reviews: 
Comments: The authors develop a model that stabilizes neural dynamics across recording sessions from multiple days. The authors first explain why this is an issue in real-life scenarios, and then develop a model that significantly improves decoding performance when transferred from one day to new days.
Strengths: I think the authors do a good job explaining what types of methods exist to tackle this problem, from models that perform alignment on the neural manifold to methods that align data in the original data space. Moreover, the authors compare their model against good baselines, and the results are extremely encouraging. I have some comments about the results, but if those are addressed the contribution can be very big, given the importance of the problem the authors try to solve in this paper. I think the paper is relatively original, but will be helped if the authors improve its clarity.
Weaknesses: Major weakness(es): \
I believe there are a few issues that together form a large reason why I would not currently recommend acceptance of this paper. First, the authors should improve the structure of the paper, while reading it I felt like the authors jumped back and forth between topics in the paper, e.g, the introduction and related work have a lot of overlap, but read disjointed. Additionally, I think the authors do not explain their intuition behind certain model choices enough, it may not be clear to readers why the authors are decomposing their dynamical latent space for example. Similarly, the theoretical verification of dynamical feature stability does not seem to be one of the main contributions of the paper, so I would move this section to the Appendix and instead focus more on how the authors are measuring stability, which is new and original for papers that try to stabilize neural dynamics across sessions. In terms of writing clarity I would lastly recommend that the authors improve their mathematical notation, the authors often do not use colloquial names but math symbols when referring to components in their model which is confusing, and the math symbols themselves are sometimes also confusing, e.g. the authors use $\mathbf{Z}, \mathbf{\tilde{Z}}, \mathbf{\tilde{Z}}_d^T, \mathbf{Z^S}, \mathbf{\tilde{Z}}_y^S $, and more all in a short section of text (L277-312). Additionally, the authors use $ \cdot^T $ as notation (e.g. $ X^T$, which is confusing because the T is normally used as a transpose operator. I think this paper could be improved tremendously by moving much of the math to the appendix (also because some of the equations are adapted from previous work, e.g. equations 1, 3, 4 are adapted from work the authors cite), and focusing on motivations for components, see [1] Section 3.3 for some tips on improvements. An intuitive explanation of many of the model’s components will greatly improve the readability of the paper and although it is good to adapt math from previous work and use it in your paper, in its current form the notation/equations reduce the clarity of the work.

Minor weaknesses:
1. The authors mention that their model is causal and thus better for real-time BCI applications. However, their model requires training data from a future day to be applicable to that day, so I don’t see how this would lead to real-time application of the current model. Specifically, the authors mention on Lines 380-382 that they use 20% of the sessions of the day that they want to transfer to as a test set. This means that without first training on the 80% of the unlabeled data on that day, the model cannot do any decoding yet. This in itself is not an issue, but one of the claims of the paper in the introduction is that “… we have the potential to meet the real-time operational requirements of BCIs”, which is a claim I would weaken given that atleast at the start of the day, the model requires training data to align first. 
2. The authors do not mention or compare to relevant work in neural population decoding [2]
3. L271-272, the authors should cite disentanglement works, e.g. [3, 4, 5] for the independence claim.
4. The authors should explain their intuition for using Chi-squared divergence more, not every reader may be familiar with Chi-squared divergence. Additionally, the authors should explain why they believe it is more accurate in measuring distribution discrepancies, since although Figure 7B (as mentioned on Line 199) does show that the author’s alignment method performs better, it is not immediately clear why this is the case and how other divergence methods compare to Chi-squared divergence.
5. Instead of using training time the authors should also (and more importantly) report the number of FLOPs since training time is very dependent on differences in hardware.
6. The authors do not compare their results with other methods for Figure 6.


Spelling/grammar:
- L144/145: “… over time as an UDA…” -> a UDA
- L175-177: “The aligned neural signals are then serve as...” -> are then provided
- L192: “This identification may help for aligning…” -> may help to align
- L245-246: “… can provide the dynamical system with an more efficient…” -> a more efficient

[1] Lipton, Z. C., & Steinhardt, J. (2019). Troubling Trends in Machine Learning Scholarship: Some ML papers suffer from flaws that could mislead the public and stymie future research. Queue, 17(1), 45-77. \
[2] Azabou, M., Arora, V., Ganesh, V., Mao, X., Nachimuthu, S., Mendelson, M., ... & Dyer, E. (2024). A unified, scalable framework for neural population decoding. Advances in Neural Information Processing Systems, 36. \
[3] Higgins, I., Matthey, L., Pal, A., Burgess, C. P., Glorot, X., Botvinick, M. M., ... & Lerchner, A. (2017). beta-vae: Learning basic visual concepts with a constrained variational framework. ICLR (Poster), 3. \
[4] Burgess, C. P., Higgins, I., Pal, A., Matthey, L., Watters, N., Desjardins, G., & Lerchner, A. (2018). Understanding disentangling in $\beta $-VAE. arXiv preprint arXiv:1804.03599. \
[5] Higgins, I., Amos, D., Pfau, D., Racaniere, S., Matthey, L., Rezende, D., & Lerchner, A. (2018). Towards a definition of disentangled representations. arXiv preprint arXiv:1812.02230.
Recommendation: 5
Reviewer confidence: 4
Soundness/correctness: 3
Clarity: 2