Place Cells as Multi-Scale Position Embeddings: Random Walk Transition Kernels for Path Planning

We thank you for recognizing our model as an interesting and novel contribution to computational neuroscience.

Weaknesses

1. Model Simplicity

Geodesic distance and complex environments: Thank you for this important concern! Our model handles sophisticated environments, including concave optimal paths, through its foundation in symmetric random walks and heat equation connections. The key insight is that transition probabilities $q(y|x,t)$ are not hand-crafted but emerge from random walk dynamics that naturally respect environmental constraints.

Theoretical foundation for complex geometries: As detailed in Section 2.2 and Appendix C, our discrete random walk converges to the heat equation with reflecting boundary conditions. Crucially, Varadhan's formula (Appendix C.4) establishes that for small $t$, our learned $q(y|x,t)$ approximates $exp(-d_g^2(x,y) / (4\alpha t))$, where $d_g(x,y)$ is the geodesic distance through the environment. This means our model implicitly encodes true shortest paths that navigate around obstacles, even when these paths are highly concave.

Empirical evidence: Our experiments can demonstrate this capability:

• The "diffraction-like patterns" in Figure 2B show how gradient fields bend around corners and through passages
• In the S-shaped and Four-room mazes (Figure 2A), our model achieves 100% success with superior efficiency (SPL > 1.0) despite inherently concave optimal paths
• The boundary layer effects (Appendix E.4) ensure obstacle-parallel flows for complex navigation.

Multi-scale robustness: Adaptive scale selection automatically adjusts resolution—coarse scales for global planning around large obstacles, fine scales for local precision—making our approach effective for concave paths requiring both global awareness and local navigation.

Additional experiment validation: Following your advice, we tested our model on the concave environment from Figure 10 of Muller et al. [1], achieving 100% success rate.

2. Additional results

We sincerely thank the reviewer for this valuable feedback. Our primary aim was to demonstrate the fundamental properties of our framework across key dimensions: biological plausibility through place cell characteristics (field size distribution, adaptability, remapping), and functional capabilities in robust path planning. We conducted experiments across multiple environment types. We selected the four-room maze as our most complex scenario because it presents multiple navigation challenges—hierarchical structure, bottlenecks, long-range planning, and multi-scale reasoning—effectively demonstrating our model's capabilities while maintaining biological plausibility.

We agree that broader experimental validation would strengthen our work. In response to your feedback, we will significantly expand the experimental section to include additional complex maze configurations with varying geometric properties (concave).

3. Remapping

We appreciate the opportunity to clarify our remapping experiment. The reviewer correctly notes that hippocampal remapping encompasses diverse phenomena (global, partial, rate remapping). In Section 3.2.4, we specifically modeled topology-dependent remapping—where removing obstacles creates novel shortcuts, introducing minimal physical but significant topological changes. This well-documented remapping type occurs when place fields reorganize to reflect new spatial relationships. Our model demonstrates that minimal fine-tuning enables embeddings to adapt and utilize new shortcuts, aligning with hippocampal responses to novel pathways. While we don't model all remapping forms, our experiment addresses a biologically relevant aspect highlighting topological adaptability. We will clarify this specific focus in our revision.

Questions

1. Goal-reaching with obstacles

Thank you for this question! Our model inherently handles goal-reaching with obstacles through Varadhan's formula, which ensures geodesic distance encoding even in complex environments. Our experimental validation demonstrates this capability: the U-shaped environment resembles Figure 15(A) in [1], while our four-room maze represents even greater complexity. Section 3.2 shows 100% success rates with superior efficiency (SPL > 1.0), indicating optimal geodesic navigation. We will add more classical environments from [1] to validate this capability further.

2. Matrix multiplication

We consider matrix squaring ($P_{2t}=P_t^2$) as a primary method for two main reasons, highlighting both computational efficiency and biological plausibility:

• Computational Efficiency for a Scale Hierarchy: Matrix squaring is exceptionally efficient for building a multi-scale hierarchy where the scales are powers of 2 (e.g., $t=2^k$). To compute $P_{2^k}$ from $P_1$, it only requires $k = \log_2(2^k)$ matrix multiplications. For example, to get $P\_{256}$ from $P\_1$ requires only 8 matrix multiplications ($P_2, P_4, P_8, \dots, P_{256}$). This is much faster than computing $P\_t$ as $P_1 \times P_1 \times \dots \times P_1$ ($t$ times).
• Biological Analogy to Preplay: This efficient generation of global representations from local transitions enables hippocampal "preplay-like" path planning without memorizing trajectories. We hypothesize this occurs through rapid, iterative computation in recurrent circuits, where each "squaring" step propagates over conceptual distance, allowing offline shortcut discovery that mirrors preplay's predictive nature.

