Complementary Coding of Space with Coupled Place Cells and Grid Cells

We sincerely thank you for your time and comments on our manuscript. While we acknowledge that some of your concerns pertain to the suitability of our work for ICLR, we hope to clarify its relevance and address the specific points raised. Below, we provide detailed responses to both your major and minor concerns.

Major concerns:

1. Suitability of the manuscript for ICLR:

We appreciate your comments on the alignment of our work with the primary audience of ICLR. However, we respectfully argue that ICLR has increasingly become a platform for computational neuroscience studies. Several recent works focusing on computational modelling of grid cells, rather than direct ML/DL contributions, have been published at ICLR. Examples include:

• Yu et al. (2021): Prediction and generalisation over directed actions by grid cells.
• Whittington et al. (2021): Relating transformers to models and neural representations of the hippocampal formation.
• Dorrell et al. (2023): Actionable neural representations: Grid cells from minimal constraints.
• Whittington et al. (2023): Disentanglement with biological constraints: A theory of functional cell types.

These papers, like ours, do not directly propose new ML algorithms or compare models to experimental data. Instead, they focus on developing theoretical insights into biological systems, advancing the field of computational neuroscience, and will eventually link to brain-inspired intelligence. Given this precedent, we believe our work fits well within the ICLR community’s interest in biologically-inspired approaches to learning and representation.

2. Experimental validation:

We acknowledge the importance of comparing computational models with experimental data to enhance biological plausibility. However, our current study focuses on developing a theoretical framework for understanding spatial representations in coupled place-grid cell networks. Integrating experimental datasets, such as those from the Buzsáki Lab, Moser Lab, or Dombeck Lab, is an excellent suggestion and could significantly extend this work in the future.

While we do not include a direct quantitative comparison with experimental data in this paper, we have identified qualitative experimental evidence that supports key aspects of our model. For instance, findings from Campbell et al. (2021) align well with our theoretical predictions. Specifically, their results state:

"In V1 and RSC, firing rates peaked 20 cm before each visual landmark (Figures 1I and 1J), with the receptive field location influencing spatial firing rate maps in V1 (Figure S1C). In MEC, the average firing rate was relatively constant over the VR track (Figures 1I and 1J)."

This observation suggests that MEC grid cells may not receive direct landmark cue-based inputs. If such inputs are present, grid cell firing rates would likely vary with proximity to landmarks. In contrast, our model predicts that landmark cues influence grid cells indirectly, mediated through place cells. In our framework, place cells first process landmark cue-based inputs, encoding this information via their attractor dynamics, which inherently stabilize the activity and reduce input variance. This stabilized information is then relayed to grid cells, explaining the lack of landmark proximity effect in their firing rates.

Additionally, Campbell et al. (2021) report that "MEC map shifts were larger when landmark inputs were less certain, although the contrast sensitivity appeared to be sharp." This observation aligns with the Bayesian principle that inputs should be weighted by their uncertainty—a core idea in our model's Bayesian integration of information between place and grid cells.

To enhance the link of our work to experimental data, we will add a discussion of these experimental correspondences in the revised manuscript, demonstrating how our theoretical framework provides a mechanistic explanation of these findings.

Minor concerns:

1. Figure 3c and 3d colors:

Thank you for pointing out the discrepancy in the figure caption. We will revise the captions of Figures 3 to “blue, red, and green,” ensuring they align with colors in the figure.

2. Figure 1d notation:

The “c” under the “π” in Figure 1d was unintended and will be removed in the revised manuscript for clarity.

We sincerely thank Reviewer xhaK for their thoughtful and detailed feedback. Below, we address each of the concerns raised, providing explanations, additional analyses, and revisions where necessary.

1. About the position-based, rather than motion-based, input to grid cells in our model:

Thanks for the thoughtful suggestion. In the present study, our primary focus is not on how grid cells perform path integration, but rather on how the coupled networks between place cells and grid cells efficiently integrate information from different modalities and they eliminate non-local errors in the absence of external environmental cues. In our model, the position-based inputs received by grid cells may represent signals from upstream cells performing path integration. These inputs could originate from other grid cells, as suggested in path integration models of grid cells (e.g., Giocomo et al., 2011), or potentially from other cell types, such as band cells (e.g., Krupic et al., 2012).

