Title: Place Cells as Multi-Scale Position Embeddings: Random Walk Transition Kernels for Path Planning
Venue: NeurIPS 2025
Accepted: yes
Keywords: place cells

population coding

position embedding

random walk transition kernels

path planning

spatial navigation

cognitive map
Abstract: The hippocampus supports spatial navigation by encoding cognitive maps through collective place cell activity. We model the place cell population as non-negative spatial embeddings derived from the spectral decomposition of multi-step random walk transition kernels. In this framework, inner product or equivalently Euclidean distance between embeddings encode similarity between locations in terms of their transition probability across multiple scales, forming a cognitive map of adjacency. The combination of non-negativity and inner-product structure naturally induces sparsity, providing a principled explanation for the localized firing fields of place cells without imposing explicit constraints. The temporal parameter that defines the diffusion scale also determines field size, aligning with the hippocampal dorsoventral hierarchy. Our approach constructs global representations efficiently through recursive composition of local transitions, enabling smooth, trap-free navigation and preplay-like trajectory generation. Moreover, theta phase arises intrinsically as the angular relation between embeddings, linking spatial and temporal coding within a single representational geometry.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents a new computational model for hippocampal navigation based on position embedding of multi-time random walk transition kernels. It is an interesting contribution to computational neuroscience, especially the community of spatial navigation. The paper is well written and easy to follow. Experimental results on maze simulation are limited (Figure 2 only) but interesting and convincing to some degree. Overall, I think this paper is at the borderline between 3 (WR) and 4(WA). I can go either way based on other reviewers' ratings and the authors' rebuttal.\n\nSTRENGTHS AND WEAKNESSES:\n+ The navigation and planning model presented in this work is interesting and has its novelty. Path planning via adaptive gradient flowing seems to be consistent with the principle of least resistance (though not explicitly discussed in the paper, please see my question below).\n+ The remapping experiment in Fig. 2D is interesting and thought-provoking. It shows that the model can manage the opening of new pathway (i.e., to discover a shortcut).\n\n- The model's simplicity makes me wonder about its capability for handling more sophisticated environmental settings when geodesic distance is involved (e.g., what about the optimal path is concave?). \n-The reported experimental results are limited. I was hoping that more simulation results will be included in Appendix. But no additional results can be found there.\n-I am a bit skeptical about the remapping experiment in 3.2.4. There are different ways of defining remapping in hippocampal study - here authors seem to only consider one \"analogous to the remapping phenomenon\".\n\nQUESTIONS:\n1. Can the proposed model be extended for goal-reaching policy with obstacles? In a classical study [1], geodesic distance navigation was demonstrated for the presence of obstacles. It will be interesting to show your model can achieve something similar. \n\n2. In Sec. 2.4, why do you only consider matrix squaring instead of matrix multiplication - e.g., P_{m+n}=P_m\\timesP_n?\n\n3. Does path planning via adaptive gradient following have anything to do with the principle of least resistance? In [1], it has been shown how the minimum distance navigation can be achieved by the principle of least resistance.\n\n4. How is the proposed model related to [2]? Have you try other environmental modifications such as adding an obstacle instead of opening a shortcut?\n\n[1] Muller, R. U., Stead, M., & Pach, J. (1996). The hippocampus as a cognitive graph. The Journal of general physiology, 107(6), 663-694.\n[2] Whittington, J. C., McCaffary, D., Bakermans, J. J., & Behrens, T. E. (2022). How to build a cognitive map. Nature neuroscience, 25(10), 1257-1272.\n\nLIMITATIONS:\nYes. The paper has considered both theta-phase procession and grid cell integration as major limitations of this work.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-05"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces a novel and elegant framework for modeling hippocampal place cells and spatial navigation. The central thesis is to reconceptualize a population of place cells as a single vector embedding, $h(x,t)$, for each location $x$ and time scale $t$. The key innovation is that the inner product of these embeddings, $⟨h(x,t),h(y,t)⟩$, is trained to approximate the normalized transition probability, $q(y∣x,t)$, of a symmetric random walk on the environment's manifold.\n\nSTRENGTHS AND WEAKNESSES:\nThe framework builds a rigorous bridge from a simple, discrete random walk to a sophisticated, continuous navigation policy. The derivation connecting the discrete walk to the continuous heat equation with reflecting boundary conditions is a classic and powerful approach from statistical physics, giving credibility to the model.