We agree with you that we can also employ $P_{m+n}=P_m\times P_n$, and we mention in our paper that "one can design any discrete sequence $\mathcal{T}={t_k,k=1,...,K}$, and calculate $P_{t}$ using matrix multiplication".

3. Least resistance

Thank you for suggesting this really interesting work! Indeed there is a strong conceptual connection between our adaptive gradient following and the "principle of least resistance" described in [1].

In Muller et al. [1], the "least resistance" refers to the path through a "cognitive graph" that minimizes accumulated "resistance," where resistance might be related to synaptic weights or connection strengths, effectively finding the path of strongest connectivity or lowest cost. In our framework, the transition kernel $q(y|x,t)$ (and its unnormalized version $p(y|x,t)$) measures the "spatial adjacency" or "connectivity strength" between locations $x$ and $y$ at a given scale $t$. Path planning via adaptive gradient ascent on $q(y|x,t)$ (i.e., moving in the direction of maximal increase in transition probability towards the goal) means we are effectively seeking the path where there is the "most probable" or "easiest" flow from the current location towards the target. The gradient $\nabla_x q(x|y,t)$ acts like a "force field" guiding the agent. Paths following this gradient correspond to areas of high "connectivity" or "flow," which can be interpreted as paths of least resistance to reach the goal at that particular scale. Our adaptive scale selection refines this: small $t$ prioritizes shortest geodesic paths (least physical resistance), while large $t$ emphasizes global topological connectivity (least topological resistance via accessible corridors). This multi-scale approach aligns with cognitive maps integrating metric and topological properties, making our method a manifestation of least resistance principles adapted to multi-scale representations.

4. Regarding Whittington et al. (2022)[2]

We thank the reviewer for highlighting the work by [2], which presents a framework showing how RNNs trained with SR objectives can learn cognitive maps. Both approaches share the goal of providing computational accounts of cognitive maps. While complementary, several key distinctions exist:

• Mathematical Foundation: Our model is explicitly grounded in the mathematics of random walks, heat diffusion, and spectral graph theory. This allows for direct connections to established concepts like geodesic distance and provides a principled mechanism for multi-scale representation via the time parameter $t$.
• Explicit Multi-Scale Kernels: While Whittington et al. implicitly learn multi-scale representations, we explicitly construct distinct transition kernels $P_t$ for various $t$ values, directly linking to dorsoventral scaling observations.
• Representational Format: Our framework proposes place cells as position embeddings $h(x,t)$ derived from the transition kernels, with an emphasis on their inner product approximating transition probabilities $\langle h(x,t), h(y,t) \rangle = q(y|x,t)$. While Whittington et al. also use population codes, our specific inner product formulation and its connection to spectral decomposition and non-negative matrix factorization (via Horn's theorem) provide a distinct mathematical basis.
• Path Planning Mechanism: Our path planning relies on adaptive gradient ascent on these multi-scale transition probability fields, which naturally produce robust navigation properties.

Experiments with environmental modifications: Thank you for this suggestion. Our current remapping experiment (Section 3.2.4) focused on shortcut discovery to demonstrate cognitive map flexibility, but we have also tested obstacle addition. In the four-room environment, we added obstacles blocking direct pathways and fine-tuned the model. Initially, rooms were connected as shown in the lower panel of Figure 2(D), allowing direct northward navigation. The agent successfully adapted by discovering alternative routes through adjacent rooms when adding an obstacle. This demonstrates bidirectional adaptability—both exploiting new connections and navigating around obstacles. We can include this experiment in the revision for a more complete picture of adaptability.

Thank you again for your thoughtful review and constructive feedback. We are committed to addressing your concerns and will revise our paper accordingly.

We are grateful that you found our framework novel and elegant.

Weaknesses

1. Computational Efficiency and Training Bottleneck

Our claim of computational efficiency primarily refers to the inference phase (path planning) and the rapid generation of multi-scale kernels via matrix squaring. We acknowledge that the training phase, which sums over all pairs of locations, has a complexity of $O(N^2)$, where $N$ is the number of lattice points. While feasible for our tested environments (40×40 grid, training under 5 minutes), this would be a bottleneck for much larger maps.

