Hierarchical Stochastic Differential Equation Models for Latent Manifold Learning in Neural Time Series

TL;DR

This paper introduces a hierarchical stochastic differential equation model for neural time series that efficiently uncovers low-dimensional manifold structures, balancing interpretability and computational scalability.

Contribution

It proposes a novel hierarchical SDE framework using Brownian bridges to model neural manifolds, improving interpretability and computational efficiency over existing models.

Findings

Accurately recovers underlying manifold structures in neural data.

Scales linearly with data length, enabling analysis of large datasets.

Validates effectiveness on synthetic and real neural recordings.

Abstract

The manifold hypothesis suggests that high-dimensional neural time series lie on a low-dimensional manifold shaped by simpler underlying dynamics. To uncover this structure, latent dynamical variable models such as state-space models, recurrent neural networks, neural ordinary differential equations, and Gaussian Process Latent Variable Models are widely used. We propose a novel hierarchical stochastic differential equation (SDE) model that balances computational efficiency and interpretability, addressing key limitations of existing methods. Our model assumes the trajectory of a manifold can be reconstructed from a sparse set of samples from the manifold trajectory. The latent space is modeled using Brownian bridge SDEs, with points - specified in both time and value - sampled from a multivariate marked point process. These Brownian bridges define the drift of a second set of SDEs, which are then mapped to the observed data. This yields a continuous, differentiable latent process capable of modeling arbitrarily complex time series as the number of manifold points increases. We derive training and inference procedures and show that the computational cost of inference scales linearly with the length of the observation data. We then validate our model on both synthetic data and neural recordings to demonstrate that it accurately recovers the underlying manifold structure and scales effectively with data dimensionality.