On Conformal Isometry of Grid Cells: Learning Distance-Preserving Position Embedding

Q3: Why might the neural data satisfy the conformal isometry over a larger scale than the simulations?

As explained in our reply to W5, the conformal isometry relationship allow different scaling factor $s$. For small $s$, it is satisfied for a large range of $\|\Delta x\|$. For big $s$, it is satisfied for a smaller range of $\|\Delta x\|$. Moreover, we can also choose different side length for the square domain. Thus the freedom of choosing $s$ and the side length causes the apparent difference between neural data and simulations.

Q4: Which modules was analyzed in Fig. 5 and Fig. 11, and was any pre-processing performed on them?

In Figure 5 (initial submission), we use rat ‘R’ day 1, module 1; and in Figure 11 (initial submission), we use rat ‘R’ day 1, module 3. There is no more pre-processing besides reading the neural data following the first chunk of https://github.com/erikher/GridCellTorus/blob/main/Analysis%20-%20Number%20of%20PCs.ipynb.

Finally, thank you for the thoughtful feedback, which has greatly enhanced the quality and clarity of our work.

References

[1] Langston, Rosamund F., James A. Ainge, Jonathan J. Couey, Cathrin B. Canto, Tale L. Bjerknes, Menno P. Witter, Edvard I. Moser, and May-Britt Moser. "Development of the spatial representation system in the rat." Science 328, no. 5985 (2010): 1576-1580.

[2] Sargolini, Francesca, Marianne Fyhn, Torkel Hafting, Bruce L. McNaughton, Menno P. Witter, May-Britt Moser, and Edvard I. Moser. "Conjunctive representation of position, direction, and velocity in entorhinal cortex." Science 312, no. 5774 (2006): 758-762.

W3(minor points): What is meant by " $\Delta x$ is constrained to be smaller than 3 grids?" By this do you mean 3 grids of the underlying space discretization? Maybe using a different word would be helpful to keep the reader from confusing this with grid cells.

Thanks for the question. Following your suggestion, we have updated the manuscript to avoid potential confusion with the terminology "grid." Instead of "3 grids," we now explicitly describe this range in terms of the corresponding physical displacement, such as 3/40 = 0.075 of the side length of the square domain, to provide better clarity.

W4(minor points): What is valid rate that is reported in Table 1?

The criteria of valid rate, which are adopted from [1,2], is the valid percentage of grid cells with gridness score > 0.37. This brings another perspective of goodness of learned grid patterns and ours achieve almost perfect hexagonal patterns for each learned cell.

W5(minor points): The real neural data analyzed by the authors satisfies conformal isometry (Fig. 5 - up to $\|\Delta x\|=0.2$) for a much larger scale than the grid cells generated by the authors' models (Fig. 4 - up to $\|\Delta x\|=0.08$). Why do the authors think this is? Could this be due to the Gardner et al. data having variability in grid properties (Redman et al., 2024) not present in the authors' model?

The conformal isometry relationship allow different scaling factor $s$. For small $s$, conformal isometry is satisfied for a large range of $\|\Delta x\|$ with low resolution. For big $s$, it is satisfied for a smaller range of $\|\Delta x\|$, but with high resolution. In our Fig. 4, we use $s = 10$ while assuming the side length of the square doain is 1. Both $s$ and the side length can vary. So the two numbers $.2$ in Fig. 5 and $.08$ in Fig. 4 do not necessary correspond to each other due to the freedom in the choice of $s$ and the side length.

W6(minor points): Which module from the Garnder et al. data set was analyzed? If I remember correctly, none of the modules had 93 grid cells. Did the authors perform some screening on which grid cells to include?

Thanks for asking about the real data. In the main manuscript, Figure 5 shows the results of rat ‘R’ day 1, module 1, which include the activations of 93 cells. We used the data from (https://figshare.com/articles/dataset/Toroidal_topology_of_population_activity_in_grid_cells/16764508?file=35078602) and follow the data processing method from this repo (https://github.com/erikher/GridCellTorus).

Q1: Why is the hexagonal torus optimal (as compared to other flat tori)?

As shown in Section 4, and as explained in our reply to W1 above, the hexagonal torus has isotropic $D(\Delta x)$, which measures extrinsic curvature. Figure 6 in Appendix shows that there are five types of 2D periodic lattice. Only hexagon lattice has six fold symmetry. Other flat tori do not have fix-fold symmetry and thus do not have isotropic $D(\Delta x)$. We then show that for fixed average extrinsic curvature, the hexagon flat torus achieves minimum average deviation from local flatness due to Cauchy-Schwarz inequality. The isotropy property of hexagon torus underlies its minimum deviation from local flatness.

Q2: How does having multiple grid modules affect the conformal isometry?

As explained in our reply to W2, for multiple grid modules, each module satisfies conformal isometry with its own scaling factor $s$. This enables the agent to represent the spatial relationships at different scales.

We appreciate your thoughtful and insightful feedback, which has greatly helped us improve our paper.