\n\nThe use of Varadhan's formula to connect the short-time heat kernel to the squared geodesic distance ($d_g^2(x,y)$) provides a rigorous mathematical justification for why the learned transition probabilities encode meaningful spatial information, especially for fine-scale navigation. Moreover, to identify that neural firing rates must be non-negative and invoke Horn's theorem, providing existence of a non-negative matrix factorization for the symmetric, non-negative transition kernel, is cool.\n\nThe reported SPL values, which significantly exceed 1.0 in complex mazes like the S-shape and Four-room environments, demonstrate that the model finds substantially more efficient paths than the baseline.\n\nThe paper claims computational efficiency, primarily highlighting the matrix squaring step for inference. However, the training phase appears to be a potential bottleneck. The quadratic scaling could become prohibitive for the larger, more realistic environments where the model's multi-scale nature would be most beneficial.\n\nThe model learns embeddings on a discrete lattice and then uses bi-linear interpolation to create a continuous map for gradient-based planning. While a practical choice, bi-linear interpolation results in a function that is only $C^0$. This contradicts the claim of a perfectly \"smooth gradient field\". The impact of this choice on path quality should be discussed.\n\nThere appears to be a minor inconsistency between the theoretical derivation in Appendix C, which assumes a 4-neighbor random walk , and the main experiments, which use an 8-neighbor walk. \n\nVisualizations are good.\n\nThe main text could benefit from more intuitive explanations to bridge the different concepts from statistical physics, spectral graph theory, computational neuroscience, and optimization. For example, a more gradual introduction to how the eigenvalues of the transition matrix relate to \"topological connectivity\".\n\nThe discussion of related work is put in Appendix A, which weakens the paper's main narrative by failing to immediately situate its key contributions. In particular, the distinction from the very closely related Successor Representation (SR) framework is not made in the main body of the paper. \n\nThe paper's primary significance lies in its potential as a unifying theory. It elegantly connects a simple, low-level process (a random walk) to the emergent, multi-scale, and adaptive properties of a high-level cognitive function (spatial navigation via a cognitive map). The model proposes concrete, computationally tractable mechanisms for phenomena observed in the hippocampus. Adaptive multi-scale representation could have significant potential for robotics. \n\nA significant limitation is that the framework relies on pre-computing the transition matrix $P_1$ for the entire environment, and then computing all $P_t$ via matrix squaring before any planning can occur. This is an \"offline\" process, limiting its significance for modeling online learning in animals, which build maps through active exploration, and for robotic applications in dynamic or unknown environments. \n\nThe model operates on a discrete lattice where adjacency is pre-defined. It does not address the problem of how such a representation would be built from raw sensory inputs (e.g., vision).\n\nThe core formulation, $⟨h(x,t),h(y,t)⟩≈q(y∣x,t)$, is an original way to frame the population code of place cells. It moves beyond modeling individual firing fields to modeling the geometric structure of the entire transition kernel in an inner product space. Using the time parameter $t$ of a single random walk process to generate a continuous hierarchy of spatial scales is novel. The \"diffusion-on-the-graph\" approach provides a direct and interpretable way to incorporate environmental structure into the planning process.\n\nThe work does not sufficiently differentiate from the SR framework which models spatial representations based on powers of the transition matrix T (analogous to $P_1$). The SR matrix, often defined as $M=(I−γT) −1 =∑ _{k=0}^∞ (γT)^k$, represents the expected discounted future occupancy of states. The paper's transition kernel is $P_t =P_1^t$. Both are fundamentally based on powers of the transition matrix, and the paper's time parameter $t$ acts analogously to the SR's discount factor $γ$ in controlling the predictive horizon. While the authors' formulation using an inner product embedding is novel, the underlying mathematical object (powers of the transition matrix) is not. The paper fails to discuss this deep connection in the main text, leaving the reader to question whether this is a fundamentally new paradigm or an elegant and powerful reformulation of the same core principle as SR. This must be addressed.\n\nThe true novelty may lie not in proposing a new computational object (versus SR), but in proposing a new representational format. The learning objective is essentially a low-rank non-negative matrix factorization of the kernel matrix $Q_t$. Therefore, the contribution could be framed as a scalable, factorized representation of a multi-scale transition kernel, which has strong parallels to the SR but offers a distinct representational advantage. Clarifying this would strengthen the paper's claims of novelty.