Subsampling, aligning, and averaging to find circular coordinates in recurrent time series

2024-12-24

AI Summary

Extracting stable circular coordinates from recurrent temporal data with non-uniform sampling—such as *C. elegans* neural activity—remains challenging due to sampling-density bias and instability in topological coordinate estimation. Method: We propose a robust cohomological coordinate decoding framework: (i) rejection sampling to correct sampling-density bias; (ii) multi-subsample Procrustes alignment followed by coordinate averaging to stabilize estimation of one-dimensional cohomological structure. This bypasses the strong uniform-sampling assumption inherent in conventional Rips complex construction. Contribution/Results: Our method achieves superior computational efficiency and coordinate stability compared to state-of-the-art approaches. Validated on both synthetic benchmarks and real *C. elegans* neural recordings, it successfully constructs an interpretable topological ring model of brain states. Furthermore, it maps distinct arc segments of the inferred circle to macroscopic behaviors—including forward locomotion and turning—providing a novel, topology-driven tool for analyzing recurrent neural dynamics.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Machine Learning: Learning with Manifolds
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We introduce a new algorithm for finding robust circular coordinates on data that is expected to exhibit recurrence, such as that which appears in neuronal recordings of C. elegans. Techniques exist to create circular coordinates on a simplicial complex from a dimension 1 cohomology class, and these can be applied to the Rips complex of a dataset when it has a prominent class in its dimension 1 cohomology. However, it is known this approach is extremely sensitive to uneven sampling density. Our algorithm comes with a new method to correct for uneven sampling density, adapting our prior work on averaging coordinates in manifold learning. We use rejection sampling to correct for inhomogeneous sampling and then apply Procrustes matching to align and average the subsamples. In addition to providing a more robust coordinate than other approaches, this subsampling and averaging approach has better efficiency. We validate our technique on both synthetic data sets and neuronal activity recordings. Our results reveal a topological model of neuronal trajectories for C. elegans that is constructed from loops in which different regions of the brain state space can be mapped to specific and interpretable macroscopic behaviors in the worm.

Problem

• Non-uniform Sampling
• Circular Trajectory Detection
• Biological Pattern Analysis

Innovation

• Stable Circle Coordinates
• Non-uniform Sampling
• Procrustes Alignment