W1: While the authors made it clear why a hexagonal torus is local isotropic, it was not clear to me when/if the authors ever showed it was optimal because it "achieve the least deviation from local flatness because it distributes the extrinsic curvature evenly over all directions" (Sec. 4.4). Providing more discussion on this (and explaining why other flat tori do not distribute extrinsic curvature evenly) would make their point of optimality more clear.

Thank you for this insightful question. Flat tori come with different sizes or surface areas, which are related to their average extrinsic curvatures. When comparing flatness of different tori, we should fix the size or average extrinsic curvature for fair comparison. Specifically, we show that for any fixed average extrinsic curvature $\int D(\Delta x) d \theta$, the overall deviation from local flatness $L(\Delta r) = \int (\|v(x+\Delta x) - v(x)\|^2 - \|\Delta x\|^2)^2 d\theta = \frac{1}{12^2} \int (D(\Delta x))^2 d\theta$ is minimized if $D(\Delta x)$ is constant over $\theta$. This is a result of Cauchy-Schwarz inequality.

After establishing isotropy of $D(\Delta x)$ for hexagon torus, we added explanation that other flat tori do not have six fold symmetry thus do not have isotropic $D(\Delta x)$. Figure 6 in Appendix shows that there are five types of 2D periodic lattice. Only hexagon lattice has six fold symmetry. For instance, square torus has 4-fold symmetry, and its $D(\Delta x)$ is not isotropic. An example of square torus is Clifford torus in 4D, $v(x) = (\cos x_1, \sin x_1, \cos x_2, \sin x_2)$, with $D(\Delta x) \propto (\cos^4 \theta + \sin^4\theta) \Delta r^4$, which is not isotropic over $\theta$. We have added the above explanation to Section 4 following your advice.

W2: While it was interesting to see that the framework developed by the authors could be expanded to include the generation of multiple modules (that were organized approximately $\sqrt{2}$ apart in grid spacing), how exactly the multiple modules tied in to the idea of conformal isometry was lost for me. In an otherwise well structured paper, it felt a little like it was tacked on at the end and the thread was lost. Providing more discussion on how having multiple modules, in addition to the need to decode place responses, affects the kinds of representations that lead to conformal isometry, would be helpful.

Thank you for this deeply insightful comment. We completely agree with you about Section 5 of our initial submission. Based on your feedback and the suggestion of Reviewer hz9p, we have taken out the key ideas and concepts in Section 5 and merge them into Section 4 as subsections. We then move the integrated model with place cells to Appendix J, partially due to space limit, but more importantly as a response to your impression that the original Section 5 feels like it was tacked on at the end and the thread was lost.

In the setting of multiple modules, each module satisfies conformal isometry with its own scaling factor $s$. The scaling factor $s$ determines the scale or resolution of the spatial representation. For small $s$, conformal isometry holds within a big range, but the module has low resolution because $v(x+\Delta x)$ and $v(x)$ are close to each other while the neuron activities are noisy with low precision. For big $s$, conformal isometry holds within a small range, but the module has high resolution because $v(x+\Delta x)$ and $v(x)$ are farther apart.

Conformal isometry is of cricial importance for path planning. For instace, planning a straight path in the open field can be accomplished easily by steepest descent on the distance between the neural representations of the target position and the current position on the path. Meanwhile, path planning has to be done in a proper resolution $s$. For instance, in playing golf, the first stroke and the final putt require very different $s$. Thus it is important to have multiple modules of different resolutions.

Decoding the self-position of the agent via place cells modeled by Gaussian kernel functions requires multiple modules, with modules of low resolutions providing the approximate global position, while models of high resolutions providing more accurate localization (a single module of big $s$ has ambiguity in position due to the periodic pattern, i.e., the same $v$ correspond to multiple locations).

We sincerely thank you for your deep insight into the real neural data. We are honored to receive guidance from an expert who not only understands theortical anlysis, but also familiar with the real neural data. We believe your suggestions have substantially improved the quality of our paper.

I now understand the authors point about it being hard to directly compare Fig. 4 with the real data in Fig. 5a, given that can vary. However, given that is connected to spatial scale, could the authors use the measured grid spacing in the real neural data to find a good value for $s$ and see if it better matches the range of isometry observed in the real data.

We appreciate the reviewer's insightful question! Indeed we agree with the reviewer that $s$ is related to the spatial grid spacing, and in response, we have conducted quantitative analysis using real neural recordings to further compare real neural data and numerical simulation. We have added a new Appendix K.2 on this quantitative comparison.

To align with the settings in our simulation, we first normalize the side length of the square domain (1.5 m) in the neural data to a unit length of 1, which is used in our numerical simulation. Based on the scale of the grid module (rat R, module 1, day 1) reported in [1], the spatial grid spacing is 0.5 m, which corresponds to 0.33 spacing after normalization (0.5m/1.5m). We also normalize $||v||=1$ in the neural data, noting that this pre-processing step was not applied in the original results shown in Figure 5.

