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

Andrew J. Blumberg, Mathieu Carrière, Jun Hou Fung, Michael A. Mandell

TL;DR

The paper introduces a robust three-stage pipeline for extracting circular coordinates from recurrent time-series data: density uniformization via rejection sampling to mitigate uneven sampling, persistent cohomology to obtain circular coordinates on subsamples, and alignment-averaging through Procrustes analysis to produce a global coordinate. This approach yields coordinates that are more robust to noise and outliers and substantially more efficient than applying persistent cohomology to the full dataset. Validation on synthetic data and C. elegans neuronal recordings reveals a topological model of worm brain-state trajectories, where loops map to interpretable behavioral states, with mutual information indicating higher informativeness for the corrected coordinates. The method thus enables scalable, unsupervised discovery of circular, recurrent structure in high-dimensional neural time series and can be extended to broader dynamical-system contexts.

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.

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

TL;DR

The paper introduces a robust three-stage pipeline for extracting circular coordinates from recurrent time-series data: density uniformization via rejection sampling to mitigate uneven sampling, persistent cohomology to obtain circular coordinates on subsamples, and alignment-averaging through Procrustes analysis to produce a global coordinate. This approach yields coordinates that are more robust to noise and outliers and substantially more efficient than applying persistent cohomology to the full dataset. Validation on synthetic data and C. elegans neuronal recordings reveals a topological model of worm brain-state trajectories, where loops map to interpretable behavioral states, with mutual information indicating higher informativeness for the corrected coordinates. The method thus enables scalable, unsupervised discovery of circular, recurrent structure in high-dimensional neural time series and can be extended to broader dynamical-system contexts.

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.

Paper Structure

This paper contains 12 sections, 5 theorems, 30 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Finding circular coordinates using persistent cohomology
  3. Density uniformization via rejection sampling
  4. Averaging coordinates with Procrustes alignment and hill climbing
  5. Experimental results
  6. Unbalanced circles and ellipses
  7. C. elegans neuronal recordings
  8. Data from Kato et al.
  9. Data from Yemini et al.
  10. Qualitative evaluation in terms of mutual information
  11. Runtime analysis
  12. Discussion

Key Result

Proposition 2.1

Let $[\tilde{\alpha}] \in \operatorname{im}(H^1(K_\bullet; \mathbb{Z}) \to H^1(K_\bullet; \mathbb{R}))$ and let $\tilde{\alpha}$ be a cocycle representative of the form $\tilde{\alpha} = \alpha + \delta f$ for some $\alpha \in C^1(K_\bullet; \mathbb{Z})$ and $f \in C^0(K_\bullet; \mathbb{R})$. Then

Figures (8)

  • Figure 1: Top: An unbalanced circle dataset colored by the "true" angle (left) or the uncorrected coordinate inferred from persistent cohomology (center). A plot of the uncorrected coordinate against the true coordinate (right); the dotted diagonal line indicates equality. Bottom: Examples of the subsampled unbalanced circle and their inferred phases (left), the final corrected coordinate obtained by aligning the coordinates assigned to subsamples (center), and a plot of the corrected coordinate against the true coordinate (right).
  • Figure 2: Top: An unbalanced ellipse dataset colored by arc length (left) or the uncorrected coordinate inferred from persistent cohomology (center). A plot of the uncorrected coordinate against the arc length parametrization (right). Bottom: Examples of the subsampled unbalanced ellipse and their inferred phases (left), the final corrected coordinate (center), and a plot of the corrected coordinate against the arc length parametrization (right).
  • Figure 3: The projection of the neuronal trajectories into PCA space (left) and the persistence diagram (right) for two different worms.
  • Figure 4: The neuronal manifold of Caenorhabditis elegans global brain dynamics, coordinatizing a cyclic locomotory gait, colored by (a) the uncorrected coordinate or (b) the corrected coordinate.
  • Figure 5: Comparison between the labels from Kato2017 and our inferred coordinates, either (a) uncorrected or (b) corrected. Points are colored according to the discrete categories of the provided labels: forward (FWD), reversal (REV), sustained reversal (REVSUS), dorsal turn (DT), and ventral turn (VT).
  • ...and 3 more figures

Theorems & Definitions (12)

  • Proposition 2.1: deSilva2011
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • Definition 4.1
  • Lemma 4.2
  • ...and 2 more