While we acknowledge that directly incorporating self-motion signals into the model could provide a more comprehensive understanding of grid cell dynamics, as a first step, which is already a novel contribution in the field, the present study will only focus on how the interactions between place cells and grid cells enhance spatial coding robustness, especially under conditions where external cues are unavailable. We will certainly consider your insightful feedback in future work.

2. About parameter sensitivity of the model:

Thank you for raising these important issues. We have done some of them in the current manuscript. For example, in Fig. 3d, we compared the network's information integration with Bayesian integration under varying noise levels. As suggested by the reviewer, to further validate the robustness of our model, we have conducted new experiments by varying the coupling strength between place and grid cells. These new results, now included in the appendix of the revised manuscript, show that the network performs near-optimal Bayesian information integration across a range of coupling strengths (see Fig. S4).

3. About demonstration of non-local error elimination of the model:

Thanks for the suggestions. We would like to clarify that some of them have been done in the current manuscript. As shown in Fig. 4a, we compared the error distributions of the coupled network and the MAP (baseline model) under an identical noise condition. Furthermore, as shown in Fig. 4b, we systematically varied the input noise strength and compared how the error variances of both decoding methods change with the increasing noise level. These results demonstrate that the coupled network significantly reduces non-local errors compared to MAP, supporting the statement that our model effectively mitigates non-local errors. As suggested by the reviewer, we will perform a statistical p-test to quantitatively assess the differences between the network decoding results and the baseline model presented in Fig. 4a. This analysis will be included in the revised manuscript.

4. About discussion about mapping the model’s feedback dynamics to known hippocampal projections:

Thanks you for the suggestion. In the current model, we consider that place cells and grid cells are directly connected. This simplified model gives us insight into understanding how place and grid cells interact with each other to improve spatial representation. We recognize that in the actual neural system, the projections from grid cells to place cells are via direct inputs from MEC Layer II/III to CA3 and CA1, and the projections from place cells to grid cells involve indirect pathways—such as projections from CA1 to MEC Layer V or the subiculum, followed by internal MEC connections back to Layer II/III. We also acknowledge that incorporating more biologically detailed structures will allow our model to better align with the anatomical evidence. In future work, we plan to extend the current model to include these specific pathways and explore their implications for spatial representation. Nevertheless, in the revised manuscript, we will add a discussion of these anatomical pathways and their potential influences on our results.

6. Comparison with experimental literature:

Thanks for pointing out these valuable references. We have read the studies by Campbell et al. (2018, 2021) and agree that they provide an excellent opportunity to align our model predictions with recent experimental findings, enhancing the connection between our theoretical framework and real data.

First, we would like to address the reviewer’s concern regarding our claim that place cells and grid cells receive independent inputs (cue-based and self-motion-based, respectively). In our model, while grid cells and place cells receive independent self-motion and environmental cues, respectively, they are connected reciprocally. This implies that the activity of grid cells is influenced by the environmental cue via the connection from place cells. In other words, the dependence of grid cells’ activity on the environment cue does not mean grid cells receive the environmental cue directly.

Actually, the findings in Campbell et al. (2021) tend to support our model. For example, the results section of their paper states: "In V1 and RSC, firing rates peaked 20 cm before each visual landmark (Figures 1I and 1J), with the receptive field location influencing spatial firing rate maps in V1 (Figure S1C). In MEC, the average firing rate was relatively constant over the VR track (Figures 1I and 1J)." This observation suggests that MEC grid cells do not receive direct landmark cue-based inputs. If they do, we would expect grid cells’ firing rates vary with proximity to landmarks. In contrast, our model predicts that landmark cue reaches grid cells indirectly, mediated through place cells. In our framework, place cells first receive the landmark cue-based input and then encode this information via the attractor dynamics, which inherently reduces the input variance by stabilizing the activity into an attractor state. This filtered information is then relayed to grid cells, which justifies why grid cells’ firing rates remain unaffected by landmark proximity.

Moreover, Campbell et al. (2021) reported that "MEC map shifts were larger when landmark inputs were less certain, although the contrast sensitivity appeared to be sharp.". This finding aligns well with the Bayesian principle that inputs should be weighted according to their uncertainty—a fundamental idea underlying our model’s Bayesian integration of information between grid cells and place cells. In this sense, our model provides a theoretical explanation for these experimental observations.