To quantitatively examine the scaling factor $s$ in the normalized neural data, we first fit a linear relationship between $||v(x+\Delta x) - v(x)||$ (after normalization) vs $||\Delta x||$, with the intercept fixed at 0. As detailed in Appendix K.2, the fitted scaling factor is $s=13.4$. In Table 2 of our paper, we show that the estimated spatial grid spacing of the learned patterns is inversely proportional to $s$. Specifically, the product of the scaling factor and the estimated grid spacing in the real neural data ($13.4\times0.33=4.42$) is comparable to the result in numerical simulation ($10\times0.41=4.1$).

To further examine the local conformal isometry, we fit a quadratic curve to the data, similar to the approach in Section 3.4. The deviation arise as $||\Delta x||$ is larger than $0.08$, and this result is in agreement with the result in our numerical simulation.

In summary, the real neural data is in agreement with results in our numerical simulation in terms of the relationship between the scaling factor $s$ and the estimated grid spacing, as well as the range of conformal isometry observed. We believe this quantitative analysis strengthens the connection between our findings and the real neural data. Thank you again for raising this important point!

I believe it would be useful to explicitly note which module the authors analyzed in paper (I did not see it anywhere in the revised manuscript, but I may have missed it somewhere). Lastly, (and very minorly) but I think it would be good to note in the paper that the authors only analyzed "pure" grid cells - rat R module 1 day 1 had a number of conjunctive grid + head direction cells, which the authors did not analyze (which is not bad, just worth noting).

Thank you for the detailed and thoughtful guidance! We have revised the paper and added the details of the data in Appendix K.1 of the revised manuscript.

Reference

[1] Gardner, Richard J., Erik Hermansen, Marius Pachitariu, Yoram Burak, Nils A. Baas, Benjamin A. Dunn, May-Britt Moser, and Edvard I. Moser. "Toroidal topology of population activity in grid cells." Nature 602, no. 7895 (2022): 123-128.

We appreciate your insightful feedback and suggestions! Based on your suggestions, we conducted additional experiments and revised our manuscript.

W1: As far as I can tell, there is no normative explanation or hypothesis that would explain why the neural manifold encoding 2D space ought to be a isometric embedding of physical space. What are the potential benefits of this coding scheme? Absent any functional benefit, it is hard to interpret the significance of these results. Is it just a coincidence that this model gives the same grid coding scheme as real networks? Or is there a deeper reason why biological and artificial networks will converge upon a conformal map of space?

We sincerely appreciate your deeply insightful question regarding the normative explanation and functional benefits of conformal isometry.

Conformal isometry provides a normative framework by ensuring that the brain's spatial representation preserves the geometry of local environment, where distances are preserved with a scaling factor $s$. In term of functional benefit, this distance-preserving property can be critical for path planning. For instance, planning a straight path in an open field can be easily accomplished by following the steepest descent on the neural representation of distance between the target position and the current position on the path. Similarly, this geometry-preserving property can also facilitate the planning of more complex paths involving obstacles and landmarks by maintaining accurate spatial relationships. Multiple grid cell modules with different scaling factors $s$ further allow path planning at differnt spatial scales or resolutions, adapting to varying navigation demands. We will study path planning in future work.

We have revised our manuscript to address your deep question. In particular, we have added a new section 6 to discuss conformal isometry for path planning.

W2: Sections 4.2 - 4.4 are confusing. Because there are three sections, it feels as though the authors are trying to list three different results, but I think all three of these sections are really just supporting the same result (i.e. that hexagonal flat torus results in minimal deviation from local isometry). I suggest that the authors merge these three subsections into one. Furthermore, it would be easier to read if the main result were formally presented as a proposition and, potentially, if the result in section 4.3 were formally presented as a lemma. Overall, I would suggest re-writing this portion of the paper to make the motivation more clear.

Thank you for this great advice. Following your advice, we have merged these three subsections into one (see the new Section 4 of the revised manuscript). Also following your advice, we have formally presented the results as propositions and theorems. We have also re-written this portion to make the motivation more clear. This indeed improves the readability of this portion of the paper.

W3: It would be useful to investigate how important assumption 3 (normalization) is to the results. Figure 3f shows that the hexagonal grids do not form when normalization is removed entirely. Real biological networks are likely normalized imperfectly. Is imperfect normalization sufficient to get hexagonal grids? That is, what if $1-\epsilon\leq\|v(x)\|\leq1+\epsilon$ for some $\epsilon>0$? How big can get before the grids fail to form?

Thank you for the valuable suggestion! To further investigate normalization constraint, we have added experiments for imperfect normalization, presented in Appendix I.2. The results indicate that hexagonal patterns are robust to mild deviations from normalization but collapse as $\epsilon$ becomes too large, highlighting the importance of approximate normalization for forming the patterns stably.

W4: There is a question about whether ICLR is the optimal venue for this kind of work, which I think will be primarily interesting to experimental and theoretical neuroscientists. I think this work will have little to no cross-over appeal to machine learning or artificial intelligence communities.