To address this, our framework supports adaptive grid resolution tied to planning scale: coarse grids for long-range planning, fine grids for short-range precision. This multi-resolution strategy aligns with our model's multi-scale nature and manages training complexity. Future work will explore adaptive grids with sampling techniques like mini-batching for enhanced scalability.

2. $C^0$ Continuity and the Gradient Field

The reviewer correctly observes that bi-linear interpolation yields a $C^0$ continuous function, whose true gradient is indeed discontinuous at grid boundaries. Our use of "smooth" was intended to be qualitative, reflecting that the resulting paths are not erratic.

Specifically, this theoretical non-differentiability poses no practical problem for our path planning algorithm. In our numerical implementation, we approximate the gradient using a finite difference method. We evaluate $q(y∣z,t)$ at multiple candidate next positions $z\in∂(x)=\\{x+\Delta r(cos⁡θ,sin⁡θ)\\}$ around the current location $x$, where $\theta$ is discretized into equally spaced directions, and select the position that maximizes $q(y∣z,t)−q(y∣x,t)$ (Equation 9). This calculation only requires the function to be defined at these points; it does not require the existence of a continuous infinitesimal gradient.

Therefore, our planner can still produce smooth, effective trajectories. We will clarify this point in the revised manuscript.

3. Inconsistency in Random Walk Definition

We thank the reviewer for accurately identifying a minor discrepancy. Appendix C's theoretical derivation uses a 4-neighbor random walk for simplicity, while experiments use an 8-neighbor walk. The core connection to a diffusion process and geodesic distance holds for any symmetric random walk, with the specific diffusion coefficient $\alpha$ varying based on the neighborhood. We will clarify this in Appendix C, noting that the choice of neighborhood affects $\alpha$ but the theoretical properties of the heat equation connection remain.

4. Intuitive Explanations for Concepts

We agree that clearer intuitive explanations are needed. We will revise the Introduction and Method sections to provide better conceptual build-up. Specifically, we'll explain that dominant eigenvalues capture "slow-changing" modes of the random walk that reveal global environmental structure—the topological layout of rooms and pathways that remains invariant under metric distortions. This demonstrates how our model naturally leverages random walk properties for robust navigation.

5. Related Work in Appendix A

We agree with the reviewer. Placing related work in an appendix was a submission choice that indeed weakened the narrative. We will move a condensed, yet comprehensive, "Related Work" section to the main body of the paper, likely after the Introduction. Crucially, this section will explicitly introduce and differentiate our framework from the Successor Representation (SR), a point elaborated in response to comment 8.

6. Offline Process Limitation

We are grateful for the reviewer's recognition of our paper as a unifying theory. The reviewer correctly notes our current framework is "offline," pre-computing $P_1$ for the entire environment. However, our model does enable "preplay-like path planning" through matrix squaring, which efficiently computes global transitions ($P_t$​) from local rules ($P_1$​) without memorizing trajectories—analogous to hippocampal preplay of novel shortcuts. Future work will address incremental $P_1$ construction and dynamic updates for online map building and adaptation.

7. Lack of Sensory Input Integration

The reviewer correctly notes our model assumes a pre-defined discrete lattice, a limitation we acknowledge in Section 4. Our focus was on computational principles linking random walks to hippocampal representations, abstracting away perception. Future work will integrate sensory processing by learning $P_1$ from observations or deriving $h(x,t)$ directly from sensory features.

8. Differentiation from Successor Representation (SR)

We thank the reviewer for this insightful observation about the connection to SR. Both frameworks build on the powers of transition matrices, situating our work within the broader literature. We will significantly revise the "Related Work" section to clarify this:

1. Shared Foundation: Both frameworks root spatial representations in transition matrices.
2. SR's Discounted Sum: SR ($M = \sum_{k=0}^\infty (\gamma T)^k$) represents expected discounted future occupancy, with $\gamma$ setting the predictive horizon.
3. Our Multi-Time Transition Kernel: Our approach uses discrete powers, $P_t = P^t$, where $t$ explicitly defines a "time step" or "spatial scale" ($\sqrt{t}$). We model probabilities at distinct time horizons, not a discounted sum.
4. Analogy of $t$ and $\gamma$: While $t$ and $\gamma$ both control predictive horizons, $t$ directly relates to a physical time scale of a random walk, aligning with distinct dorsoventral place field scaling.

