Modeling Latent Neural Dynamics with Gaussian Process Switching Linear Dynamical Systems

Summary

This paper introduces a new approach to model the low-dimensional latent dynamics of a collection of neurons over time. Their approach balances the two desiderate of capturing complex nonlinear dynamics while remaining interpretable. Specifically, they introduce the Gaussian Process Switching Linear Dynamical System, which models the latent state as a Gaussian process with a specific novel kernel function, which interpolates between different linear dynamics akin to a switching linear dynamical system. This mitigates some of the pathologies of rSLDS and also provides uncertainty estimates for the dynamics. The authors validate their method both on synthetic and experimental data from neural recordings.

Strengths

This paper tackles an important topic in neuroscience and provides interesting methodological advances continuing a line of research on inferring the latent dynamics of neuronal populations. The paper is overall well-written and provides a concise and clear exposition. For example, Figure 2G very clearly illustrates the problems that recurrent switching linear dynamical systems have. The methodology seems sound to me and is motivated clearly in terms of the issues with rSLDS. The experiments are well-executed and illustrative and importantly also use real data from two animal experiments tackling questions about behavior and decision-making. In addition, the authors provide code for their method and for two of the experiments.

Weaknesses

The biggest weakness the paper currently has in my opinion is that it is hard to judge what the computational trade-offs are between the different methods. While the authors provide estimates for the overall compute they use in all experiments (see Appendix E), there is no comparison between different methods. I would have liked to see an analysis that compares GP-RBF, rSLDS and gpSLDS not only as a function of the number of trials or number of steps simulated forward, but also in terms of their predictive performance as a function of training time / budget. Otherwise, it is hard to judge whether these methods were compared fairly. One aspect that would make the paper stronger, but is not strictly necessary in my opinion, is to add another baseline, which is different from the two approaches that were combined in gpSLDS, e.g. something like LFADS (Sussillo et al, 2016). - Sussillo, D., Jozefowicz, R., Abbott, L., & Pandarinath, C. (2016). LFADS - Latent Factor Analysis via Dynamical Systems. In Advances in Neural Information Processing Systems (NeurIPS)

Questions

- Figure 3: GP with RBF kernel seems to perform better with respect to forward simulation, why do you think that is? - Could you comment on how expensive the different approaches are relative to each other in terms of training and prediction?

Rating

7

Confidence

4

Soundness

4

Presentation

4

Contribution

3

Limitations

While the paper discusses limitations of their method throughout the paper, this is a bit sparse. I think the paper could be improved by a dedicated limitations paragraph in the discussion in Section 6.

Summary

This paper proposes a new model called Gaussian Process Switching Linear Dynamical System (gpSLDS). The model is more interpretable and infers more stable latent compared with the alternative rSLDS. Particularly, the weird oscillations of the latent can be avoided due to the newly proposed Smoothly Switching Linear (SSL) kernel. Extensive experimental results on one synthetic and two real-world datasets shows the benefit of gpSLDS compared with the one with the traditional RBF kernel and the rSLDS.

Strengths

* This proposed method is detailed in math and intuitive with some instructive explanations. * One synthetic and two real-world experiments are done, which is good.

Weaknesses

* Some of the presentations can be improved. * Lacks some complexity analysis or results. This might be an important factor to be considered for long sequences. See questions.

Questions

* Typo: Line 129, function's slope of * Figure 1A, if different colors are different samples (should be clearly state if so), then it is meaningless to show these samples unless kernel parameters are told. * Line 177, $\boldsymbol z_m \in \mathbb{R}^D$? And it seems like this $\boldsymbol z$ is different from the $z_j$. * What does the conjugacy of the model mean in line 193? * How about the effect of choosing $K$ and $J$. At least some explanations or guidance and the corresponding effects are necessary for them. * How about the algorithm complexity. Computing GP is time consuming, which might hinder the computing efficiency for long sequences.