Thank you for raising this important point. We believe that ICLR is not solely focused on artificial intelligence (AI) and machine learning (ML). It emphasizes representation learning, a theme that spans multiple disciplines, including neuroscience, cognitive science, and computational biology. Our work investigates the geometric principles underlying spatial representations in the brain, contributing directly to understanding how neural systems encode information, leading to insights that inform and inspire advancements in representation learning.

Grid cell-related research has already found a home at ICLR and very similar conferences such as NeurIPS, bridging the neuroscience and ML communities. Examples include:

1. Cueva, Christopher J., and Xue-Xin Wei. "Emergence of grid-like representations by training recurrent neural networks to perform spatial localization." ICLR (2018).
2. Gao, Ruiqi, Jianwen Xie, Song-Chun Zhu, and Ying Nian Wu. "Learning grid cells as vector representation of self-position coupled with matrix representation of self-motion." ICLR (2019).
3. Dorrell, William, Peter E. Latham, Timothy EJ Behrens, and James CR Whittington. "Actionable neural representations: Grid cells from minimal constraints." ICLR (2023).

Related studies presented at NeurIPS, including the papers shown below, also made signifcant contributions exploring neural representations.

4. Sorscher, Ben, Gabriel Mel, Surya Ganguli, and Samuel Ocko. "A unified theory for the origin of grid cells through the lens of pattern formation." NeurIPS (2019).
5. Gao, Ruiqi, Jianwen Xie, Xue-Xin Wei, Song-Chun Zhu, and Ying Nian Wu. "On path integration of grid cells: group representation and isotropic scaling." NeurIPS (2021).
6. Schaeffer, Rylan, Mikail Khona, and Ila Fiete. "No free lunch from deep learning in neuroscience: A case study through models of the entorhinal-hippocampal circuit." NeurIPS (2022).
7. Schaeffer, Rylan, Mikail Khona, Tzuhsuan Ma, Cristobal Eyzaguirre, Sanmi Koyejo, and Ila Fiete. "Self-supervised learning of representations for space generates multi-modular grid cells." NeurIPS (2024).

Our paper aligns with ICLR’s mission by focusing on learning representations, a core theme of ICLR. By exploring the conformal isometry hypothesis, we offer a biologically inspired framework for efficient and robust spatial encoding, relevant to both neuroscience (normative explanations for grid patterns) and machine learning (novel representation models).

Q1: As mentioned above, I would like to know whether there is a normative explanation or theory behind the conformal hypothesis of grid cell coding. Is there a potential benefit (e.g. in terms of energy efficiency or decoding accuracy) for this kind of representation?

Thank you for this profound question, which we addressed above in our reply to W1. Just to summarize, having a neural representation that perseves the geometry of the local environment at multiple scales can be of critical importance for path planning and navigation.

We have added the above explanation as a new Section 6: Discussion: conformal isometry for path planning, and we shall investigate path planning in our future work. Thank you so much for this crucial question.

Q2: Put differently, is there a justification for why the objective function defined in equations 3 and 4 is a good proxy for the constraints faced by real biological neural systems?

Thanks for the insightful question. Equation (4) ensures that the neural representation is updated as the agent moves in the environment. This is important for path integration, so that the agent is aware of its self-position based on the accumulation of its movements. It is also important for path planning, where the agent plans a sequence of movements from the current position to the target position. Equation (3) ensures that the learned neural representation satisfies conformal isometry to preserve the geometry of the local environment, where the notion of "local" depends on the scaling factor $s$. As mentioned above, such a neural representation is probabably indispensible for path planning in navigation.

Thank you again for your valuable feedback in helping us enhance our paper significantly!

Q1: It would be valuable, however, to see a comparison between the proposed conformal model and other standard models of grid cell activity, as this would provide more context on the advantages or limitations of the proposed approach.

As discussed in our reply to W3, in this work, we demonstrate that:

1. Contrary to the claims in previous works, the emergence of hexagonal grid patterns does not depend on interactions with the place cell system.
2. Many assumptions in prior works, such as specific place cell formulations or task-driven objectives, are not necessary for the emergence of grid patterns.
3. A minimalist framework based on conformal isometry provides a more robust and biologically plausible explanation for grid cell activity.

Q2: Could the authors expand on how this hypothesis might be experimentally tested in biological systems or how it could inspire new architectures in machine learning?

Thanks for these great questions! In the paper, we tested conformal isometry hypothesis in both simulated experiments and real biological data. In section 3.5, we analyze conformal isometry on real neural recordings, and Figure 5(a) shows a clear linear relationship between $\|v(x)- v(x+\Delta x)\|$ and $\|\Delta x\|$. Also, in Figure 5(b), we tested normalization assumption.

In the machine learning field, our learned high-dimensional representations may serve as a positional embedding used by other deep learning models. For example, [7] used grid cell representations in 3D Camera Pose, and such representation are also used for modeling absolute positions and spatial relationships of places in geographic applications in [8].

