Place Cells as Multi-Scale Position Embeddings: Random Walk Transition Kernels for Path Planning
Authors: Minglu Zhao
Dehong Xu
Deqian Kong
Wenhao Zhang
Ying Nian Wu
Venue: NeurIPS 2025 poster ()
Keywords: place cells
population coding
position embedding
random walk transition kernels
path planning
spatial navigation
cognitive map
Primary area: general_machine_learning

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.

Place Cells as Multi-Scale Position Embeddings: Random Walk Transition Kernels for Path Planning
Minglu Zhao' Dehong Xu Deqian Kong',, Wen-Hao Zhang Ying Nian Wu! UCLA UT Southwestern Medical Center
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 cognitive map of adjacency. The combination of non-negativity and inner-product structure naturally induces sparsity, providing 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 dorsoven- tral 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 single representational geometry.
Introduction
Place cells in the hippocampus central to spatial cognition, firing selectively at specific locations as animals navigate their environment [1, 2]. Rather than modeling individual place fields, we propose viewing place cell populations non-negative position embeddings derived from spectral decomposition of multi-step transition kernels that collectively encode spatial relationships. This population-leve approach captures the hippocampus's role in forming cognitive maps, integrating metric and topological properties for flexible navigation [3].
Hippocampal place cells exhibit dynamic behaviors, adapting firing fields to environmental changes [4], scaling along the dorsoventral axis [5], and engaging in preplay before exploration [6]. These properties suggest multi-scale, adaptive representation that supports navigation across diverse contexts- from narrow passages to open terrains. The hippocampus ability to predict novel paths, in preplay, and maintain robust navigation under environmental deformations further underscores its computational sophistication [7-9].
We formalize these insights by modeling place cell population vector embeddings derived from multi-step symmetric random walk. The transition matrix admits spectral decomposition, enabling construction of non-negative embeddings h1(x, 7) such that the inner product between embeddings approximates the normalized transition probability: (h( r), (y,,,) = g(yla, ( r), where 1x r) is the embedding at location z, and q 7) is the symmetric transition probability over time step Remarkably, combining this inner product structure with non non-negativity and orthogonality constraints
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Project page: httpp://sites..oone www//viwwwpllcc-cellll..... Code: http://gihuub.oommnnnguuuuu Cell.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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induces emergent sparsity- the embeddings naturally develop disjoint support sets, explaining why place cells exhibit localized firing fields without explicit regularization. The time parameter vf defines a spatial scale hierarchy, mirroring hippocampal dorsoventral scaling [5]. ((y[2, 7) defines spatial adjacency between z and y at scale or resolution vT, and the pairwise adjacency relationships (g(vlar, -), Vr, v) are reduced into individual embeddings (h(1,, r), Vx) that collectively form multi- scale, Euclideanized, and sparsified cognitive map of the environment. The Euclideanization emerges from inner product expansion of the transition kernel, the sparsification arises naturally from non- negativity of the place cell responses and the orthogonality between embeddings with zero inner product, and the multi-scale hierarchy enables adaptive navigation across spatial resolutions. Efficient matrix squaring (P2r = P2) computes global transitions from local ones (P1) without past trajectory memorization, enabling preplay- liike shortcut detection [10, 7].
Our framework uses gradient ascent on (3)a,, = (M(x; r), h(y +)) with adaptive scale selection, choosing the time scale maximizing the gradient for trap-free, smooth trajectories. This produces robust navigation with properties like boundary avoidance, liffraction-like passage guidance, aligning with hippocampal navigation [11, 9, 3]. Continuous interpolation ensures smooth gradient fields, supporting natural paths. Due to the Euclideanization, path planning amounts to gradient descent on Euclidean distance at the selected scale, enabling what can be called "straight forward" path planning.
Beyond spatial coding, we propose a novel theoretical framework where theta phase naturally emerges from the population embedding structure, unifying spatial representation with temporal dynamics and offering a principled account of theta phase precession.
Appendix A provides an extensive review of related work.
Our contributions include: (1) reconceptualizing place cells as non-negative embeddings from spectral decomposition encoding transition probabilities at multiple scales; (2) demonstrating emergent sparsity explaining localized place fields; (3) modeling multi-scale, Euclideanized, and sparsified cognitive maps via time scale parameter /F for transition probabilities; (4) employing efficient matrix squaring to build up multi-scale spatial relationships; (5) introducing adaptive gradient ascent for Euclideanized path planning at selected scale; (6) proposing theta phase formulation from population embeddings, and (7) demonstrating biological plausibility through properties like preplay. Bridging connectionist models [12, 13] and cognitive map theories [2, 9], our framework offers a scalable, biologically inspired model of hippocampal spatial navigation.
2 Method
2.1
Multi-Step Random Walk Transition Kernel
The foundation of our approach is symmetric random walk on discrete lattice over the environment (c.g.. 40 x 40 lattice), with a subset of lattice points belonging to the obstacles. This random walk serves as a mapping policy rather than a goal-reaching policy. Remarkably, this purely random exploration policy leads to optimal path planning without explicit policy optimization. We use 7 to denote the time step of this random walk mapping policy, to avoid confusion with the time of the planned trajectory. T plays the role of scale, which adaptively selected during path planning.
For a location & = (i, j) on a 2D lattice, let N(x) be the set of its unobstructed neighbors, i.e., neighbors that do not belong obstacles (e.g.. 4 nearest neighbors), we define the one-step transition probabilities as:
For each unobstructed neighbor y of x: p(ulE, = 1) = Pmove (e.g., 1/4 in the case of 4 nearest neighbors)
For self-transition: p(z|æ, T = 1) = 1- 1 Pmooe
where [2(x) is the number of unobstructed neighbors of æ. A critical property of this formulation is that the transition probabilities are symmetric, meaning P(y|ær = 1) p(zly, = 1) for all locations x and y. This symmetry ensures that the random walk process preserves the bidirectional nature of spatial relationships, which is essential for creating a well-defined proximity metric.
The above random walk defines r-step symmetric transition kernel p(y jz, 7) over time step r, which measures the spatial adjacency between and y at spatial scale or resolution captured by т.

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2.2 Heat Diffusion, Geodesic Distance and Topological Connectivity
Our discrete random walk model establishes connection to the continuous heat equation. For small spatial discretization dæ and temporal discretization dt with dr Vdt, our discrete random walk converges to the heat diffusion equation with reflecting boundary conditions as dx - 0 [14, 15]:
Op(yla, 7) = aV p(ypo, T)
on
Ω \
Rabstucles
ÔT Oplula, T) = 0 on 0 oasstaaes
On
where 52 is the whole region, Pobstacls is the region of obstacles, and doossuiles is its boundary. a the diffusion coefficient (different one-step transition p(y]2z, T = 1) in discrete case leads to different a, e.g., a 1/4 for 4-nearest neighbor random walk).
fundamental result from heat diffusion theory relates the short-time behavior of the heat kernel to geodesic distance. For small values of the heat kernel has the asymptotic form [16, 17]:
Lg y)
T)
exp
4TTT
4aT
where (2, y) is the geodesic distance. For open domains, (z( 2) becomes Euclidean distance, and p(y|2;, (x, (arr, Gaussian distribution with variance 2&t standard deviation V2a7, demonstrating that / correspond to the spatial scale. We take square root of r to emphasize this scaling relationship between time and space.
This result demonstrates that (z, y) = - log (tyl, r) approximates the squared geodesic distance for small values of t, providing a fundamental connection between random walks and geodesic distances in complex environments.
As increases, the transition probability incorporates additional information about path multiplicity and global connectivity, creating a multi-scal representation of spatial relationships. In particular, for large т, the transition probability is dominated by cigenvalues that are close to 1, whose eigenvectors depend on topological connectedness rather than local geometry.
Appendix D provides detailed background on heat equation. Appendix G explains topological properties for large T.
2.3 Place Cells as Non-negative Spectral Embeddings
We formalize place cell population as vector embedding h(x, 7) E R", where n is the number of place cells (e.g.. = 500). The inner product between embeddings approximates the normalized transition probability kernel:
(4) where glyl:, T) p(yll,, plula,,)////pc/ p(ylu,T) is the normalized transition probability, so that [|h(,,, T) due normalization. For each cell i, h,(2,,) r) 0 for biological plausibility, and h2(2, r) is the response map or profile of cell i at spatial scale vF.
ss T(ylar,
The above formulation can be derived through spectral decomposition of the transition matrix. Since the one-step transition matrix P1 is symmetric by construction, its powers P, = PÍ admit an eigen- decomposition [18]: P. QA=QT where is orthogonal (QTQ= 7) and A diag(A, .n, contains eigenvalues 0 s A I (assuming the random walk is irreducible and aperiodic).
From this spectral decomposition, we can construct spectral embedding by defining: H,( T) x (x()) where Q; is the i-th column of Q. This yields an exact representation of the transition probability through inner products:
plylx,7) = Σ
(H(a,,) ((y,)
(5)
where Hz(x, 7) is the i-th element of the vector H (I, For the normalized transition probability q3y2,, r), define normalized embeddings:
H(I,T)
T(I,T)
Nspee (ar, T) = |H (H,
(6)
Vp(x|z,7)
3

