Introduction
A central goal in computational neuroscience is to develop models capable of discovering latent structure within noisy, high-dimensional neural signals. By identifying hidden relationships within neural recordings, we can begin to understand, predict, and control the behaviors of the underlying systems. Modeling neural time-series is challenging due to the range of temporal dynamics present. For example, there may be gradual short-term fluctuations, abrupt shifts in response to external stimuli, and long-term global drifts resulting from changes in baseline activity levels [ 32 , 39 , 35 , 10 , 6 ] .

Although black-box approaches based on deep learning are available [ 30 , 17 , 34 ] , their complexity often obscures the relationships learned from the data, making it difficult to extract scientific insights from these models. As a result, practitioners may favor time-varying linear state-space models which offer mathematically interpretable representations by approximating complex dynamics with simple locally linear regimes [ 22 ] . However, obtaining latent variable estimates that are robust to dynamical noise and system nonlinearity in these state-space models is challenging. When applied to neural time-series, latent variable estimation may become unstable due to inflexible model formulations or noise-sensitive inference procedures. This can incorrectly produce disparate results for signals generated from the same underlying system.

For example, switching linear dynamical systems (SLDS) and related models [ 38 , 40 ] segment time-series into discrete linear dynamical states, providing a piecewise linear approximation of the underlying system while highlighting coherent groups of activity. However, the assumption of discrete components can be a poor modeling choice for neural signals that contain continuous-valued fluctuations, such as gradual or random changes to the system speed as seen during neural ramping activity [ 27 ] or Lévy walk dynamics in the cerebral cortex [ 23 ] .
As demonstrated in [ 26 ] and our experiments in Section 4.3 , when applied to real-world datasets, inference of the switching variables can result in rapid, random oscillations between the discrete modes, indicating that the model is unable to identify meaningful structure in the data.

To address the limitations of discrete states, the decomposed linear dynamical systems (dLDS) model [ 26 ] learns a dictionary set of linear dynamical regimes, referred to as dynamic operators (DOs), that can be modified and combined through a linear combination of sparse coefficients. By allowing coefficients to be time-varying and continuous-valued, dLDS naturally captures both gradual changes by adjusting the coefficient magnitudes and abrupt shifts by changing the set of active DOs over time.
Inference is accomplished by optimizing over a cost function that encourages data reconstruction while also constraining the structure of the dynamic coefficients to be sparse and temporally smooth.

Unfortunately, there are two critical shortcomings that prevent the robust inference of the latent variables in dLDS. First, the cost-based inference procedure is sensitive to noise, because of the regularization term encouraging temporal smoothness. This term sequentially propagates errors from noisy coefficient estimates over the length of the time series. Consequently, the model may produce inconsistent coefficient estimates on similar signals and have poor multi-step inference performance, indicating that the learned dynamics are unable to generalize well beyond a single time step. Second, the original latent dynamics model lacks a method for accurately representing systems with multiple fixed points, causing DO coefficients to oscillate or switch arbitrarily in a way that may not align with the underlying process. To learn an effective decomposed model in practice, we require a strategy that provides robust estimates of latent variables despite the presence of noise and system nonlinearity.

In this work, we address these limitations by introducing the probabilistic decomposed linear dynamical systems (p-dLDS) model. Our approach improves robustness of latent variable estimation while maintaining the richness of a decomposed dynamical systems model.
First, we propose a probabilistic inference procedure that reduces the model’s sensitivity to temporal noise by accounting for uncertainty in the latent variable estimates over time. Namely, we introduce time-informed hierarchical variables that encourage both sparse and smooth model coefficients. We devise a variational expectation maximization (vEM) procedure to perform inference and learning over this probabilistic structure. Second, we incorporate a time-varying offset term to model systems that orbit multiple fixed points. While we analytically identify model degeneracies with this formulation, we propose an additive decomposition strategy that prevents convergence to trivial solutions.

Through several synthetic examples, we demonstrate how these contributions lead to improved accuracy and robustness of latent variable estimation despite difficult noise conditions. We extend these results to an empirically-derived brain-computer interface experiment [ 7 ] , showcasing robustness to highly nonlinear observation functions and the ability to extract meaningful insights from the learned latent variables. Finally, we illustrate how our method effectively identifies interpretable and coherent structure in a clinical neurophysiology dataset where previous models are unsuccessful.