Finally, we sincerely thank you for your insightful comments and suggestions that have greatly helped us improve our paper.

References

[1] Cueva, Christopher J., and Xue-Xin Wei. "Emergence of grid-like representations by training recurrent neural networks to perform spatial localization." ICLR (2018).

[2] Banino, Andrea, Caswell Barry, Benigno Uria, Charles Blundell, Timothy Lillicrap, Piotr Mirowski, Alexander Pritzel et al. "Vector-based navigation using grid-like representations in artificial agents." Nature 557, no. 7705 (2018): 429-433.

[3] Sorscher, Ben, Gabriel Mel, Surya Ganguli, and Samuel Ocko. "A unified theory for the origin of grid cells through the lens of pattern formation." NeurIPS (2019).

[4] Whittington, James CR, Timothy H. Muller, Shirley Mark, Guifen Chen, Caswell Barry, Neil Burgess, and Timothy EJ Behrens. "The Tolman-Eichenbaum machine: unifying space and relational memory through generalization in the hippocampal formation." Cell 183, no. 5 (2020): 1249-1263.

[5] Nayebi, Aran, Alexander Attinger, Malcolm Campbell, Kiah Hardcastle, Isabel Low, Caitlin S. Mallory, Gabriel Mel et al. "Explaining heterogeneity in medial entorhinal cortex with task-driven neural networks." NeurIPS (2021).

[6] Schaeffer, Rylan, Mikail Khona, and Ila Fiete. "No free lunch from deep learning in neuroscience: A case study through models of the entorhinal-hippocampal circuit." NeurIPS (2022).

[7] Zhu, Yaxuan, Ruiqi Gao, Siyuan Huang, Song-Chun Zhu, and Ying Nian Wu. "Learning neural representation of camera pose with matrix representation of pose shift via view synthesis." In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 9959-9968. 2021.

[8] Mai, Gengchen, Krzysztof Janowicz, Bo Yan, Rui Zhu, Ling Cai, and Ni Lao. "Multi-scale representation learning for spatial feature distributions using grid cells." arXiv preprint arXiv:2003.00824 (2020).

[9] Gao, Ruiqi, Jianwen Xie, Xue-Xin Wei, Song-Chun Zhu, and Ying Nian Wu. "On path integration of grid cells: group representation and isotropic scaling." NeurIPS (2021).

[10] Xu, Dehong, Ruiqi Gao, Wen-Hao Zhang, Xue-Xin Wei, and Ying Nian Wu. "Conformal isometry of lie group representation in recurrent network of grid cells." arXiv preprint arXiv:2210.02684 (2022).

We are grateful for your insightful comments and suggestions! Following your suggestions, we have conducted additional experiments and revised our manuscript accordingly.

W1: A primary limitation of this paper is that the explored neural networks are restricted to ReLU, GELU, and Tanh activation functions. Could the authors consider extending their framework to support a broader range of activation functions?

We thank the reviewer for raising this point. We have added experimental results for Leaky ReLU and Swish activation functions. Please see Appendix I.1 of the revised version for detailed results. These experiments demonstrate that the emergence of hexagonal patterns remains robust across most commonly used activation functions in modern machine learning models, highlighting the generality and flexibility of our framework.

W2: Could the authors clarify the structure of neural network studied in this paper? There is no mention about the number of layers. Additionally, the reader could benefit from explaining why the authors selected the studied activation functions.

Thanks for the question. As discussed in Section 3.1 Assumption 2, we provide 3 different parameterizations of the transformation function $F$, and both the linear and nonlinear models are one layer. We want to demonstrate that the emergence of hexagonal patterns is agnostic to the specific forms of transformation model $F$. No matter which form of transformation (linear or nonlinear) we use, the grid patterns emerge under the conformal isometry, even with the simplest structure of nerual network. We adopted the commonly used activation functions in our initial submission, and as explained above, we have also added experiments with other activation functions.

W3: The paper would benefit from a deeper discussion on how the conformal isometry hypothesis compares to alternative theories of grid cell patterns which are missing in the current submission.

We appreciate the opportunity to clarify the relationship between our paper and prior works.

Understanding grid cells using machine learning models [1-5] is a research topic that is of interest to both machine learning and computational neuroscience communities. Pioneering papers [1,2] adopted recurrent neural networks and used the objective of path integration to learn grid cells, and the models also involved interactions with place cells at the output layer where the place cells are often modeled by predefined functions, e.g. Gaussian tuning curves [2,9,10]. However, hexagon grid patterns do not consistently emerge from the learned models [6]. Thus extra assumptions or hypotheses are necessarily to explain the emergence of hexagon grid patterns.

One hypothesis is to assume non-negativity of the grid cell responses coupled with Difference-of-Gaussians tunning curves for place cells, as in [3,5]. However the biological plausibility of center-surround Difference-of-Gaussian assumption is questioned [6].

Another hypothesis is conformal isometry hypothesis proposed in [9,10]. However, these papers learned multiple modules of grid cells with specific forms of transformation models, and they made extra assumptions about place cells and the interactions between grid cells and place cells.

