Stiefel Manifold Dynamical Systems for Tracking Representational Drift

Understanding neural dynamics is crucial for uncovering how the brain processes information and controls behavior. Linear dynamical systems (LDS) are widely used for modeling neural data due to their simplicity and effectiveness in capturing latent dynamics. However, LDS assumes a stable mapping from the latent states to neural activity, limiting its ability to capture representational drift-gradual changes in the brain's representation of the external world. To address this, we introduce the Stiefel Manifold Dynamical System (SMDS), a new class of model designed to account for drift in neural representations across trials. In SMDS, emission matrices are constrained to be orthonormal and evolve smoothly over trials on the Stiefel manifold-the space of all orthonormal matrices-while the dynamics parameters are shared. This formulation allows SMDS to leverage data across trials while accounting for non-stationarity, thus capturing the underlying neural dynamics more accurately compared to an LDS. We apply SMDS to both simulated datasets and neural recordings across species. Our results consistently show that SMDS outperforms LDS in terms of log-likelihood and requires fewer latent dimensions to capture the same activity. Moreover, SMDS provides a powerful framework for quantifying and interpreting representational drift. It reveals a gradual drift over the course of minutes in the neural recordings and uncovers varying drift rates across dimensions, with slower drift in behaviorally and neurally significant dimensions.