Rating

6

Confidence

4

Soundness

2

Presentation

3

Contribution

3

Limitations

/

Summary

The paper explores latent state inference and parameter learning within a switching stochastic dynamical system. In this context, the dynamics are represented by a stochastic differential equation, with the drift function modeled as a Gaussian process. Notably, the paper introduces a novel kernel for this Gaussian process—a mixture of linear functions that captures the system’s switching behavior. The inference process employs a variational inference framework. To validate its effectiveness, the paper conducts evaluations using both synthetic and real neuroscience data.

Strengths

The proposed method offers a fresh perspective on inference for switching dynamical systems. It introduces a novel Gaussian process (GP) kernel that captures the switching behavior between linear functions. The experimental results are convincing and fair, and the paper is well-written and easy to follow. Additionally, the authors suggest a minor modification to the variational inference (VI) algorithm to achieve faster updates.

Weaknesses

My primary concern with this work lies in how the latent dynamics, denoted as $f$, are modeled as independent draws from the same Gaussian process (GP) prior. In my view, this implies that the posterior over latent components should be independent in each component. Consequently, learning becomes challenging when only single components of the system are observed. This limitation could pose a substantial problem, especially in classical tracking scenarios where only specific state components are observable. Additionally, I perceive the contribution as somewhat marginal. Essentially, the paper introduces a new GP kernel, but beyond that, the impact seems limited. Choice of Linear Kernel: I wonder about the rationale behind using a linear kernel. Why not explore switching between nonlinear kernels? Inference Algorithm Analysis: Lastly, a detailed examination of the variational inference algorithm would provide valuable insights. Additionally, a synthetic example for comparison could help demonstrate the algorithm’s performance. For instance, the authors could select a linear stochastic differential equation (SDE) and perform closed-form parameter estimation using methods like the Expectation Maximization (EM) algorithm. In this scenario, latent state inference could be achieved through RTS smoothing.

Questions

- Could the authors provide further details on how the presented approach could incorporate a multi-dimensional correlated Gaussian process prior?

Rating

6

Confidence

3

Soundness

4

Presentation

3

Contribution

3

Limitations

-

Summary

This paper introduces the Gaussian Process Switching Linear Dynamical System (gpSLDS), a novel approach for modeling latent neural dynamics. This model extends the Gaussian process stochastic differential equations framework by incorporating a new kernel function that supports smoothly interpolated locally linear dynamics. This innovation allows the gpSLDS to maintain the expressiveness needed for complex systems while enhancing interpretability, addressing the limitations of the rSLDS. The paper's contributions include the development of the gpSLDS model, introduction of a novel kernel function to balance expressiveness and interpretability, and a new learning algorithm that improves the accuracy of kernel hyperparameter estimation. The model's effectiveness is shown through applications to both synthetic and real neuroscience data, showing superior performance compared to existing methods like the rSLDS. This advancement provides a more robust framework for understanding neural computation and dynamics.

Strengths

1. The gpSLDS introduces a novel kernel function within the Gaussian process framework that uniquely addresses the trade-off between model expressiveness and interpretability in the analysis of neural dynamics. This model contribution is original in the using of switching dynamics within Gaussian process, allowing for smoothly interpolated transitions between local linear regimes. 2. The methodology presented in the paper is well-developed. The paper proposes a GP-SDE model with a well-conceived kernel that facilitates a locally linear interpretation of complex dynamic behaviors. 3. The paper is well-written with a clear structure that guides the reader through the problem motivation, model formulation, and experiments.

Weaknesses

1. The proposed model effectively transforms a linear dynamical system into a Gaussian Process with a well-designed kernel. While this adaptation supports smoothly interpolated locally linear dynamics, it does compromise computational efficiency, shifting from linear time to cubic time cost.

Questions

N/A

Rating

7

Confidence

3

Soundness

3

Presentation

3

Contribution

3

Limitations

The authors should explore and discuss the limitations of the gpSLDS, including specific scenarios or conditions under which the model may underperform or fail.