Our paper is built on [9,10], and we have achieved scientific reductionism by isolating grid cell system of a single module, without making any assumptions about the place cells as well as the interactions between grid cells and place cells. We have also achieved generality by being agonistic of the form of the transformation model. This enables us to focus solely on the conformal isometry hypothesis and demonstrate its crucial role in the emergence of hexagon grid patterns. Moreover, we also provide neuroscience evidence and theoretical evidence for this hypothesis.

In our work, we study conformal isometry hypothesis with explicit scaling factor $s$, whereas in [9,10], $s$ is only implicit in their learned models. $s$ determines the scale or resolution of the spatial representation of a grid cell module.

We also provide theoretical understanding of the role of conformal isometry for the emergence of hexagon grid patterns, by showing that the hexagon flat torus has minimal deviation from local conformal isometry.

We have revised our manuscript in response to your feedback by incorporating the Relation to Past Works into the main text of the paper, now presented in Section 5 of the revised version.

Dear Reviewer 982F,

Thank you again for your insightful suggestions, which have significantly helped strengthen this work! We have tried our best to address your questions (see our rebuttal above), and revised our paper by following suggestions from all reviewers. As the deadline of manuscript revision is approaching soon, we would love to know if you have any follow-up questions or areas needing further clarification, so that we could reflect in the revision timely.

Your insights are very valuable to us, and we stand ready to provide any additional information that could be helpful. If you feel we have adequately addressed your concerns, we kindly request you to consider raising the score.

Best, Authors

Q6: More clarifications are needed for Figure 2. How exactly are Figure 2b(ii) and (iii) constructed?

Thanks for the suggestion. For the topological analysis in Figure 2b, we qualitatively followed the methodology used by [3] and [7] (https://github.com/ganguli-lab/grid-pattern-formation/blob/master/inspect_model.ipynb). Specifically, in Figure 2b(i), we employed spectral embedding [8] and observed that the emergent grid cell population states fell on a 3D toroidal manifold. Then we computed the mean power spectrum of grid cell rate maps by applying the Fourier Transform and averaging the power across samples. In Figure 2b(ii), we visualized the results as a scatter plot, which revealed that cells with similar periods were tightly clustered, exhibiting nearly identical periods and orientations. The results clearly demonstrated that the Fourier power was hexagonally distributed along three principal wavevectors: $k_1, k_2, k_3$. Following the approach of [3], we projected the activity manifold onto three axes defined by the phases calculated for each unit. This projection revealed the presence of three distinct rings, as shown in Figure 2b(iii), confirming that the attractor manifold exhibits a 2D twisted torus topology.

Q7: Is the finding on the proportion of the scale of patterns and the s parameter unique and new from this paper?

Yes, we believe our paper is first to study the $s$ parameter defined in equation (2) for conformal isometry and the scale of hexagonal grid patterns. And as presented in Table 2, the estimated scale of the patterns is proportional to $1/s$, which has not been carefully dicussed in the previous literature.

Q8: What is the impact of $\lambda$ on the presented analysis and results?

The $\lambda$ used in our loss function is a hyper-parameter balancing the scale of two loss terms. Our experiments suggest that the results are not very sensitive to this hyper-parameter.

Finally, we want to express our deep gratitude for your insightful comments, questions and suggestions that have greatly helped us improve our paper.

References

[1] Cueva, Christopher J., and Xue-Xin Wei. "Emergence of grid-like representations by training recurrent neural networks to perform spatial localization." ICLR (2018).

[2] Banino, Andrea, Caswell Barry, Benigno Uria, Charles Blundell, Timothy Lillicrap, Piotr Mirowski, Alexander Pritzel et al. "Vector-based navigation using grid-like representations in artificial agents." Nature 557, no. 7705 (2018): 429-433.

[3] Sorscher, Ben, Gabriel Mel, Surya Ganguli, and Samuel Ocko. "A unified theory for the origin of grid cells through the lens of pattern formation." NeurIPS (2019).

[4] Whittington, James CR, Timothy H. Muller, Shirley Mark, Guifen Chen, Caswell Barry, Neil Burgess, and Timothy EJ Behrens. "The Tolman-Eichenbaum machine: unifying space and relational memory through generalization in the hippocampal formation." Cell 183, no. 5 (2020): 1249-1263.

[5] Nayebi, Aran, Alexander Attinger, Malcolm Campbell, Kiah Hardcastle, Isabel Low, Caitlin S. Mallory, Gabriel Mel et al. "Explaining heterogeneity in medial entorhinal cortex with task-driven neural networks." NeurIPS (2021).

[6] Schaeffer, Rylan, Mikail Khona, and Ila Fiete. "No free lunch from deep learning in neuroscience: A case study through models of the entorhinal-hippocampal circuit." NeurIPS (2022).

