Comparing neural population responses based on pairwise $p$-Wasserstein distance between topological signatures

Real-world data are often encoded in high-dimensional representations. Moreover, it is often unclear which coordinates and metrics can be meaningfully justified. Topological properties are well-suited for characterizing the structure of such high-dimensional data point-cloud: they are generalized to high-dimensional surfaces; they are also invariant under different coordinates and robust to the choice of metrics. Our work aims to compare point-clouds based on their topological properties and is motivated by emerging open problems in neuroscience to analyze the high-dimensional neural population response. A crucial gap in related works is that they have not considered how these neural population responses can be appropriately compared, which is key to understanding neural representations.We develop a topology-based approach and apply it to compare neural population responses in the mouse retina to different visual stimuli. We use nonlinear dimensionality reduction to obtain a lower-dimensional neural manifold of retinal ganglion cell population activity. Topological features are then extracted using persistent homology and represented as persistence diagrams. Finally, we compute the pairwise p-Wasserstein distance between these persistence diagrams. Our experiments show that in terms of topological structures, the neural population response to low-frequency gratings is significantly different from other types of flow stimuli, informing further neuroscientific investigations into this selective preference. Moreover, the p-Wasserstein distance induces a metric space of persistence diagrams where standard statistical objects are well-defined, allowing statistical inference on a distribution of persistence diagrams for the respective neural population responses, such as the expected diagram and the variance over the diagrams. The proposed approach can be used to compare neural population responses arising from a variety of artificial and biological neural networks.