In the revised manuscript, we will incorporate a discussion of these findings to highlight how the model’s predictions align with experimental results. In future work, we will quantitatively compare our model predictions with the experimental data from Campbell et al. (2021).

5. Comparison with existing models:

Thanks for the detailed comments and insightful suggestions. We appreciate the opportunity to clarify and expand on the comparisons between our model and the existing literature.

• First, regarding comparisons with classic attractor networks. Actually, our coupled network belongs to classic attractor models, such as we used 1D CANN (i.e., line attractor network) for place cells and multiple ring attractors for grid cells. In the revised manuscript, we will include discussion of models that incorporate landmarks into attractor networks, such as those proposed by Ocko et al. and Campbell et al.. Unlike these models, which rely on external environmental cues (e.g., landmarks) to correct path integration errors, our work focuses on how place cells contribute to improving grid cell coding in the absence of external information—such as during navigation in darkness. Specifically, in our model, error correction is achieved through the storage of historical information in the attractor dynamics of the place cell network.

• Second, following the reviewer’s suggestion, we have added a direct comparison with the "constrained range model" proposed by Sreenivasan & Fiete (2011) in Appendix of the revised manuscript (Fig. S5). The results show that both models can eliminate non-local errors under small, constrained decoding ranges. However, when the decoding range increases, our model continues to perform robustly, while the constrained range model fails. This distinction arises from that our model leverages the recurrent dynamics of the place cell network to store the history information, enabling error correction without sacrificing spatial coding capacity. In contrast, the constrained range model mitigates non-local errors by limiting the spatial range, which inherently sacrifices the coding capacity.

• Regarding the total coding range of grid cells, both our model and the model proposed by Sreenivasan & Fiete employ phase combination coding, resulting in equivalent coding capacity theoretically. However, in practice, since the constrained range model requires decoding within a small, constrained range to avoid non-local errors, its true coding capacity becomes very limited; whereas, our model always achieves large effective coding range as it relies on the coupling between grid and place cells.

We sincerely thank Reviewer iXH2 for their thoughtful and detailed feedback, which has greatly helped us identify areas to refine and improve our manuscript. Below, we address each of the concerns raised, providing explanations, additional analyses, and revisions where necessary.

Weakness 1:

• Noise dependence across neurons.

Response:

• Thank you for raising pointing this issueout. Indeed, as discussed in the study by Abbott & Dayan (1999), neural noise is not independent across neurons in experimental results obtained from in vitro slice preparations. In our work, we assumed independent noise to simplify the network dynamics, allowing for a concise correspondence with the gradient-based optimization of the posterior (as shown in Eqs. (17) and (18) in the main text). However, including neural correlation does not change our main conclusions. To confirm this, we conducthave also conducted simulations includingtroducing correlated noise in the network dynamics, where the noise covariance is modeled as, $〈(r_i-f_i )(r_j-f_j )⟩=\sigma^2 [\delta_ij+c(1-\delta_ij )]$ with $c=0.15$, which falls within the range of experimentally experimentally observed noise correlations ($c\in(0.1,0.2)$) as reported by Zohary et al. (1994) and Shadlen et al. (1996). These simulations confirmed that our main conclusions remain unchangedrobust: (1) the network continues to integrate information in a Bayesian manner (see new Fig.S2 a&b in the revised Appendix), and (2) in the absence of positional information ( $I_p=0$ ), the network can still eliminate non-local errors caused by grid cell coding, enabling more robust spatial coding (see new Fig.S2 c in the revised Appendix).

Weakness 2:

• Parametric assumptions of the model.

Response:

• Thank you for raising these important issues regarding parameter assumptions in our model. We address them one by one below:

1. Unimodal Gaussian tuning curve for place cells (Equation 1): By definition, place cells are neurons that exhibit localized place fields in an environment. While it is true that some hippocampal neurons show multiple place fields in large environments (e.g., Fenton et al., 2008; Rich et al., 2014), a significant proportion of place cells have single, localized place fields (e.g., see Fig. 4b in Rich et al., Science, 2014). Additionally, neurons exhibiting multiple fields may result from remapping, where the animal perceives a large environment as multiple sub-environments. In our model, the Gaussian tuning curve is a mathematical approximation to facilitate theoretical tractability. We believe that alternative bell-shaped tuning curves (e.g., sigmoidal or cosine tuning) would yield similar results and do not qualitatively affect the model's conclusions, as confirmed by many previous modelling studies.
2. Gaussian noise model (Equation 5): In our firing-rate model, each unit represents a population of neurons encoding the same position (e.g., place cells) or sharing the same phase (e.g., grid cells). While individual neurons typically exhibit Poisson spiking behavior, the firing rates averaged across a population approximate a Gaussian distribution due to the central limit theorem. This mean-field approach is widely used in modeling study and provides a biologically reasonable simplification. Thus, assuming Gaussian noise is consistent with the aggregated firing rate behavior of neuronal populations and does not diminish the validity of our results.
3. Independent grid cell modules: Experimental evidence suggests that grid cell modules are anatomically separated, with very weak interconnectivity (Hafting et al. 2005). Therefore, studies such as Sreenivasan and Fiete (2011) and Agmon and Burak (2020) treated different grid cell modules as independent in their theoretical models. Additionally, introducing weak random connections between grid cell modules won’t significantly affect our simulation results. Overall, we would like to point out that these parametric assumptions are common in theoretical modelling studies and they are not critical to the core conclusions of our work. They mainly contribute to facilitating a tractable theoretical framework for analyzing the coupled dynamics of place cells and grid cells. We hope that our responses and revisions have addressed the concerns of the reviewer. Thanks again for the valuable comments.

Weakness 3:

• Loss function of "Optimal Decoding"

Response:

• Thanks for bringing this issue to our attention. We have revised the manuscript to clarify the relationship between the MAP estimator and its associated loss function. Specifically, in lines 220–221 of the revised main text, the original statement "Mathematically, optimal decoding is achieved by maximum a posterior (MAP)" has been modified to: "Maximum a posteriori (MAP) decoding is a widely used method (references) and proven to be the optimal decoding strategy under a 0-1 loss function." This revision provides a more accurate and complete explanation of our approach. We sincerely appreciate your feedback on this important detail.

Weakness 4:

• Direct comparison between network dynamics and Gradient ascent of posterior

Response:

• Thanks for this suggestion. In Fig. 3b of our paper, we compared the decoding results obtained from the network with those obtained via gradient-based optimization of the posterior (GOP). But as you pointed out, we did not explicitly include the trajectory of the network activity during the decoding process (before reaching the steady state) and compare it to the trajectory of the gradient ascent-based optimization. We have now added examples illustrating the network trajectory and its alignment with the gradient ascent trajectory. These results will be added to the revised Appendix for completeness (see the new Fig. S2).

Weakness 5:

• More citations and references

Response:

• Thank for pointing out these issues. We have addressed them in the revised manuscript. Specifically:

1. A citation to Fiete et al. (2008) was added in section 2 to explain the importance of co-prime factors (see line 154).
2. A reference to the Cramér–Rao bound was included in section 2.2 (lines 162–163): "We use Fisher information, whose inverse is the lower bound of decoding variance of any unbiased estimator (commonly known as the Cramér–Rao bound (Cramér, 1946))."
3. The citations to MacKay's textbook were standardized throughout the manuscript to ensure consistency, and the reference has been updated to indicate that we are specifically referring to Chapter 3.

We appreciate your feedbacks, which help us to improve the clarity and accuracy of the manuscript.

Weakness 6:

• Discussion about other works of Bayesian intepretation of attractor networks

Response:

• Thank you for pointing out these two relevant studies, both of which discussed the relationship between attractor networks and Bayesian inference. These works share some common ground with our study, and we will add a paragraph discussing them in the revised manuscript. However, there are significant differences between our work and these studies:

1. Our model is not a single-layer attractor neural network, rather it consists of a line attractor network formed by place cells, which is reciprocally coupled with multiple ring attractor networks formed by grid cells
2. The focus of our paper is not on proposing a Bayesian interpretation of attractor dynamics. Instead, we investigate: 1) how the interactions between the place cells’ network and the grid cells’ networks enable the efficient integration of multimodal information, where the reciprocal connections between place and grid cells convey the correlation prior; 2), how the coupling with place cells help grid cells to resolve the non-local error problem.

To our best knowledge, our results are novel in the study of how place cells and grid cells interact with each other to improve spatial representation in the brain, and they are valuable contributions to the field. We hope that our replies have addressed all concerns of the reviewer and could persuade the reviewer to raise the score.