[7] Schaeffer, Rylan, Mikail Khona, Tzuhsuan Ma, Cristobal Eyzaguirre, Sanmi Koyejo, and Ila Fiete. "Self-supervised learning of representations for space generates multi-modular grid cells." NeurIPS (2024).

[8] Saul, Lawrence K., Kilian Q. Weinberger, Fei Sha, Jihun Ham, and Daniel D. Lee. "Spectral methods for dimensionality reduction." (2006).

[9] Gao, Ruiqi, Jianwen Xie, Xue-Xin Wei, Song-Chun Zhu, and Ying Nian Wu. "On path integration of grid cells: group representation and isotropic scaling." NeurIPS (2021).

[10] Xu, Dehong, Ruiqi Gao, Wen-Hao Zhang, Xue-Xin Wei, and Ying Nian Wu. "Conformal isometry of lie group representation in recurrent network of grid cells." arXiv preprint arXiv:2210.02684 (2022).

Q3: It was reported briefly how this work differs from Gao et al. 's. (2021) and Xu et al. (2022) on their study on conformal isometry, please discuss the importance of the difference. Why the proposed approach of only modeling the grid cells is better than the previously studied combined model?

Unlike Gao et al. (2021) and Xu et al. (2022), which rely on combined models involving both grid cells and place cells, our work isolates the grid cell system and focuses solely on its internal dynamics. By removing dependencies on place cells, isotropic Gaussian kernels, and interactions between grid cells and place cells, we focus solely on the intrinsic properties of grid cells and the role of conformal isometry.

Our model is agnostic to the specific form of the recurrent transformation $F$, while Gao et al. (2021) only study a specific neural architecture. This flexibility demonstrates that the emergence of hexagonal patterns is robust across various transformation models, suggesting that conformal isometry is a more fundamental principle.

Additionally, our study contributes a theoretical understanding of why hexagonal patterns arise naturally in grid cells by minimizing deviation from local conformal isometry. This theortical analysis, detailed in Section 4, offers a deeper geometric interpretation of grid cell patterns, which is not touched in Gao et al. (2021) and Xu et al. (2022).

Furthermore, as mentioned above, we study the conformal isometry hypothesis with explicit scaling factor $s$, while it is only implicit in the learned models in Gao et al. (2021) and Xu et al. (2022). This explicit treatment offers greater clarity regarding how spatial resolution and range are governed within the hypothesis.

Therefore, we believe that our work achieves scientific reductionism and generality compared to in Gao et al. (2021) and Xu et al. (2022), and it also provides deeper theoretical understanding of the conformal isometry hypothesis and its connection to hexagon grid patterns.

Q4: Could the authors please explain the choice of 1.25 and 3 in the experiment design in Section 3.3.

Thanks for the insightful question. We assume the one-step displacement in the agent's motion is less than 3 grids, i.e., 3/40 = 0.075 of the side length of the square domain. We assume the transformation model holds within this range. We have updated the manuscript to avoid potential confusion.

The value 1.25 specifies the range of $s \|\Delta x\|$ within which the conformal isometry property holds with minimal deviation. Figure 4 in Section 3.4 of our initial submission shows $\|v(x+\Delta x) - v(x)\|$ versus $\|\Delta x\|$ for $s = 10$. It can be seen that conformal isometry holds nearly exactly within the range $s \|\Delta x\| \leq .8$, and deviation starts to kick in beyond this point.

To examine the impact of varying the range of $s \|\Delta x\|$, we conducted additional experiments by increasing and decreasing the range. As shown in Appendix I.3, we varied the range and found that the emergence of hexagonal patterns remains stable.

Q5: Three models are chosen, showing that the grid pattern still emerges. Can the authors clarify if their results are fully independent of the choice of such models?

Thanks for pointing this out. From the theortical perspective, we hope to illustrate that the emergence of hexagon grid patterns is agnostic to the parametrization of the transformation model $F$. Our theoretical results in Section 4 do not depend on specific form of the transformation model. And in the empirical experiments, to show this generality on transformation model $F$, we choose 3 different models including linear and nonlinear ones with various of activation functions, such as ReLU, GeLU and Tanh. Additionally, we have added experimental results for Leaky ReLU and Swish activation functions. Please see Appendix I.1 of the revised version for detailed results. All of these models can lead to the hexagonal patterns. Figure 2(a) shows the major results of linear models and Figure 3(b-d) shows the learned patterns in two different nonlinear models with different activations.

Q1: It would be nice to know the consequence of their model assumptions on what we know about neural representations in the brain; their model seems to have removed some assumptions that the literature seems to think are needed for grid cells to arise.

Thank you for the insightful question. As explained above, our work focuses on the minimalistic setting of a single grid cell module that satisfies conformal isometry at a specific scale, and we show that hexagon grid patterns naturally emerge in this minimalistic setting.