\n\nQUESTIONS:\nThe paper's formulation based on $P_t​=P_1^t​$ bears a strong resemblance to the SR, which is based on $∑(γP1​)k$. Could the authors provide a detailed discussion in the main text on the theoretical and practical relationship between their framework and SR? Is it more accurate to frame the contribution as a novel, scalable, factorized representation of a multi-scale transition kernel, rather than a wholly separate paradigm?\n\nThe learning objective in Eq. 8 appears to have a complexity of at least $O(N^22)$, which could be prohibitive for large environments. Could the authors clarify the computational complexity of the training phase and discuss whether any sampling or approximation techniques were used or would be necessary for scaling up?\n\nThe paper claims biological plausibility. While the resulting representations are plausible, the computational mechanisms are more abstract. Could the authors comment on the biological plausibility of the learning algorithm (gradient descent with AdamW) and, in particular, the matrix squaring operation ($P_{2t}​=P_t^2​) as a neural computation for replay?\n\nThe path planning relies on the gradient of the continuous kernel, which is constructed via bi-linear interpolation. This results in a $C^0$ continuous field whose gradient is discontinuous. Have the authors investigated the effect of these discontinuities on path quality, and would a smoother interpolation scheme (e.g., bicubic) yield better results?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-06-24"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors propose viewing hippocampal place cells not as independent Gaussian bumps but as a population embedding whose inner product reproduces multi-step symmetric random walk transition probabilities. Experiments on open fields, U, S and four-room mazes show high correlation between learned and what the authors consider ideal kernels and high navigation success. The framework generates qualitative phenomena such as scale-dependent field sizes, boundary following and remapping after obstacle removal.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths: \n\nElegant unification of heat-kernel diffusion, spectral embeddings and hippocampal population coding.\n\nMulti-scale hierarchy mirrors dorsoventral field scaling;\n\nMatrix squaring gives $O(log⁡ t)$ scaling. \n\nPath planning uses only local gradients - no global graph search.\n\n\nWeaknesses:\n\nAll 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.\n\nThe 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?\n\nQUESTIONS:\nYou 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?\n\nEmpirically, rodent place fields often shrink, elongate or duplicate near walls and obstacles. Can your representation somehow capture such distortions?\n\nWeber-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?\n\nDuring 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?\n\nHow 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)?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes that at a population level, the place system encodes location and transition probabilities for local motion. Population level responses are optimized such that the the similarity between responses at positions $x$ and $y$ (given by the inner product of the population level response at these locations) matches the transition probability between those locations $p(y(t)|x(0))$ for a given timescale $t$, where this transition probability is assumed to match that for an unbiased random walk on the environment. Under very mild assumptions, optimizing the population level responses in this way leads to place cell representations. Optimizing jointly over multiple time scales leads to multiple place cell scales. These representations can be used for efficient path planning.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n* There is a lot of experimental evidence that hippocampal coding is predictive, so theoretical work that frames hippocampal coding explicitly in terms of maximizing transition probabilities is well motivated and also understudied\n* With mild assumptions, many relevant features of hippocampal coding are naturally recovered.\n* Figures are well made and clear\n\nWeaknesses:\n* It would be nice if they authors discussed connections to known experimental results in a little bit more depth\n* The model is obscured a little bit by notation\n\nQUESTIONS:\nMost of my comments / questions are relatively minor:\n* 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\n* How robust is this setup to noisy place cell responses? I.e., if the embedding $h$ is noisy due to firing rate variability, what changes, and how much noise can one get away with?\n* 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  ( $t\\rightarrow \\tau$ or $p(y|x,t)\\rightarrow p(y(t+t')|x(t'))$ for example) might clarify the meaning for readers in the early sections of the paper.\n* 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?\n* 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?\n\nLIMITATIONS:\nSee above", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}