Key Distinctions and Novel Contributions

1. Biologically-Constrained Factorization with Emergent Sparsity: Our primary novelty lies in learning non-negative vector embeddings $h(x,t)$ such that $\langle h(x,t), h(y,t) \rangle = q(y|x,t)$, which is a biologically-constrained matrix factorization of the transition kernel. Crucially, we show that orthogonality + non-negativity automatically induces sparsity: When $q(y|x,t) \rightarrow 0$ for distant locations, the constraint $\langle h(x,t), h(y,t) \rangle = 0$ combined with $h(x,t), h(y,t) \geq 0$ forces disjoint support sets—explaining why place cells are naturally sparse and localized without explicit regularization.
2. Multi-Scale Architecture and Adaptive Navigation: Unlike SR's single discount factor, we explicitly model multiple scales $t \in {2^k}$ simultaneously via efficient matrix squaring ($P_{2t} = P_t^2$), directly mirroring hippocampal dorsoventral organization. Our adaptive gradient ascent selects optimal $t^*$ at each navigation step, enabling smooth, trap-free trajectories.

The reviewer's insightful comments help clarify our contribution: we provide a scalable, biologically plausible realization of multi-scale transition kernels that bridge abstract SR principles with concrete neural population dynamics. The geometric insight that orthogonality + non-negativity = sparsity offers a new theoretical understanding of why spatial representations are naturally sparse and localized.

We will incorporate proper SR citations and discussion in our revision.

Questions

1. Relationship to Successor Representation (SR)

As detailed in our response to reviewer Weakness 8, we will add a dedicated "Related Work" discussion. We will clarify that both frameworks derive spatial representations from transition matrices—SR uses discounted sums while we use discrete powers for multi-scale probabilities. Our core novelty lies in the representational format (non-negative embeddings $h(x, t)$) and explicit multi-scale hierarchy with adaptive navigation, offering a biologically interpretable paradigm.

2. Computational Complexity of Training

See our response to Weakness 1. While the training is feasible for current environments, this scales quadratically. We will acknowledge this limitation and discuss that future work will explore sampling techniques (e.g., mini-batching pairs of $(x,y)$ during optimization) to reduce the complexity per iteration to $O(B)$, where $B$ is the batch size, making it scalable to much larger environments.

3. Biological Plausibility of Mechanisms

While AdamW is an abstract optimization algorithm used to demonstrate learnability, biological learning mechanisms (e.g., Hebbian-like plasticity) may perform analogous optimizations. For matrix squaring ($P_{2t} = P_t^2$), which computes multi-step transitions from local rules, we propose that this could be mediated by recurrent neural circuits. Repeated, rapid "spreading activation" within hippocampal-entorhinal networks could effectively simulate multi-step transitions, akin to how the brain "preplays" and discovers novel shortcuts offline. This is an interpretation of the function that such neural circuits could perform, rather than a literal "matrix squaring" computation.

4. Effect of $C^0$ Continuity on Path Quality

See our response to Weakness 2. Our path planner does not compute an analytical gradient but a finite difference approximation, taking a small but non-infinitesimal step to find the direction of ascent. This numerical approach effectively sidesteps the issue of discontinuities. At every point, including the boundaries between grid cells, the planner can evaluate the transition probability $q(y∣z,t)$ at each candidate position and determine the next step.

Because our method does not rely on true differentiability, we did not observe any degradation in path quality attributable to these discontinuities. Our "near-optimal trajectories" validate this robustness. While smoother interpolation schemes remain theoretically interesting, our approach proves effective for high-quality navigation. We will clarify our numerical approach in the revision.

Thanks for your detailed feedback. We will incorporate all your insightful comments in our revision.

We appreciate your recognition of our theoretical framework's strong motivation, its recovery of key hippocampal features, and the clarity of our figures.

Weaknesses

1. It would be nice if they authors discussed connections to known experimental results in a little bit more depth

We appreciate this suggestion. In the revised manuscript, we will expand our discussion of connections to experimental results. For instance, in our response to Reviewer wLru's third question, we elaborate on how our multi-scale hierarchy naturally reproduces the systematic widening of place fields with distance from salient cues, consistent with the Weber-Fechner law. We also discuss in our response to Reviewer wLru's second question how our model spontaneously captures the empirically observed distortions (shrinking, elongating) of place fields near boundaries and obstacles. Furthermore, the adaptive scale selection mechanism inherently mirrors the dorsoventral axis scaling observed in the hippocampus. These points will be highlighted in the relevant sections to more explicitly link our theoretical findings to neurobiological evidences.

