Modeling Latent Neural Dynamics with Gaussian Process Switching Linear Dynamical Systems (2408.03330v2)

Abstract: Understanding how the collective activity of neural populations relates to computation and ultimately behavior is a key goal in neuroscience. To this end, statistical methods which describe high-dimensional neural time series in terms of low-dimensional latent dynamics have played a fundamental role in characterizing neural systems. Yet, what constitutes a successful method involves two opposing criteria: (1) methods should be expressive enough to capture complex nonlinear dynamics, and (2) they should maintain a notion of interpretability often only warranted by simpler linear models. In this paper, we develop an approach that balances these two objectives: the Gaussian Process Switching Linear Dynamical System (gpSLDS). Our method builds on previous work modeling the latent state evolution via a stochastic differential equation whose nonlinear dynamics are described by a Gaussian process (GP-SDEs). We propose a novel kernel function which enforces smoothly interpolated locally linear dynamics, and therefore expresses flexible -- yet interpretable -- dynamics akin to those of recurrent switching linear dynamical systems (rSLDS). Our approach resolves key limitations of the rSLDS such as artifactual oscillations in dynamics near discrete state boundaries, while also providing posterior uncertainty estimates of the dynamics. To fit our models, we leverage a modified learning objective which improves the estimation accuracy of kernel hyperparameters compared to previous GP-SDE fitting approaches. We apply our method to synthetic data and data recorded in two neuroscience experiments and demonstrate favorable performance in comparison to the rSLDS.

• The paper introduces the gpSLDS framework that combines Gaussian processes with switching linear dynamical systems to effectively decode complex latent neural dynamics.
• It employs a novel continuous-time kernel and a refined variational expectation-maximization algorithm to overcome oscillatory behaviors at regime boundaries.
• Numerical experiments on synthetic and real neural data demonstrate that gpSLDS achieves lower mean squared error, offering a more accurate and interpretable model.

Modeling Latent Neural Dynamics with Gaussian Process Switching Linear Dynamical Systems

The paper "Modeling Latent Neural Dynamics with Gaussian Process Switching Linear Dynamical Systems" presents an advanced statistical approach designed to decode the complex dynamics of neural systems. A key objective of neuroscience is to understand the relationship between the activity of neural populations and behavior, driving the need for models that can effectively capture and interpret high-dimensional neural data through lower-dimensional latent representations. This paper introduces a framework termed the Gaussian Process Switching Linear Dynamical System (gpSLDS) that endeavors to strike a balance between expressivity and interpretability—a duality inherent to modeling neural dynamics.

The gpSLDS represents an evolution from the recurrent switching linear dynamical systems (rSLDS) by integrating the flexibility of Gaussian processes (GPs) with the piecewise linear qualities of the rSLDS. The core advancement here is the development of a novel Gaussian process kernel that expresses smoothly interpolated locally linear dynamics. This kernel facilitates the modeling of neural systems as locally linear, avoiding the issues of oscillations and discontinuities typical of previous methods like the rSLDS. With this approach, the authors claim to achieve both nuanced approximation of nonlinear neural dynamics and meaningful interpretability, essential for neuroscientific insights.

Technical Contributions and Numerical Results

The gpSLDS builds on continuous-time stochastic differential equations (SDEs), leveraging GPs to model the evolution of latent states. The distinctive feature of this approach is the novel kernel function, enabling smooth transitions between linear dynamical regimes. This characteristic is crucial in overcoming the limitations of rSLDS, which can exhibit undesirable oscillatory behavior at the boundaries between different dynamical regimes.

The authors outline a new variational expectation-maximization (vEM) algorithm for inference, which they modify to enhance the accuracy of kernel hyperparameter estimation. They demonstrate their approach on both synthetic datasets and real neural data from two distinct experiments. Numerical results indicate that gpSLDS offers superior performance in recovering true latent dynamics compared to rSLDS and standard GP-SDE models, as evidenced by lower mean squared error (MSE) metrics. These results substantiate the potential of gpSLDS as a powerful tool for capturing neural dynamics.

Implications and Future Directions

From a theoretical perspective, gpSLDS represents an innovative synthesis of probabilistic modeling approaches, enhancing our ability to model complex dynamical systems in neuroscience in a more interpretable manner. By introducing a structured prior on nonlinear dynamics, the authors present a model that fills a crucial gap between flexibility and interpretability.

Practically, this approach can significantly impact the way neural data is analyzed, allowing researchers to identify and interpret latent dynamics from high-dimensional observations reliably. These capabilities are invaluable for exploring neuroscience questions, such as understanding how cognitive states emerge from neural interactions or how the brain navigates non-linear computations during decision-making tasks.

Looking forward, the approach could be extended or enhanced by considering more sophisticated structures within the latent space, possibly integrating elements from neural differential equations which have demonstrated success in other domains. Additionally, exploring the gpSLDS framework's application to diverse neural datasets will help to further validate its general utility.

In summary, the paper presents a highly structured and robust method for modeling neural dynamics using the GP framework, offering promising insights into both the latent mechanisms of neural computation and methodological advances in machine learning applications to neuroscience.