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

2025-07-29

AI Summary

Modeling and interpreting low-dimensional manifold structures in high-dimensional neural time series remains challenging due to their intrinsic nonlinearity and temporal complexity. Method: We propose a hierarchical stochastic differential equation (SDE) framework that innovatively integrates Brownian bridge SDEs with multivariate marked point processes to enable continuous, differentiable latent-space dynamics modeling. Sparse sampling coupled with an SDE-driven observation mapping ensures efficient manifold trajectory reconstruction. Contribution/Results: The method achieves both interpretability and computational efficiency—its inference complexity scales linearly with sequence length, markedly outperforming conventional nonlinear dimensionality reduction and black-box deep SDE approaches. Extensive validation on synthetic benchmarks and real neural recordings—including electrophysiological data from macaque motor cortex—demonstrates accurate recovery of underlying manifold geometry and dynamics, alongside strong scalability.

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: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Modeling low-dimensional neural dynamics from high-dimensional time series
• Balancing computational efficiency and interpretability in manifold learning
• Recovering latent manifold structure with scalable inference procedures

Innovation

• Hierarchical SDE model for manifold learning
• Brownian bridge SDEs define latent dynamics
• Linear computational scaling with data length