2. The model is obscured a little bit by notation

We agree with this and will make sure to clarify the notation and explanations throughout the paper, especially in the early sections. As suggested by your third question, we have changed the notation for the time scale from $t$ to $\tau$ to avoid confusion with the agent's real-time movement. This change, along with revised explanations, aims to make the model's concepts more accessible.

Questions

1. The meaning in the abstract is a little bit unclear before reading the paper, it might be good to use fewer equations in the abstract

We thank the reviewer for this suggestion. In the revised manuscript, we will reduce the number of equations in the abstract and focus on a more descriptive and intuitive explanation of our core ideas. We will emphasize the conceptual framework of our model – viewing the hippocampus as a predictive engine that learns multi-scale representations of space – and how this leads to the emergent properties we observe.

2. How robust is this setup to noisy place cell responses? I.e., if the embedding is noisy due to firing rate variability, what changes, and how much noise can one get away with?

This is an excellent question. To address this, we performed additional experiments by adding Gaussian noise to the learned embeddings, $h$, during the path-planning phase. And we test the success rate of path planning under multiple noise levels in different environments. Since $\|h\|^2=1$ due to normalization, to be more systematic, we measure the noise levels with $\sigma^2*n$ in the table, where $n$ is the number of cells. Our preliminary results shown in the table below indicate that the model is robust to moderate levels of noise.

3.The setup was a little bit confusing on first reading. t is meant to represent a time scale for which transition kernels are optimized for, but at first it may be mistakenly be thought that the transitions that are optimized for are changing (and growing) with time for an agent, which might lead to confusion. Changing text slightly to emphasise what is meant by time t or changing the notation slightly (t->$\tau$ or p(y|x,t)->p(y(t+t')|x(t')) for example) might clarify the meaning for readers in the early sections of the paper.

We are grateful to the reviewer for pointing out this potential source of confusion. To improve clarity, we will take the reviewer's suggestion and change the notation for the time scale from $t$ to $\tau$ throughout the paper. This will help to distinguish the time scale of the transition kernels from the time variable of the agent's movement. We will also revise the text in the early sections of the paper to explicitly state that $\tau$ represents a fixed time scale for which the representations are learned.

4. There is experimental evidence of place cells that respond to location and travel direction (i.e., they respond when an agent travels through a particular location from North to South but not from others). Would you expect to see place cells with this property if assumptions on the transition kernel being symmetric are lifted, and the underlying random walk is biased in some way?

This is a fascinating question that points to an exciting future direction for our work. The reviewer is correct that our current model, with its symmetric transition kernel, would not produce direction-tuned place cells. However, we strongly agree that lifting the assumption of a symmetric transition kernel is a promising way to model such cells.

If the underlying random walk were biased (e.g., due to a prevailing wind, a slope in the environment, or an animal's natural tendency to move in a particular direction), the transition probabilities would become asymmetric. In this case, we could extend our framework to learn $p(y|x, t) = ⟨h(x, t), w(y, t)⟩$ to accommodate asymmetry, where $w(y, t)$ can be interpreted as connection weights that differ from the source embeddings $h(x, t)$. This formulation would naturally capture directional biases while maintaining the inner product structure that makes our model computationally tractable.

With such asymmetric transition probabilities, our model would learn embeddings that reflect this asymmetry, and we would expect to see the emergence of place cells that are tuned to both location and direction of travel. We will add a discussion of this point to the future work section of the paper and acknowledge this as a key extension of our framework.

5. The representations are optimized for a random walks on space, where $p(y(t)|x(0))$ is gaussian with $\sigma \sim \sqrt{t}$. However, many exploration strategies are not truely random walks, but biased in some way. Would one expect this scaling relation to change?

The reviewer raises a very important point. While the random walk is a useful starting point, we agree that real-world exploration is often biased. We do expect the scaling relation to change with different exploration strategies. For example, a Levy flight, which involves occasional long-distance jumps, would lead to a different scaling of the transition probabilities.

Our framework is flexible enough to accommodate different exploration strategies. The key is that as long as the exploration strategy can be described by a transition kernel, our model can learn the corresponding representations. We believe that investigating how different exploration strategies affect the learned representations is a rich area for future research. We will add a paragraph to the discussion section to address this point and speculate on how the scaling relation might change under different, more realistic exploration models.

