain regions. We also propose connectivity and signal processing rules for building uniform hypergraphs and argue that each multivariate interdependence metric could define weights in a hypergraph. As a proof of concept, we investigate the most relevant three-point interactions in the human brain by searching for high-order “hubs” in a cohort of 100 individuals from the Human Connectome Project. We find that, for each choice of multivariate interdependence, the high-order hubs are compatible with distinct systems in the brain. Additionally, the high-order functional brain networks exhibit simultaneous integration and segregation patterns qualitatively observable from their high-order hubs. Our work hereby introduces a promising heuristic route for hypergraph representation of brain activity and opens up exciting avenues for further research in high-order network neuroscience and complex systems.
2:50 - 3:00 PM EDT
Topological feature selection for time series: an example with C. elegans neuronal data
Lightning Talks - 11th Floor Lecture Hall
Speaker
Johnathan Bush, University of Florida
Session Chair
Katie Morrison, University of Northern Colorado
Abstract
Neurons across the brain of the model organism C. elegans are known to share information by engaging in coordinated dynamic activity that evolves cyclically. Takens' theorem implies that a sliding window embedding of time series, such as neuronal activity, will preserve the topology of an orbit of the underlying dynamical system driving the time series. These orbits are then quantifiable by the persistent homology of the sliding window embedding. In this setting, we will describe a method for topological optimization in which each time series (e.g., a single neuron's activity) is assigned a score of its contribution to the global, coordinated dynamics of a collection of time series (e.g., the brain).
3:00 - 3:10 PM EDT
The Directed Merge Tree Distance and its Applications
Lightning Talks - 11th Floor Lecture Hall
Speaker
Xinyi Wang, Michigan State University
Session Chair
Katie Morrison, University of Northern Colorado
Abstract
Geometric graphs appear in many real-world datasets, such as embedded neurons, sensor networks, and molecules. We investigate the notion of distance between graphs and present a semi-metric to measure the distance between two geometric graphs via the directional transform combined with the labeled merge tree distance. We introduce a way of rotating the sub-level set to obtain the merge trees, and represent the merge trees using a surjective multi-labeling scheme. We then compute the distance between two representative matrices. Our distance is not only reflective of the information from the input graphs, but also can be computed in polynomial time. We illustrate its utility by implementation on a Passiflora leaf dataset.
3:10 - 3:20 PM EDT
Structure Index: a graph-based method for point cloud data analysis
Lightning Talks - 11th Floor Lecture Hall
Speaker
Julio Esparza Ibanez, Instituto Cajal - CSIC (Spanish National Research Council)
Session Chair
Katie Morrison, University of Northern Colorado
Abstract
A point cloud is a prevalent data format found in many fields of science, which involves the definition of points in an arbitrarily high dimensional space. Typically, each of these points is associated with additional values (i.e. features) which require interpretation in the representation space. For instance, in neuroscience, neural activity over time can be pictured as a point cloud in a high-dimensional space. In these so-called neural manifolds, one may project different features onto the point cloud, such as any relevant behavioral variable. In this context, understanding if and how a given feature is structured along a point cloud can provide great insights into the neural representations.
Here, I will introduce the Structure Index (SI), a graph-based metric developed to quantify how a given feature is structured along an arbitrarily high-dimensional point cloud. The SI is defined from the overlapping distribution of data points sharing similar feature values in a given neighborhood of the cloud. Using arbitrary data clouds, I will show how the SI provides quantification of the degree of local versus global organization of feature distribution. Moreover, when applied to experimental studies of head-direction cells, the SI is able to retrieve consistent feature structure from both the high- and low-dimensional representations. Overall, the SI provides versatile applications in the neuroscience and data science fields. We look to share the tool with other colleagues in the field, in order to promote community-based testing and implementation.