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which satisfy:
p <()22,))
(Mspec (1) T), Mspec (v,
vp(0 T) p(ylu.
p(1/2, 7) is constant in the open field and is smooth in general, so q is essentially scaled version of p-
However, these spectral embeddings may contain negative components, conflicting with the biological constraint that neural firing rates must be non-negative. Horn's theorem [19, 20], which is built upon the above spectral decomposition, guarantees the existence of a non-negative l(z,, T), such that (h(x, r), hy,, 7)) g ar, 7) for non-negative matrix factorization.
This formulation represents a fundamental shift from modeling individual place cells to modeling the place cell population as distributed position embedding. The embedding vector (æ, r) represents the firing rates of all place cells at location x for spatial scale vT, capturing the idea that it is the pattern across the population-nol the activity of any single cell-that encodes location.
The time parameter /T serves as a fundamental unit of spatial resolution or scale, with larger values producing broader, more diffuse representations, and smaller values producing more localized repre- sentations. This naturally mirrors the variation in place field sizes observed along the dorsoventral axis of the hippocampus 21J.
See Appendix C for more details on spectral analysis. Appendix provides analytical results for open field, where q0y|2, 7) is Gaussian, and elements of hizz, r) exhibit Gaussian profiles over z.
2.4 Matrix Squaring, Learning, and Continuation
Let Pi be the one-step transition matrix for the random walk on a discrete lattice. Pi depends on obstacles in the environment and amounts to local one-step exploration, detailed in subsection 2.1.
We calculate Pr for a discrete set of r, T= (r = 2*, I,.... K}, via Par P?? The matrix squaring is very efficient for calculating P. for T, and these P. correspond explorations of different spatial scales vT, where the adjacent spatial scales in TT have ratio v2, mirroring grid cell module scaling (~1.4-1.7) [22]. addition to matrix squaring, one can design any discrete sequence T {44, k = 1,..., K}. and calculate P, for T using matrix multiplication.
We learn a separate population of place cells h(r,7) for each T, by minimizing the least squares eлror:
C
Σ [a(ulz, r) (11 k(y,))))))))) 22,11
(8)
We learn A(2, r) over the discrete lattice, where Σ.,y in (8) is over all pairs of points on the lattice. We optimize this objective using the A optimizer [23], where after each iteration, for each æ, set the negative elements of hzz, r) to O, and then normalize h(x, r) so that [[A(z, +)ll = 1.
The learning reduces pairwise adjacency relationships (g(v) x. r), Var, y) into individual embeddings (h(2, 7), Vr.,, which collectively form map of the environment.
After learning,
use bi-linear interpolation to make h2x,7) continuous map over As result, و -) = (1( T), (y) also becomes continuous, approximating the transition kernel of the continuous heat equation. The continuous gyylar, 7) can then elegantly guide path planning in continuous space instead of discrete lattice.
2.5 Euclideanized Cognitive Map
Geometrically, (h(x, r), kly, r)) is the cosine of the angle between the normalized vectors h(z, 7) and h(y, r). Moreover, we have
- [1(2, hlu, 7)1 =1 - (h(x,T);,(,,TT)) = -0v/2,,7
That is, the angle or the Fuclidean distance between (x,,7) and h(y, encodes proximity or adjacency between and y. Thus, our method geometrizes transition probability.

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(h(2, 7), Vx) is 2D manifold in the high-dimensional embedding space. This 2D manifold is an embedding of the 2D physical space. Because of the flexibility endowed by the high nigh-dimensional embedding space, the path between and y the manifold is essentially "Fuclideanized" even though the physical path is far from being Euclidean.
2.6 Emergent Sparsity from Non-negativity and Orthogonality
The conjunction of non-negativity and orthogonality induces emergent sparsity in the representá- tional code. Let h(2, 7) R. denote the representation position z and scale vF, such that (h(x, (y,,,yy)))
g pr 7), where q(yl ær, r) measures spatial adjacency at scale VT. If two positions z and are non-adjacent (g(y) Ir, T) O), the corresponding vectors must be orthogonal. Because all components of h(x 7) are non-negative, orthogonality implies disjoint support-no com- ponent can be active in both representations. Consequently, the joint requirement of non-nen non-negativity and orthogonality forces most components to be zero, yielding sparse, localized activity patterns. As result, each place cell fires at a localized place, and the population of place cells collectively tile the environment. The symbolic representation thus emerges automatically from population-base representation.
The scale parameter modulates this sparsity: for small т, most position pairs are orthogonal, producing highly localized "place fields"; ; for large т, overlaps increase and place fields broaden. Thus, the locality of place-cell activation follows directly from the geometry of non-negative, inner- product spectral embeddings. Appendix F provides technical details.
2.7 Straight Forward Path Planning with Adaptive Scale Selection
We adaptive gradient following for goal-directed navigation. When navigating from current position x to a target location y. we select the next position Tnex from the neighborhood 0(x) where 0(x) = (= direction in j0, 2T)). We use 0(x) for continuous x for path planning to differentiate from N N(x) in the discrete lattice for random walk.
Ar(cos 0, sin 0)). Ar is the step size, and 0 is discretized into equally spaced
The path planning algorithm is as follows:
Compute the gradient of the normalized transition probability for each neighbor a(x) and each scale T:
A = (() <0l|, r) (h(v:7), - h(x,,
(10)
Select the scale 7" that provides the strongest directional signal:
arg max max A(z,7) r)
(11)
TET #EO(2)
Choose the next position that maximizes the gradient the selected scale:
arg max A (2, 7*)
(12)
xE0(x)
Note that in the above algorithm, z and are continuous, because h(x, r) is made continuous in with bi-linear interpolation after learning discrete lattice.
A(Z,T) measures the reduction in the angle squared Euclidean distance. The path planning seeks maximal reduction in angle or Euclidean distance. Therefore we call it the straight forward path planning the Euclidcanized embedding space, where the vector h(,, 7) rotates straightly (ly, on the path.
This approach selects the scale r" that provides the clearest guidance for the current navigation step. Intuitively, larger scales provide better guidance for distant goals, while smaller scales offer more precise navigation for nearby goals. The adaptive scale selection mechanism automatically finds this optimal scale at each step, similar to choosing the most appropriate "ruler" for measuring at the current distance.
Now consider the idealized continuous limit where Ar + 0. is continuous, and 7 con- tinuous. The path planning algorithm follows the gradient V=g(vlz,7) Veg(zlg,7) where
5