Thank you again for your comprehensive review. We will revise our paper by incorporating all your insightful comments and suggestions to strengthen our paper.

Thank you for noting our elegant unification of diffusion, spectral embeddings, and hippocampal coding. We are grateful that you recognize our multi-scale hierarchy and the efficiency of matrix squaring for local gradient-based path planning.

Weaknesses

1. All tasks are small 2-D lattices with full observability. No comparison with stronger planners (A*, DWA, agents based on successor representations, deep RL) or with SLAM-style real-world data.

We thank the reviewer for this important point. The primary focus of this work is to introduce a biologically-motivated theoretical framework and demonstrate that several complex features of hippocampal coding can emerge naturally from the simple principle of learning multi-scale transition probabilities. Our goal was to establish a proof-of-concept for this new perspective rather than to develop a state-of-the-art planner for robotics.

We agree that a comprehensive comparison with established planning algorithms is a crucial next step. To demonstrate this we have conducted a comparison with A* on Success weighted by inverse Path Length (SPL) as suggested.

The results show that while A* remains the optimal planner, our method achieves highly competitive performance. Our objective $q(y|x, t)$ captures geodesic distance at small t and topological connectivity at large t. For short ranges, our method follows geodesic shortest paths; for long ranges, it prioritizes topological connectivity.

We thank the reviewer again for the suggestion and will add this to our revised manuscript.

2. The model requires the entire transition matrix to form $q$ before learning; this is unrealistic for animals and for large robotics maps. Can the factorisation be updated online from sampled transitions?

The reviewer raises a critical point about the biological and practical plausibility of our learning rule. While the current implementation uses batch learning on a pre-computed transition matrix for simplicity, the underlying mathematical framework is fully compatible with online learning. We envision an online implementation where the embeddings, $h$, are updated incrementally based on single-step transitions sampled by the agent as it explores the environment. This would involve a stochastic gradient descent update to minimize the error between the observed transition and the one predicted by the current embeddings. This would be a more biologically plausible and scalable approach. We are actively developing this online version of the model and will revise the discussion to elaborate on this possibility and its implications for online learning and adaptation.

Questions

1. You initialised the one-step transition matrix using the four‐nearest neighborhood on the lattice. From a neurobiological standpoint, do head-direction modulation or the hexagonal symmetry of grid cells argue for a different neighborhood structure?

This is a very insightful question! The four-neighbor lattice is indeed a simplification. Our framework is flexible and can accommodate more complex neighborhood structures. For instance, one could easily define the one-step transitions on a hexagonal lattice, which would more closely align with the known symmetry and isotropy of grid cell firing fields. Furthermore, incorporating head-direction information could be achieved by modulating the transition probabilities, making transitions in the agent's current heading direction more likely. We hypothesize that these modifications would lead to representations that even more closely mirror the geometric properties of grid and place cells, and we will discuss this in the future work section as a promising extension.

2. Empirically, rodent place fields often shrink, elongate or duplicate near walls and obstacles. Can your representation somehow capture such distortions?

Thank you for this insightful question. Yes, our model can naturally capture these phenomena. The presence of a boundary or obstacle alters the local transition statistics of the random walk. A state next to a wall has fewer available transitions than a state in the middle of an open field. Since our model learns the embeddings directly from these transition probabilities, the learned representations for states near boundaries will be different from those for states in open areas. This results in distortions are shown in the learned place fields in Figure 2. This is an emergent property, not an explicitly programmed feature, which we believe is a key strength of our model.

3. Weber-Fechner law predicts that the tuning width of place cells grows with distance from salient cues or landmarks (at least in the absence of visual inputs that could provide anchoring). Can your hierarchy reproduce a systematic widening of fields with distance?

This is an insightful question that connects directly to the multi-scale nature of our model. Our hierarchy of representations, learned at different time scales $t$, provides a principled explanation for this phenomenon. The place fields corresponding to longer time scales ($t$) are inherently wider, as they represent transition probabilities over greater distances. If we assume that the brain utilizes representations learned at shorter time scales near salient landmarks (for fine-grained navigation) and longer time scales further away, our model directly predicts a systematic widening of place fields with distance from these anchor points. We will add a discussion of this point to the paper, as it provides a novel theoretical account for this well-documented experimental finding.

4. During inference, you evaluate at continuous coordinates, yet learning was performed on a discrete 40 by 40 lattice. Could you elaborate on how you interpolate or extrapolate between lattice points?

