Subsampling, aligning, and averaging to find circular coordinates in recurrent time series
This work addresses the sensitivity to sampling density in topological data analysis for neuroscience, offering an incremental improvement for analyzing recurrent patterns in data like C. elegans neuronal activity.
The paper tackles the problem of finding robust circular coordinates in recurrent time series data, such as neuronal recordings, by introducing an algorithm that corrects for uneven sampling density through subsampling, aligning, and averaging, resulting in a more robust and efficient coordinate than existing approaches.
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.