Subspace Identification Methods for Extracting Behaviorally Relevant Neural Dynamics

Neural population recordings provide a unique window into how the brain generates behavior, yet transforming these high-dimensional signals into interpretable dynamical models remains a major challenge. A central difficulty is that neural activity reflects a mixture of intrinsic dynamics, behavioral components, and externally driven responses, and standard modeling approaches do not reliably separate these components. Non-preferential modeling extracts dominant patterns from neural activity without regard for behavioral relevance, and therefore often captures structure that is unrelated to the behavioral output of interest. Preferential modeling addresses this limitation by prioritizing the identification of latent states that are directly relevant to behavior, yielding more accurate and interpretable models of neural population dynamics. Methods that account for behavior but not external inputs, however, confound input-driven structure with intrinsic behavior-related dynamics, leading to systematically imprecise estimates. This thesis provides a rigorous and self-contained analysis of two preferential subspace identification algorithms for linear neural population dynamics: Preferential Subspace Identification (PSID) and its input-driven extension, Input Preferential Subspace Identification (IPSID). Both algorithms are derived within the linear state-space identification framework of Van Overschee and De Moor, with the projection geometry examined in detail at each stage. Non-preferential baselines, Non-preferential neural Dynamic Modeling (NDM) and its input-driven counterpart INDM, are introduced to contextualize the advantage of preferential identification. The theoretical exposition is complemented by numerical simulations on randomly generated state-space models, which validate the theoretical guarantees of each method. PSID reliably recovers behaviorally relevant dynamics across a diverse set of models, while IPSID achieves more precise eigenvalue recovery and higher behavioral decoding accuracy in the presence of external inputs. Direct comparison confirms that ignoring measured inputs systematically confounds intrinsic and input-driven dynamics, establishing the practical necessity of IPSID in input-driven settings.