We apologize for not making this clearer in the text. To obtain a continuous representation for planning, we use bi-linear interpolation. After the embeddings, $h$, have been learned for every point on the discrete lattice, the embedding for any continuous coordinates is computed as a weighted average of the embeddings of the four nearest lattice points. The contribution of each lattice point's embedding is weighted by the continuous point's proximity to it. This is a standard method for creating a smooth, continuous function from a discrete grid. We will add a clear description of this interpolation process to the methods section. The ability to interpolate the learned $h$ naturally enables our method to plan smooth paths.

5. How might the model integrate self-motion cues (path integration) with the learned transition kernel, and what predictions would this make for place-cell dynamics during real-world locomotion (e.g., phase precession)?

This is a forward-looking question touching on an exciting frontier. We propose that self-motion cues and path integration can be naturally incorporated through the grid cell system. The modules of grid cells form distance-preserving embeddings with different metrics, and their recurrent connections enable path planning, as demonstrated in recent work on neural path integration and spatial representation learning [1,2,3]. In our appendix, we discuss the interactions between place cells and grid cells, where grid cells preserve distance and serve as Euclidean coordinates, while place cells preserve proximity $q(y|x, t)$ and guide path planning.

Regarding theta phase precession, we have developed an interesting theoretical framework detailed in Appendix F of our submission. Phase precession describes how the timing of place cell spikes advances relative to the theta rhythm as an animal crosses a place field. In our proposed theory, the theta phase of a place cell depends on the inner product between the embedding of the current location and the embedding of the center of the place field of this place cell. Therefore, the theta phase encodes adjacency to the center of the place field. Since place fields of different place cells tile the environment, the theta phases give the agent the awareness of its adjacency to different places.

[1] Sorscher, B., Mel, G., Ganguli, S., & Ocko, S. (2019). A unified theory for the origin of grid cells through the lens of pattern formation. Advances in neural information processing systems, 32.

[2] Gao, R., Xie, J., Wei, X. X., Zhu, S. C., & Wu, Y. N. (2021). On path integration of grid cells: group representation and isotropic scaling. Advances in Neural Information Processing Systems, 34, 28623-28635.

[3] Xu, D., Gao, R., Zhang, W., Wei, X. X., & Wu, Y. N.(2024). On Conformal Isometry of Grid Cells: Learning Distance-Preserving Position Embedding. In The Thirteenth International Conference on Learning Representations.

Thanks again for your thoughtful evaluation and constructive suggestions. We will revise our paper by incorporating all your insightful comments and suggestions to improve the clarity and rigor of our work.

To contextualize results from A*, it might be useful to show values for random or some other type of exploration (for example, for a four-room environment, the difference between the two algorithms is 0.6 and while I agree that it's expected for A* to perform better, it would be informative to have more baselines to better interpret the magnitude of the difference). I'm confident this is something the authors can easily do for the final version of the manuscript.

Thank you for the valuable advice! Following your advice, we have compared our method with several baselines in the Four-Room environment, with each trial limited to 50,000 steps. The results show that baseline approaches like the Bug algorithm (without oracle guidance) and Random Walk fail to solve the most of the challenging scenarios.

Regarding question 2, it would be informative if you could demonstrate the approach on realistic data, e.g., some of the classical work from O'Keefe & Neil Burgess (Fig. 2 in https://www.nature.com/articles/381425a0)

Thank you for directing us to this classical paper! Following the experimental design in [4], we have implemented two key environments: a rectangular environment (40×80 grids, analogous to their HR condition) and a larger square environment (80×80 grids, analogous to their LS condition).

Using t=128 iterations, our method successfully reproduces key phenomena reported in their Figures 1 and 2: (1) field elongation in the rectangular environment along the extended dimension, and (2) field dissection into two components in the larger square environment.

The emergence of elongated patterns in HR and dissected fields in LS demonstrates that our method can model the geometric determinants of place field structure described in this seminal work. These results show clear correspondence with their experimental observations of how environmental geometry shapes hippocampal spatial representations. Due to rebuttal policy constraints, we are not allowed to present addtional plots, but we will include these comparative analyses and visualizations in the final version of the manuscript.

[4] O'Keefe, J., & Burgess, N. (1996). Geometric determinants of the place fields of hippocampal neurons. Nature, 381(6581), 425-428.

Thank you so much for your insightful guidance!