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q(512, r) (arly. =) due to symmetry. The planed trajectory is the gradient ascent flow: dr(t)/dt v.q(((t))) r()). where is the time on the planed trajectory. r(t) arg max, [Vag(z(t)l) r) is the optimal scale at time / with the maximal gradient.
The gradient-based navigation framework has the following key properties that ensure computational efficiency and biological plausibility:
Adaptive scale selection dynamically adjusts VT x d(a.y) for precise navigation in the open environment, mirroring hippocampal spatial tuning [11].
2. The unimodal smooth gradient field Vag(< r), with unique maximum at the goal, ensures trap-free paths, aligning with hippocampal navigation [11, 9].
3. Planned paths match the shortest path for small but prioritize topological connectivity for large
т, reflecting cognitive map robustness [9, 3].
4. Near obstacles, V= 7) flows parallel to boundaries, preventing collisions, akin to hippocam- pal obstacle avoidance [11].
Diffraction-lik patterns guide trajectories through passages, resembling hippocampal maze navigation [11, 9].
Topological invariance maintains navigation under environmental deformations, mirroring hip- pocampal place cell encoding [3, 9].
7. Matrix squaring from local transitions predicts shortcuts, embodying hippocampal preplay [7. 8j.
These properties, detailed in Appendix G, underpin the framework's 's alignment with neural mecha- nisms.
2.8 Theta-Phase Procession Based on Angle-Phase Duality
The hippocampus not only encodes spatial relationships but also organizes temporal dynamics through theta phase precession, where place cells fire at specific phases of the theta rhythm as an animal traverses their fields [24, 25]. We extend our population embedding framework to model this phenomenon.
As explained above, the non-negativity and inner product structure of the position embeddings induces localized sparse patterns, i.e., each place cell only fires localized field around place, and together the place fields of all the place cells tile the environment. Let arg max, h2(2, T), i.c., the center of the place field for cell The theta phase of cell can defined terms of the angle between h(z, r) and (P,, T). As x approaches from H4, the angle changes from π. /2. Appendix H provides details.
the angle changes from π to O, and as moves away
2.9 Integrating Grid Cells
For scientific reductionism, we focus on place cells without incorporating grid cells. A natural extension would explore the relationship h(2, r) (F))g(z), where g(x) represents the vector of grid cell activations and ((T) is a learned transformation matrix, potentially unifying our framework with the complementary roles of place and grid cells in spatial representation [9]. Appendix provides
the formulation.
Experiments
We design experiments to evaluate both the biological plausibility of our position embedding frame- work and its functional capabilities in path planning. All experiments are conducted a simulated environment. We first examine the positional representation and population density profiles in open field. Then we create more complex environments by adding obstacles. For implementation details, Appendix J; For additional experiment visualizations, see Appendix K.
3.1 Place Cell Representations in Open Field
We begin our numerical experiments in a simple open field environment to demonstrate fundamental properties. The environment consists of 40 x 40 lattice grid. For the transition kernel,

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A
B
T == 16
7
= 256
Start
Target
1004
120
Time
C
Response Value
max
T=4
== 16
64
256
Figure I: Place Cell Representations and Navigation in Open Field Environment. (A) Goal-directed path planning trajectories with adaptive scale selection (selected scale is color coded, red for big scale and blue for small scale). (B) Normalized transition probability kernels qyyla, 7) multiple scales with gradient vector fields. y is fixed at the center of the environment. (C) Learned activation patterns of h(x,, 7) different scales across randomly chosen cells, exhibiting Gaussian-like firing fields centered at specific locations within the open environment.
employ a 3 X 3 neighborhood, meaning each position has at most 8 neighboring points, with transition probability Pmove 1/9 distributed uniformly among neighbors well as self. Within this environment, we learn h(x, r) embeddings across all lattice points. For hyperparameters, the total number of place cells set to 500. The temporal dynamics are captured across multiple scales using the time parameter T = 2", where k = 1,2, 11, allowing us to analyze system behavior across spatial scales.
3.1.1 Multi-Scale Transitions
To evaluate the normalized transition probability kernel ( [r, r). we visualize the learned Gaussian- like patterns across multiple temporal scales in Figure 1 (B). Each panel depicts the transition probability distribution for time scales = 16, 64, 256 with position y fixed at the environment center.
The gradient fields reveal a critical temporal dependency: as т increases, gradient magnitudes intensify for locations distant from the center, while maintaining directional integrity toward the target. This multi-scale property is particularly significant for path planning applications, as it enables multi-resolutin navigational framework.
3.1.2 Spatial Activation Patterns
Our model trained by minimizing Equation 8. The optimization results demonstrate remarkable fidelity in approximating normalized transition probabilities through inner products of population embedding vectors with correlation coefficients above 0.9 for all scales. Figure (C) illustrates the learned activation patterns of h(z, r) at scales - 4, 16, 64, 256 for randomly chosen cells. The spatial receptive fields of individual units emerge naturally from our training process, exhibiting characteristic Gaussian-like firing fields centered at specific locations within the open environment. This spatial tuning closely resembles the dell-docu properties of hippocampal place cells observed in rodent navigation studies [1, 2].

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3.1.3 Path Planning and Adaptive Scale Selection
To evaluate the navigational capabilities of our model, we implemented the gradient-based path planning. We randomly selected start and target locations within the open field environment. At each step, the agent evaluates potential next positions by sampling orientation from 36 equally spaced directions within [0, 2x) and a fixed radius Ar one grid unit. Even though our model is learned in 40 x 40 discrete lattice, in path planning, representation of its location (1 T) is calculated by linear interpolation of the 4 closest neighbors. Movement direction determined by maximizing the gradient g 2, 7) q(y|2, 7) where y the larget location, x is the current position, and represents each candidate next position. And for each step, we choose the with the largest gradient. Our results in Figure 1 (A) demonstrate that the learned model consistently generates near-optimal trajectories.
reach continuous positions æ, where the
A key feature of position embedding model is the adaptive scale selection mechanism, where T" is selected with maximal gradient. The mechanism is illustrated in Figure (A), where different colors represent the selected time scale 7* at each navigational step. This reveals systematic progression from coarse to fine scales as the agent approaches its goal. This pattern emerges naturally from the gradient fields associated with different scales.
When navigating to distant goals, larger scales (T 2 128) are selected, which provides clearer guidance for long-range planning. As the agent reaches medium distances from the goal, the selected scales transition to intermediate values (T = 16 to r 64), balancing directional guidance with increasing spatial resolution. In the final approach phase, the system converges the smallest available scales (r = 2 to r = 8), which offered the most precise local guidance.
This adaptive scale selection mirrors the progressive engagement of different regions along the dorsoventral axis of the hippocampus during navigation, as observed in rodent studies [5, 21].
3.2 Place Cells in Complex Environments
To investigate the robustness and adaptability of our method, we extend our analysis to complex environmental geometries that more closely resemble naturalistic navigation scenarios. These envi- ronments incorporate obstacles and boundaries that fundamentally alter the adjacency relationships between locations, requiring the model to learn representations that respect environmental constraints rather than simple Euclidean distances.
3.2.1 Environment and Experiment Setup
To evaluate performance across different environments, we implement three distinct mazes. Detailed visualization shown in Figure
U-shaped Maze: A corridor structure with single 180-degree turn, creating a simple non-convex navigation challenge.
S-shaped Maze: A serpentine corridor with multiple turns, requiring longer detour paths around obstacles.
Four-room Maze: A compartmentalizee environment with four chambers connected by narrow doorways, representing hierarchically structured space.
To accommodate environmental complexity, we modify random walk dynamics incorpo- rate spatial constraints imposed by obstacles and boundaries. For any location I. transitions to neighboring locations y that are prohibited by the obstacles setting p(y|z, 7 1) for these for- bidden transitions. Consequently, the self- colf-transition probability is adjusted to maintain normalization p(z| - 1)
[z)) Pmuve. The key difference between open field and obstacle-containing environments lies in the value of (z). In open fields, N (z)) consistently includes all 8 neighboring locations (except at boundaries). With obstacles, N(x) represents only the subset of accessible neigh- boring locations from position æ, which could be smaller than 8. We maintain Pinoe = 1/9 across all environments for consistency. Despite these modifications to the underlying transition dynamics, our learning approach remains consistent. We compute the normalized transition probabilities y(ylar, 7) based on these resstrin dynamics and train our model learn the embedding function h(z, r) using the same objective function as in the open field.
8