The conformal isometry hypothesis may have profound consequence on our understanding of neural representation in the grid cell system. The hypothesis suggests that spatial representation in the brain preserves the geometry of local environment, where distances are preserved with a scaling factor. This distance-preserving property can be essential for path planning. For example, planning a straight path can be easily accomplished by following the steepest descent on the distance between the neural representations of the target position and the current position. Additionally, such geometry-preserving representations enable the planning of more complex paths involving obstacles and landmarks. Multiple grid cell modules with different scaling factors $s$ further allow path planning at differnt spatial scales or resolutions, adapting to varying navigation demands. We will study path planning in future work.

We have revised our manuscript based on our response to your question. In particular, we have added a new section 6 to discuss conformal isometry for path planning. We shall investigate it in our future work.

Q2: Can the authors also discuss what parameter or assumption in the model may break the emergence of hexagonal patterns? For example, can the authors conduct an ablation study where an assumption is violated or added, resulting in no longer optimal hexagons? This study may bring much value. For example, the authors emphasize the importance of the normalization step, which plays a crucial role in finding the correlation between terms in the theoretical analysis. Possible to see numerical analysis in the absence of such constraint?

We appreciate your suggestion to investigate which parameters or assumptions in our model are critical in our framwork. We conduct numerical ablation studies to explore the significance of each assumption, as detailed in Section 3.3 of our initial submission and summarized below:

1. Normalziation Assumption: In our experimental results shown in Figure 3(f), removing normalization resulted in irregular and non-hexagonal patterns, supporting its necessity for hexagonal structures.
2. Conformal Isometry Assumption: When this assumption is removed (e.g., by eliminate the loss term $L_1$), grid patterns fail to emerge, as shown in Figure 3(h).
3. Transformation Assumption: The recurrent transformation $F(v(x),\Delta x)$ is necessary to encode self-motion effectively. Removing this assumption, which is implemented by removing $L_2$, leads to degraded patterns (Figure 3(g)).
4. Non-Negativity Assumption: The non-negativity constraint $v(x)>0$, while biologically plausible, was not strictly necessary for the emergence of hexagonal patterns (Figure 3(a) and (e)). Without this constraint, the patterns still emerge in both linear and nonlinear models.

Thank you for your insightful comments and questions. We have revised our paper and carried out additional experiments based on your suggestions and questions.

W1: I am not an expert in grid cell research; hence, it is a bit hard to fully appreciate and evaluate the significance of their minimalistic assumptions to model grid cells and study the conformal isometry hypothesis. This suggests a more in-depth discussion on why this proposal is better than prior works. Table 1 briefly shows that the framework offers better grid patterns for learning. Moreover, the paper discusses that their assumptions are less strict than prior works. These comparisons should be included in-depth in the main paper (in this version, only the intro discusses). Moreover, the first half of the paper is organized well; however, the theoretical analysis can be organized better. I recommend combining Sections 4 and 5.

Thank you for this important question. We have added a discussion in the revised manuscript (Section 5) to better explain our work relative to prior work.

The study of grid cells using machine learning models [1-5] is a topic of interest to both machine learning and computational neuroscience communities. Pioneering papers [1,2] adopted recurrent neural networks and used the objective of path integration to learn grid cells. Their models also involved interactions with place cells at the output layer, and the place cells are often modeled by predefined functions such as Gaussian tuning functions [2,9,10]. However, hexagon grid patterns do not consistently emerge from the learned models [6]. Thus extra assumptions or hypotheses are needed to explain the emergence of hexagon grid patterns.

One hypothesis is to assume non-negativity of the grid cell responses coupled with Difference-of-Gaussians tunning functions for place cells, as in [3,5]. However, the biological plausibility of center-surround Difference-of-Gaussian assumption is questioned [6].

Another hypothesis is conformal isometry hypothesis proposed in [9,10]. However, these papers learned multiple modules of grid cells with specific forms of transformation models, and they made extra assumptions about place cells and the interactions between grid cells and place cells.

Our paper is built on [9,10], and we have achieved scientific reductionism by isolating grid cell system of a single module, without making any assumptions about the place cells as well as the interactions between grid cells and place cells. We have also achieved generality by being agonistic of the form of the transformation model. This enables us to focus solely on the conformal isometry hypothesis and demonstrate its crucial role in the emergence of hexagon grid patterns. Moreover, we also provide neuroscience evidence and theoretical evidence for this hypothesis.

In our work, we study conformal isometry hypothesis with explicit scaling factor $s$, whereas in [9,10], $s$ is implicit in their learned models. $s$ determines the scale or resolution of the spatial representation of a grid cell module.

We also provide theoretical understanding of the role of conformal isometry for the emergence of hexagon grid patterns, by showing that the hexagon flat torus has minimal deviation from local conformal isometry.

We have revised our manuscript in response to your feedback by incorporating the Relation to Past Works into the main text of our paper, now presented in Section 5 of the revised version. Additionally, we have reorganized the theoretical section for improved clarity. Following your advice, we have integrated the key ideas and concepts from Section 5 of the initial submission into Section 4 and moved the experimental details to the appendix to better emphasize the central theme of the paper.

W2: Minor points

Thank you for the detailed points! We have revised our manuscript accordingly.