Stochastic Dynamical Systems Based Latent Structure Discovery in High-Dimensional Time Series

Abstract

The brain encodes information by neural spiking activities, which can be described by time series data as spike counts. Latent Variable Models (LVMs) are widely used to study the unknown factors (i.e. the latent states) that are dependent in a network structure to modulate neural spiking activities. Yet, challenges in performing experiments to record on neuronal level commonly results in relatively short and noisy spike count data, which is insufficient to derive latent network structure by existing LVMs. Specifically, it is difficult to set the number of latent states. A small number of latent states may not be able to model the complexities of underlying systems, while a large number of latent states can lead to overfitting. Therefore, based on a specific LVMs called Linear Dynamical System (LDS), we propose a Reduced-Rank Linear Dynamical System (RRLDS) to estimate latent states and retrieve an optimal latent network structure from short, noisy spike count data. This framework estimates the model using Laplace approximation. To further handle count-valued data, we introduce the dispersion-adaptive distribution to accommodate over-/ equal-/ and under-dispersion nature of such data. Results on both simulated and experimental data demonstrate our model can robustly learn latent space from short-length, noisy, count-valued data and significantly improve the prediction performance over the state-of-the-art methods.