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A
Start Thaget
Target
Target
Time Scale (t)
B
Response
Start Target
Decision Point
C
pathway
16
T=4
Figure 2: Place Cells in Complex Maze Environments. (A) Path planning through obstacle-containing environments. (B) aopologically-infomeee transition kernels q(ylæ, 7) with gradient fields. Target is marked as red point. (C) Randomly sampled place cell profiles at multiple spatial scales. (D) Remapping with environmental modification.
3.2.2 Navigation Efficiency
We evaluated our model's navigation capabilities across multiple environments, conducting 50 trials per environment with randomly sampled start and goal positions from non-obstacle areas. Our model achieved 100% success rate across all tested environments.
We compare against the Bug algorithm [26] as baseline since prioritizes finding feasible paths rather than strictly optimizing for the shortest path. The Bug algorithm moves directly toward the goal until encountering an obstacle, then follows the obstacle's 's boundary until can resume its direct path. We implement the gracle-enhancee version [27] that knows which foundary-followinn direction yields the shortest path for each obstacle, establishing an upper bound on performance.
To quantify efficiency, we use Success weighted by inverse Path Length (SPL.) [28], where val- ues near 1.0 indicate near-optimal paths and val- exceeding 1.0 suggest our method found shortcuts the Bug algorithm missed. Details of SPL calculation appear in Appendix J. sim- pler environments (U (U-shape), our method per- forms comparably the oracle-enhanced Bug algorithm. However, in complex environments (S-shape and Four-room), our approach signif- icantly outperforms the baseline, discovering more efficient paths that the Bug algorithm fails to
Table I: Path planning results.
Environment
Success
SPL (1)
Open field
100%
0.991 0.04
U-shape
100%
0.919 0.25
S-shape
100%
1.392 0.98
Four-room
100%
1.519 1.17
identify even with oracle assistance.
Table 2: Path planning in Four-Room Room environment.
Metrics
Bug (w/ Oracle)
Bug (w/o Oracle)
* Search
Random Walk
Our Method
Success (%)
100%
14%
100%
2%
100%
SPI. (1)
0.66 0.51
0.15 + 0.00
1.08 + 0.50
0.01 0.00
1.00

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We also compared our method with several different baselines in the Four-Room environment, with each trial limited 50,000 steps. The results in Table 2 show that baseline approaches like the Bug algorithm (without oracle guidance) and Random Walk fail to solve most of the challenging scenarios.
3.2.3 Transition and Learned Profile in Complex Environments
To evaluate the adaptive capabilities of our approach, we examine the transition kernel g(yla; 7) and the resulting place cell activation profiles h(z,7] 7) in environments containing complex obstacles. Figure 2 (B) presents the normalized transition probability distributions g(y|æ, r) for strategically selected locations, with a large scale to reveal long-range spatial relationships. The diffraction like patterns, where probability flow encounters obstacles, reveal how spatial information propagates through available pathways while respecting environmental constraints.
The gradient fields illustrate navigation potential, showing how agent could efficiently pass obstacles. Importantly, these transition probabilities capture the true topological structure of complex environments rather than simple Euclidean distances, a critical property for realistic navigation where direct paths are often blocked. The emergent probability gradients provide a natural mechanism for guiding optimal path planning that automatically adapts environmental geometry.
We visualized place cell profiles at multiple spatial scales across complex maze environments in Figure 2 (C). Each panel displays small subset of non-overlapping place cells at different scales т, emphasizing the emergent sparsity and localized nature of individual receptive fields. Despite increased environmental complexity, the learned place cell population maintains comprehensive spatial coverage throughout accessible regions. At smaller scales, place fields are tightly localized; larger scales, fields broaden while maintaining their localized tiling structure. This multi-scale coverage property demonstrates the model's robust ability to develop effective spatial representations regardless of environmental geometry, critical feature for reliable navigation in diverse settings.
3.2.4 Remapping Properties
We evaluated our model's 's ability to adapt to environmental modifications, analogous to the remapping phenomenon observed in rodent place cells. Previous neurobiological studies have shown that hippocampal place fields reorganize in geometry-dependent ways when familiar environments are altered through elongation restructuring.
We conducted two-phase experiment with the environment shown in the upper panel of Figure (D). After training our model to convergence (2000 iterations) on this initial configuration, we modified the arena by removing several obstacles to introduce a novel shortcut path, as shown in the lower panel of Figure 2 (D). The intervention created minimal physical alteration but significantly changed the environment's 's topological structure.
We then fine-tuned our pre-trained model on this modified environment with just 50 iterations at reduced learning rate (5e-4), allowing the position embeddings adapt efficiently to the new spatial configuration. When testing path planning to identical target locations, the fine-tuned model successfully identified and utilized the newly available shortcuts critical decision points where pathways diverged.
4
Conclusion
This paper reconceptualizes hippocampal place cells as population embeddings approximating multi- step random walk transition kernels, where (h( T), h(y,T)) = q(yl xr, r), offering a biologically plausible model of spatial navigation [29, 30]. The time parameter v/ defines a multi-scale represen- tation, mirroring dorsoventral place field scaling [5], adaptability [4], and remapping [31]. Gradient ascent on 4(y|2; 2z r) with adaptive scale selection produces trap-free trajectories, guided by boundary avoidance, diffraction-lide like passage navigation, aligning with hippocampal cognitive maps [11, 9, 3]. Efficient matrix squaring (P2r = P?) computes global transitions from local ones (Pi) without past trajectory memorization, enabling preplay-like shortcut detection [10, 7, 8). Bridging connectionist. models [12, 13] and cognitive map theories [2, 9]. our framework captures hippocampal navigation's dynamic properties, providing scalable, computationally efficient model for complex environments.
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Acknowledgments
Y. W. is partially supported by NSF DMS-2415226, DARPA W912CG25CA007 and research gift funds from Amazon and Qualcomm.
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Contents
1 Introduction
1
2 Method
3 Experiments
4 Conclusion
10
A Related Work
15
B Limitations
17
C Spectral Analysis of Multi-Step Random Walk
17
D Heat Diffusion with Reflecting Boundaries (Background)
19
Open-Field Environment
20
Emergent Sparsity from Non-negativity and Orthogonality
22
G Path Planning Properties
23
H Theta Phase Modeling
28
I Grid Cells Integration
29
J Implementation Details
30
K Additional Experiment Results
30
Sections and subsections that contain mostly background materials are marked in the titles. The remaining sections and subsections contain novel developments.
A Related Work
A.1 Place Cell Models and Representations
The study of hippocampal place cells has a rich history since their discovery by [1]. Computational models of place cells have evolved from simple Gaussian tuning curves [29] to more sophisticated approaches. Several models have explored population-level representations of place cells, including manifold embeddings [32] and latent space models [33]. However, these approaches typically treat place cells as separate entities encoding specific locations rather than collective embeddings encoding transition probabilitics.
Matrix factorization approaches to neural population activity have been applied to various brain regions [34, 35], though rarely with the specific mathematical connection random walk processes proposed in our work. Recent work by [13] proposed successor representation framework for place cells, which shares some conceptual similarities with our transition probability approach but differs in the specific mathematical formulation and implementation.
Hidden Markov Models (HMMs) have also been applied model hippocampal spatial coding. The Causal Graph model (CSCG) [36] treats space as latent sequence and uses structured graph with cloned nodes disambiguate aliased sensory observations in different contexts. Related work [37] further develops sequence-based models of hippocampal function.
A.2 Multi-scale Spatial Representations
The variation in place field sizes along the dorsoventral axis of the hippocampus has been extensively documented [5, 21, 22], but computational models that explicitly address this multi-scale organiza-
15

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tion remain limited. Models incorporating scale in place cell representations include hierarchical approaches [38], wavelet-like representations [39]. and scale-space theories [40] borrowed from computer vision.
Our approach differs by deriving the multi-scale representation directly from the time parameter of random walk process, providing a principled connection between scale and exploration time that has been previously exploited place cell models.
A.3 Navigation and Path Planning
Biologically-inspir navigation algorithms have drawn on various hippocampal properties. Vector- based navigation models [41, 42] and successor representation approaches [38, 43, 44] have demon- strated effective navigation capabilities. Diffusion-based path planning algorithms [45, 46] share mathematical similarities with our heat equation formulation but lack the direct connection neural representations.
Recent work by [47] and [48] has emphasized the role of population coding in navigation but without the specific inner product relationship and transition probability framework we propose.
A.4 Successor Representation
Our work is closely related to the Successor Representation (SR) [49, 50], as both frameworks build upon powers of transition matrix to construct representation of space. The SR is typically defined as discounted sum, M ox= (+T)), which yields the expected discounted future occupancy of all states from any given starting state, and the discount factor y implicitly sets single, predictive temporal horizon. In contrast, directly, P. Pr, to explicitly model the transition probabilities at multiple, distinct time horizons, where the parameter corresponds directly to the time scale of random walk. This multi-scale representation aligns more closely with the observed functional gradient of place field sizes along the hippocampal dorsoventral axis.
approach utilizes the discrete powers of the transition matrix
The primary novelty of our framework lies in learning a biologically-conntraamme non-negative matrix factorization of these transition kernels, where vector embeddings (x,T)) are learned such that their inner product reconstructs the transition probability, (l( r), h(y, 7)}- = g(y (z, 7). crucial theoretical insight is that the combination of this inner product objective with non-negativity and orthogonality constraints for distant locations gives rise to emergent sparsity; the embeddings are foreed have disjoint support sets, providing geometric explanation for the localized firing fields of place cells without explicit regularization. This multi-scale architecture confers significant functional advantages, enabling adaptive gradient-ascent based navigation policy that selects the optimal scale r" at each step to generate smooth, trap-free trajectories. This approach is not only more direct navigation mechanism that bypasses the need for explicit value
scalable, but also offers function computation common in SR-based reinforcement learning agents.
A.5 Inner Product Spaces in Neural Representation
Inner product spaces as a basis for neural computation have been explored by several researchers [51, 52]. More recently, [53] proposed that neural populations represent probability distributions through their activation patterns, with some conceptual overlap with our approach.
Navigational planning inner product spaces has connections to kernel methods machine learning [54] and information geometry [55]. though these connections have been underexplored in neu- roscience. Our work bridges these fields by explicitly relating inner products between place cell populations transition probabilities derived from random walk processes.
A.6 Heat Equation and Diffusion Models in Neuroscience
The connection between neural dynamics and diffusion processes has been explored in various contexts [56]. The specific relationship between heat kernels and geodesic distances, which forms foundation for our approach, has strong connections to manifold learning [57] and dimensionality reduction techniques [58].
16

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Our work builds upon these ideas by applying them specifically to place cell populations and spatial navigation, providing novel bridge between diffusion processes, neural representations, and navigational behavior.
B
Limitations
B.1
Lack of Sensory Integration
Our current model does not incorporate sensory inputs for spatial navigation[ [1. Real hippocampal place cells integrate multimodal sensory information including visual landmarks, self-motion cues from path integration, and boundary detection. This sensory integration is fundamental to how place cells form their spatial receptive fields and anchor representations to environmental features.
Without sensory processing, our model cannot explain how place fields emerge during initial exposure novel environments. In our framework, position embeddings h(2r, r) are learned from pre-defined transition probabilities that already encode environmental structure, rather than being built up through sensory experience. We also cannot model perceptual anchoring, where place cells maintain stable firing relative to visual landmarks, or how sensory conflicts (such as navigation in darkness) affect place cell stability. Future extensions incorporating vision-based inputs, self-motion integration, and boundary vector cell signals would significantly enhance biological plausibility.
B.2 Limited to 2D Static Environments
Our experiments and analyses are restricted to two-dimensional static environments, whereas biologi- cal spatial navigation operates in three-dimensional space and continuously changing contexts[5, 59]. Studies in flying bats have revealed volumetric place cells encoding 3D positions, and terrestrial animals navigating multi-level structures require firee-dimensional representations. Extension 3D introduces complications our framework does not address: vertical movements and gravity-dependent asymmetries. different place field scaling across dimensions, and substantially increased computa- tional complexity. Dynamic environments would require time-varying transition probabilities and continuous updating mechanisms our framework does not provide.
B.3
Online Sequential Learning
Another limitation is lack of online learning over sequential experiences model temporal dynamics and memory consolidation[7, 8]. Our batch optimization learns h(2, r) across all spatial locations simultancously, fundamentally differing from how biological place cells develop through experience. Biological place cell formation is incremental: initial weak or unstable spatial tuning gradually sharpens through repeated exploration via eetivity-dependee framework does not capture this gradual emergence from unstructured initial conditions. Our matrix squaring P2r P² achieves global integration instantaneously based on environmental structure, not through iterative learning over aleep-wake cycles analogous to biological replay.
synaptic plasticity. Our
Despite these limitations, principles underlying hippocampal spatial navigation, particularly the role of multi-scale transition probabilities, emergence of sparsity from geometric constraints, and the connection between random walk dynamics and cognitive maps.
framework offers valuable theoretical insights into computational
C Spectral Analysis of Multi-Step Random Walk
This appendix provides detailed spectral analysis of the multi-step random walk transition kerne,, establishing connections between our random walk model and position embeddings.
C.1 Eigendecompposition of the Transition Matrix (Background)
Since the one-step transition matrix Pi is symmetric by construction, it admits an eigendecomppsition:
P = QAQ".
(13)
17

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with orthogonal Q (QTQQ = I) and diagonal A = diagí A1..... An), where 0 S X; S 1, where we assume the random walk is irreducible and aperiodic. The multi- transition matrix
PT = QATQ"
(14)
The eigenvalues are typically ordered as 1 = A > 2 A 2.., with the first eigenvalue corresponding to the stationary distribution and subsequent eigenvalues capturing spatial patterns at increasing levels of detail.
C.2 Position Embeddings from Spectral Decomposition
The spectral decomposition provides natural position embedding. If we define:
Hz( T) = x77222(())
(15)
in the discrete case where Qi(x) is the i-th column of Q, or:
Hi(x, = e
(16)
in the continuous (where X; and Φs are respectively eigenvalues and eigenfunctions of the Laplacian), then the transition probability can be expressed as:
p(yla,7) - (H(,, ((,
(17)
This provides a closed-form expression for position embeddings that exactly reproduce the transition probabilities through inner products.
C.3 Normalization and Non-Negative Embeddings
To obtain normalized embeddings, we define:
(x,,))
H(r, r)
hupec (I, [a,
(18)
vp(xla,T)
This normalized embedding satisfies:
plyle, T)
(Ispea TT, Ntspe (V,
<(v12,
(19)
Vp(a/z, T) p(uly.T)
However, hop (I,7) 7) may contain negative components, which conflicts with the biological constraint that neural firing rates must be non-negative. This is where Hom''s 's theorem becomes relevant.
C.4 Non-Negative Matrix Factorization (Background)
Horn's theorem [19] provides the theoretical foundation for obtaining non non-negative embeddings from our transition matrices: If A is a symmetric matrix with non-negative entries, then there exists non-negative matrix B such that A BBT.
We provide sketch of the proof:
Proof sketch of Horn's 's theorem. Given a symmetric non-negative matrix A, consider its spectral decomposition A UEUT. Let A Σ. Faunff where a, are eigenvalues and u; eigenvectors.
For each outer product unu (which may have negative entries), we can express it as a linear combination of non-negative rank-1 I matrices: u.u," E, Co where vj are non-negative vectors and Cj are coefficients.
By appropriate selection of vy (e.g.. using vertices of the hypercube defined by the signs of u.), can ensure that сj 2 0 when o, > 0. This allows us to express as a sum of non-negatio non-negaaive rank- matrices, which can be arranged as = BB" where B has non-negative entries.
For our normalized transition matrix Q [g( r) which is symmetric with non-negative entries, Horn's theorem guarantees the existence of a non- non-negative matrix H, such that Q, = H, The of Hy provide our desired non-negative position embeddings h(z, r).
18

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D Heat Diffusion with Reflecting Boundaries (Background)
This appendix provides detailed mathematical derivation of the connection between our discrete random walk model and the continuous heat equation, establishing the relationship between transition probabilities and geodesic distances.
D.1 From Discrete Random Walk to Reflecting Heat Equation
We begin with a discrete random walk on a two-dimensional integer grid. For simplicity, we assume the one-step transition p(y|æ, 7 1) of the random walk is to move one of nearest neighbors with Pmove = 1/4. We assume this simplest p(y r = 1) in our theoretical derivations in all the relevant sections in the Appendix. Similar results can be obtained for more general p(yla, T 1), with different diffusion coefficient a.
Let (i.j) z denote discrete spatial coordinates, and & Zzo denote discrete time steps. To connect with continuous diffusion, we introduce spatial and temporal units:
i dx. y dx, k dt
(20)
where dx is the spacing between adjacent grid points and dt is the time step. Following standard diffusion scaling, we set dr = vdl, which ensures convergence to a well-defined limit as dx + 0 [14].
The discrete random walk has the following transition probabilities:
P(('',''(('.)) = 1) = Pmove = for each unobstructed neighbor (i',5") of (i,j)
(21)
p((i.))((.),k = 1- Pmove 11 N (i,
(22)
where (i.,)) is the number of unobstructed neighbors of location (i,j) (maximum 4 in 2D grid with 4-connectivity).
D.2 Derivation of the Heat Equation
To derive the continuous limit, for interior points (those not adjacent to obstacles), the discrete update rule is:
p(i,j, + = p(i,j,A)( 4pmove) + Pmove ( +
1;)
4p(i, k)] =p(i,j,k) )) + Pmooe + 1, + ( (23)
Dividing both sides by dt and using Prnove = 1/4:
p(i,j,k k 1) p(i,i,k)
1 (p(i + 1,J. 1b) p(i - 1, + + 11,1) 1, (1.
dt
dt
(24)
Using the standard finite difference approximation for the Laplacian [60]]
v'p(i, dz2
lp(i (p(i 1. + p(i 1.j. p(i,j 1, p(i, 1,
4p(i.j,))
(25)
Substituting dx. = di and taking the limit as dt - 0:
p(i. j. 1) p(i.j,k) 4
lim d0
(²p(z,
(26)
dt
This yields the heat equation with diffusion coefficient a = 1/4:
0p(z, y. a VV
(27)
ØT
19

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D.3 Reflecting Boundary Conditions
Our discrete random walk enforces reflecting boundary conditions, preventing probability flow into obstacles, mirroring hippocampal obstacle avoidance [11]. Consider boundary point (i.j) with an obstacle I, so the number of valid neighbors is N(i. j) (points (i - 1..). (i. + 1).
(i,j - 1)). The self-transition probability is Pstay = N(G.] 1. redistributing probability that would flow the obstacle back to (i, j).
The probability update for (i:j) at time step &s + is:
1 pl.ji, - ((.).. + (p(i - 1.5, + +, +())) 1,1
(28)
This can be rewritten as:
- +( - + [p(i- 1, +(, p(i,j j- - -3((i,,k)) (29)
In finite difference methods, the reflecting condition
= 0 is enforced using ghost point at (i + 1. setting p(i + 1, k) = p(i,j. k) to ensure zero normal flux [60]. Substituting into the interior Laplacian update (equation 30):
8p 8m
(i+1.j)
= (p(i k, + p(i k,J,k) + pli.j 1, p(i.j 1,
Ap(i,j,A) (30)
yields equation (29), as ((i++,jj,)) p(i, reduces the neighbor terms three. This substitution ensures the boundary update aligns with the random walk's s mechanism, where Paay assigns zero probability obstacle transitions, maintaining probability conservation and enabling smooth navigation around obstacles, as observed in hippocampal place cell activity [11].
D.4 Connection to Geodesic Distance
Varadhan's 's formula [16] establishes a deep relationship between the heat kernel and geodesic distance. For complete Riemannian manifold M with heat kernel p(z, y, r), Varadhan proved that:
lim - 47 log p(x, T) (x, 7-0
(31)
where dy (ar, y) is the geodesic distance between points and y.
While Varadhan's is original large deviation principle [16] applies smooth manifolds without bound- aries, extensions to domains with reflecting boundaries [61] ensure that the short-time behavior the heat kernel ply, T|æ:, 0) reflects the geodesic distance dy y), the shortest path within Ω avoiding obstacles. For our normalized transition probability: similar asymptotic relationship holds:
d (z
lim -T log d(ul,, T)
(32)
4a
where a 1/4 is the diffusion coefficient. This follows because p(æ|2, T) and p(yly, 7), influenced by boundary reflections, have asymptotic forms that cancel in the logarithm as T + 0, leaving the geodesic term dominant.
As increases, our distance metric transitions from approximating geodesic distance to incorporating more global aspects of the domain's 's connectivity, creating multi-scale representation that seamlessly integrates local metric information with global connectivity structure. Section G explains eigen analysis of connectivity.
E Open-Field Environment
In unbounded, obstacle-free environments (open fields), the symmetric random walk simplifies to an isotropic diffusion process, offering theoretical justification for the Gaussian tuning of place cells
20

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observed in such settings 29]. As the grid discretization refines (dr - 0, dt + 0, dr = dtt.,
the transition probability ply x converges the heat equation fundamental solution 2D free space:
(-IM - x 1?
plu K
exp
(33)
4T
4GT
where 1/4 is the diffusion coefficient, and lyy - - is the squared Euclidean distance. Since
p(z|z, T) Amor is cesition-indepeddeee the normalized transition probability becomes:
1
p(y x.. r)
|ly x=1? 4at
ll x013
((ya,,)
exp
exp
(34)
Vp(a x. p)
This Gaussian kernel, with variance 0² 2at = 7/2,, reflects the diffusive spread of the random walk and mirrors the approximately Gaussian firing fields of hippocampal place cells in open environments. the position embeddings h(z, -) must reproduce this Gaussian
Given ( (z, 7) (h(z, h/, r)). decay. We construct la(2,T) with Gaussian components, providing a mathematical basis for why place cell population activity exhibits Gaussian profiles in open fields.
Theorem (Gaussian Embeddings in Open Fields). In an unbounded 2D open field, where 4(y12, r) exp - ). there exists a position embedding h(I,T) R with non-negative components such that (h(x,T), h(y,T)) 4(y12, T), and each component h((2,) is Gaussian function of x with variance T 2 per dimension.
2
Proof. The transition kernel ((y|2, T) = exp (- =b===l is a positive definite Gaussian kernel with variance 7/ r/2 (since 4ar for 1/4), consistent with the Gaussian tuning of place cells in open fields. We construct h(x, r) E I2 directly as:
|| (_11=-14) mill?
h,(x, cxp
(35)
where 14 E 2 are fixed anchor points (e.g., a uniform grid), C >> are constants, and the variance per dimension is 7/2. The inner product is:
| p4 12
b1 mill3
(h(2,T),
h,(2, (),
Σ & exp
(36)
T
Rewrite the exponent:
2
x +
||x mil 2 + 11 pill ||x y/1² +
Mi
(37)
2
80:
(M,, = exx
(- || [_1xx-y/p 1/2
2 exp
2 8t 2
14g
(38)
1=1
For a dense set of μui (e.g., grid with spacing < TT), the sum approximates a constant over a local
region around (x + /2/ as the terms exp - form kernel density estimate. Set
of +. as n + co, the sum converges to constant C' (proportional to the density of μ4), yielding:
|| yl
(h(x,7), ((,,)))))))))))) se Ceexp
(39)
Adjust C' = by scaling (e.g., = Cn ensuring (h( -), hly, 7)) = q(v 7). Each h, T)
Gaussian with variance T 7/2 per dimension, and hą( 7) 2 0, satisfying the theorem.
This open-field case not only validates our model against biological data but also serves as baseline to explore how environmental structure (e.g., obstacles) perturbs these Gaussian properties, as observed in constrained settings [62, 63].
21

---

Emergent Sparsity from Non-negativity and Orthogonality
Let ((a,(a E R denote the non-negative representation of position x E at scale vr, and let
(h(z, 7), h(y.
g(y x,
(40)
where q(- 1,,7) 2 0 symmetrie multi-step transition kernel (adjacency scale vr), Define the by
(undirected) transition graph G,
(z, E F
=
ạ x, T) > 0
(41)
Note that this transition graph is for steps instead of one step. Write supp(v) = {i E [m] 2s > for the support of non-negative vector 0 R7. and put
0}
S,(T)
{ E X: ((x,T) > 0} (the active set of coordinate i at scale 7)
(42)
Let deg, (I)) denote the degree of a in Gr. (
max, deg, (ar) the maximum degree, and
p(T) =
deg, (z)
(43)
X
the mean neighborhood size. Lemma 2 -iganiigativ + orthogonality + disjoint support). If ((zz)),,((v.7 E R2 and
(h(x, h(y 7)} then
supp(h(I,T)) n supp(h(y,T)) = Ø
(44)
Proof. By non-negativity,
n
(h(x,T), +)) Σ hz h.(,,
(45)
a sum of non-negative terms. Hence each term vanishes: h, T) h, r) = 0 for all i [n]. and
thus coordinate is simultaneously positive in both vectors.
Proposition 3 (Clique structure and sparsity bound). Assume (40) and that
q(v x, r) whenever (x. $ E,
(46)
Then for each coordinate i [n]:
ii (T) induces clique in Gr; i.e. every pair x, E S.(T) is adjacent in Gr
iii Consequently,
|S,(T) S min deg, (a) & 1+ A(T) #ES,(T)
(47)
(iii) The average number of active coordinates per position obeys
< n + A(T))
Σ
[M 7)
| Σ [S,(T))
(48)
| xex
|2|
Moreover, since ((T) = max, (xg,(x) and p(T) is the mean degree, any a priori bound of the form A(T) < C P(T) (with graph-dependen constant C 2 1) yields
n1 + C p(T)]
Σ
[1(xx,))
S
(49)
X
X
Proof. (i) Take any x; y E S,(T) with x + y. Then h4(x,,)) 7) > 0 and hz 0, so by Lemma they cannot be orthogonal. By the hypothesis q(y .r,)) (h(æ, r), h(,, T)}, orthogonality occurs exactly when (I, & E.. Therefore (x,y) E Ez. Since the choice of x, was arbitrary, (T) induces clique.
22

---

(i) Fix any x" E Si(T), By (i), every y S((T) {x"} must be adjacent to x", so S((T) S JuN (a*), where W (x*) the neighborhood of x²" Thus [S((T) 1+g (a*). Minimizing over x² E (T) gives the first inequality; the second follows from deg, (x*) - (T)).
(iii) The identity
Σ SEX
(1(a llo = Σ |t((T)
(50)
is a double-counting equality (both sides count the number of pairs (x, i) with (2, T) > 0). Using
(ii) and summing over yields Σ, [T.(r) S [1 + A(7)], and dividing by | establishes (48). an
priori comparison ( I C(T) holds for the graph family under consideration, the final bound follows immediately.
Geometric specialization. If X' C R is seseprrteedd (minimum inter- inter-point distance 2 > 0) and G- is geometric threshold graph
(x, E Er
| - v S V
(51)
then each closed neighborhood fits inside ball B(x, F) of radius v7. A standard packing argument implies
deg, (x) + S C.
(52)
for constant C. depending only the ambient dimension m. Consequently,
[S((T)] S =C
vF 77
m
Σ X
S
dCm |
vF
m
(53)
Thus, as r increases, the allowable clique size (hence the bound on average support) grows polyno- mially with the geometric volume of the /F-ball, quantitatively linking scale to reduced sparsity (broader fields).
Corollary 4 (Scale-dependent localization), Assume Gr is monotone in the sense that < T2 implies Fr,, S Frrg. Then A(T) and p(T) are non-decreasing in т, and the upper bounds in Proposition 3 are non-decreasing in i.e, expected sparsity decreases with scale, corresponding to larger place fields.
Proof. Monotonicity En, C Er, implies degr, (x) S degr, (a)) for all x, hence A(T) S AT2) and p(T) S p(r2). Apply Proposition 3 (iii).
Interpretation. Lemma 2 enforces disjoint supports for non-adjacent locations; Proposition shows that each coordinate can only support a clique of mutually adjacent positions and bounds the size of that clique by local neighborhood size. In geometric environments, the bound scales with the volume of a /T-bal,, making the --iieldssize relationship explicit. Corollary then formalizes the empirical trend: smaller higher sparsity (smaller fields); larger = lower sparsity (larger fields).
G Path Planning Properties
This appendix elaborates the properties of the gradient- gradient-based navigation framework introduced in Section 2.7.
G.1 Scale Transition Dynamics
The optimal scale -" = arg max, [ T) adapts dynamically to the agent's 's distance from the
goal, ensuring efficient navigation. In an open field, we formalize this selection with the following theorem.
23

---

Theorem 5 (Optimal Scale Selection in an Open Field). In an open field with Gaussian transition kernel p(zzyy, 7) 4har exp
|| ) where a = 1/4, the optimal time scale 5" that maximizes 4a0
the gradient magnitude V =. with g(xlu, r) p(a/y, T)/ vp(ac/e, p(uly, is:
(x, dl (xx v)
(54)
4a
where d(z, y) = ||-- yll is the Euclidean distance.
Proof. The transition kernel is:
(- de 4
)
p( = 4TaT
exp
d d(x,y)
(55)
Normalization gives:
p( =(ylu, 4xaT"
/p p(ulU.+)
(56)
4TOT
Thus:
(<>l,,T) p( exp
d2
(57)
4ar
Tstert
The gradient is:
(=q(z(y,T)= = exp
(-4er) (- (-2(s
d2
2(x - -y)
2 - y
exp
d²
(58)
4ar
2aT
4aT
The magnitude is:
d2
exp
(59)
200
dar
Maximize f(7)
2ar exp
lar
de
Compute:
d²
In f(T)
In
In
(60)
20
4aT
Differentiate:
i In
d
d
(61)
4a
4a
For = 1/4, d. The second derivative at +*:
8 In ʃ
d
d²
d2
at
2,
<
00
(62)
20r 72
4a
2at
OT
72
confirms a maximum.
In an open field, 7* x d w). so 7* decreases as the agent approaches the goal, focusing on finer spatial scales for precision.
G.2 Properties of the Gradient Field
The gradient field Vag(zly, 7) drives navigation. highlight three properties ensuring a smooth, free-free path to the goal y.
First, p(xly, satisfies the heat equation with reflecting boundary conditions:
Op(xly,7) 7) (V2llp))) a 1/4, ØT
Op(zly, T) 0 0 ahstalles
(63)
On

---

For fixed y and T, p(zly, 7) smooth in as the heat kernel (c.g., p(zly, 7) 4лa exp in an open field) is infinitely differentiable [15]. It has unique maximum at y/-
-| 4a
Second, p(ar|a, r) is smooth in for fixed In open field, p(r x 7) बसकт is constant, while in general, p()[2,7) varies smoothly due to the heat kernel's differentiability. reflecting the domain's geometry near obstacles.
Third, since the random walk is symmetric, Vaq(z r) = V.4(y/2, 7). The gradient field of (3y22, is smooth, (()12, r) q<<ly; inherits the smoothness of p(zly, r), and has unique maximum at x
These properties ensure that Vaq(ylæ, -) forms smooth field with unique maximum at the goal, producing a continuous, trap-free path toward y. This mirrors the hippocampus's efficient navigation, where place cells encode smooth, goal-directed trajectories [11]-
G.3 Planned Path vs. Shortest Path
The planned path follows the gradient Vzg(arl 7), and p(æly, r) satisfies the heat equation with reflecting boundary conditions. We compare this path to the shortest (geodesic) path from z to the goal y.
In an open field, the planned path is a straight line. The transition kernel is:
|| (-1m--uli y11?
p(x 7) = 4haT
exp
a = 1/4
(64)
4at
Since p(x|x,7) p(ylu.T) 1
the normalization gives:
4war
| 3/1 2
7( =exp
(65)
4CT
The gradient is:
V.g(+)y,))
2 --- 20t
exp
||x - y1(²
x -(x ((y
(66)
4at
Thus, dr(()) V Vag(</y,77 T) [64].
follows a straight line from x to y, matching the Euclidean shortest path
de
For small T. g(<ly, 7) aligns with the geodesic distance:
(4Ta7)d/2 exp (- dg( v)
1
(67)
40t
where dg(x,,) y) is the geodesic distance [16]. Since q(aly,7) 7) p(aly, r) in open field, V.g(wly, r) x - aa, v). ensuring the planned path follows the shortest route, even around obstacles.
For large т, ((rl.. r) emphasizes global topological connectivity over local geometric details, often producing paths superior to the shortest. The transition matrix P, PT has spectral decomposition:
P. = QATQ" Q"
(68)
where Q is orthogonal, and A diag(A A, ,.. A,,) contains eigenvalues 0 I A S 1 [57]. For large T, dominant eigenvalues (A; x 1) amplify global connectivity:
p(zlu: [Prlay
Σ xQ((())
(69)
$:1,111
The eigenvectors Q; corresponding A; encode topological features, such major pathways and passage connectivity, invariant to deformations like stretching [57]. Thus, 229(21y.7 prioritizes routes with high connectivity (e.g., wider corridors or multiple paths), potentially deviating from the shortest path to favor robust, flexible trajectories [10]. For example, in maze, the planned path may choose a longer more reliable route through a well-connected passage, avoiding narrow or risky shortcuts.
25

---

This topological focus aligns with neuroscience observations, where hippocampal place cells encode abstract connectivity (e.g.. room layouts) over precise metries, enabling robust navigation complex environments [11,9 11,9,33 9J. 3]. Paths driven by topology are often "better" than the shortest, as they prioritize accessibility and adaptability, reflecting cognitive strategies in spatial tasks.
The planned path's 's alignment with the shortest path for small T and its topological robustness for large mirror the hippocampus's 's ability to balance precision and global structure, ensuring efficient navigation across diverse environments.
G.4 Boundary Layer Effects
Near obstacles, the gradient field ( aligns parallel to boundaries, preventing collisions. This behavior arises from the reflecting boundary condition of the heat equation governing p(zly, T). with Öp(xlyy, 7)m = 0 on obstacles The condition ensures the normal component of V.p(æly, 7) vanishes at boundaries, creating a flow parallel to obstacles, with strength modulated by the goal's position and time scale T.
G.5 Diffraction-I.ist Patterns
Diffraction, in the context of our navigation framework, refers to the bending of the gradient field V.q(<ly, r) around obstacles, such as corners narrow passages, analogous to how waves bend around edges in optics or acoustics. This phenomenon arises because the transition probability p(zly, r). governed by the heat equation, diffuses probability mass through available paths, concen- trating gradients toward openings like passages or around corners. These diffraction-like patterns guide trajectories through nigh-gradient regions, ensuring efficient navigation in complex environ- ments.
The recursive nature of the random walk generates diffraction- patterns:
p(zly, + +) Σ 7())),t) 1)
(70)
This convolution reflects the diffusion of probability from y to through intermediate points 2, allowing p(=ly,T) to "bend" around obstacles by accumulating contributions from multiple paths [15]. Near corner or passage, the heat kernel explores paths that wrap around the obstacle, creating gradient field that converges toward navigable routes.
For narrow passage of width w, the transition probability approximates the geodesic distance dy (2.y) y) through the passage. For small to moderate T:
(- ((E,))
q( z exp
(71)
4aT
The gradient is:
8 dz v)
Vaq(oly,7)
exp
V Vod((E,))
(72)
20T
4aT
Since dz y) points along the geodesic path through the passage, V V2q(aly, 7) converges toward the opening, creating funneling effect. The width of this convergence region scales with /T, the heat kernel's diffusion spreads over distance proportional /4aT. For small the gradient tightly focuses on the passage, ensuring precise navigation. For large t, the broader diffusion enables early detection of passages from greater distances, smoothing trajectories by integrating multiple nearby paths.
At sharp corners, diffraction-like patterns emerge similarly. The heat kernel accumulates probability along paths bending around the corner, creating a curved gradient field that guides the agent past the obstacle. Unlike passages, corners lack single geodesic path, so the gradient field spreads, resembling optical diffraction patterns where intensity peaks near edges. This spreading more pronounced for large т, as the heat kernel explores longer, circuitous routes.
These patterns ensure robust navigation complex environments. The funneling effect through passages and curved trajectories around corners mirror hippocampal navigation, where place cells
26

---

encode paths that navigate mazes or avoid obstacles [11, 9]. The 7-dependent scaling reflects the hippocampus's ability to adjust focus, balancing precision for small with global awareness for large enhancing adaptability in varied spatial contexts [16].
G.6 Topological Invariance
Topological invariance ensures that navigation behavior, driven by the gradient field Væq(zly, T). remains consistent under continuous deformations (c.g.. stretching, bending) that preserve the environment's 's connectivity, such as the number of passages or loops. This robustness mirrors the hippocampus's 's cognitive maps, which encode abstract spatial relationships invariant to physical distortions [3]-
The transition probability p(rly, r) arises from a random walk on a discrete grid, with one-step transition matrix Pj. For time т, the transition matrix is:
P= PT
(73)
with spectral decomposition:
= QA'
(74)
where Q is orthogonal (Q=Q=1), 1), and = .ii An) contains eigenvalues 0 S X S 1 [57].
The eigenvectors Q; and eigenvalues X; encode topological features, such connectivity (number of connected components), bottlenecks (e.g., passages), and cycles. The second eigenvalue (A2) quantifies global connectivity, with higher values indicating tighter bottlenecks, while eigenvectors capture local structures like loops [15].
For large т, dominant eigenvalues (A2 R 1) amplify global topological features:
P Σ xQOo
(75)
EA,1
emphasizing coarse connectivity (e.g.. major pathways between regions). The eigenvectors corre- sponding to X; = 1 represent low-frequency modes that partition the environment into connected regions, invariant to metric distortions like stretching corridor. This because Q, depends on the graph's 's adjacency structure, not precise distances, ensuring stability under deformations that preserve connectivity [57]. For example, stretching a corridor adjusts transition probabilities proportionally, but the eigenvectors Qi retain the same connectivity patterns, they reflect the graph's 's topology.
The gradient field is V2q(aly, r), where p(zly, 7)
[Prlay Since P, 's eigenvectors are topolog- ically invariant, V V=r(r1y, T) preserves navigation paths (e.g., through passages) across equivalent environments. For large т, the dominance of A ensures that Vzg(zly, r) prioritizes global connectivity, guiding trajectories along major routes regardless of local geometry.
This invariance aligns with hippocampal navigation, where place cells encode connectivity (e.g., room layouts) over precise metrics, enabling robust path planning under environmental changes [9]. The focus topology for large 7, driven by eigenvectors tied to X; reflects cognitive maps that emphasize structural relationships, as observed in rodent navigation 3].
G.7 Hippocampal Preplay and Shortcut Detection
Our path planning framework discovers novel shorteuts using only local exploration, mirroring hippocampal preplay, where neural sequences predict unexplored paths [7, 8]. This process relies on efficient computation of transition probabilities and adaptive scale selection, without memorizing past successful paths.
The one-step transition matrix P1 defines probabilities of moving between adjacent grid points. .2....,K)
Multi-step transition matrices P. = PT for scales T = 2* (k = 1, 1,,,... K) are computed via matrix squaring:
P2 = P?. P P?, Ps P?,
Po+ P
(76)
requiring log, 7 operations.
The path planning algorithm adaptively selects 7" arg max, V V.q(lly T) T)ll, guiding trajectories via Vag(zly, For regions A and B connected by unexplored passage, local exploration
27

---

defines P1. Matrix squaring yields Pr, and adaptive r-selection ensures q(abb, > 0, discovering the shortcut without prior traversal.
This wrocess-matri squaring to compute P, and adaptive 7-selection hippocampal preplay, predicting connectivity from local data, akin place cell sequence pre-activation. The framework's 's efficiency and reliance on P1 enhance its biological plausibility, supporting robust navigation.
H Theta Phase Modeling
We propose local definition of theta phase that embeds spatial adjacency within the phase angle, capturing the observed precession range of 270 90" while enabling navigation across a tessellated environment. This formulation leverages an external 8 Hz theta rhythm for temporal pacing, where I denotes real time during navigation, distinct from the random walk time scale т, and aligns with distributed hippocampal computation without requiring individual cells to explicitly encode spatial centers.
H.1 From Inner Product Angle Theta Phase
The emergent sparsity from non-negativity and orthogonality constraints endows each cell in /(x,, T) a localized place field, so that each cell has a place field centered 14:- The place fields of all the cells tile the whole environment. At the current position rtt) at real time t, the adjacency between r(t) and each μ4 is measured by the angle between h(x(t), r) and h(p, r), and this angle can then be embedded as the theta phase.
Specifically, define the adjacency of the current position st) the field center of the i-th place cell at scale T:
(t ()) 7))
(77)
where ()) F" is the population embedding (firing rates of n place cells) at position xt), and hlp,, represents the embedding at the center μ4, where
h;(ps, max (x, =Ex
(78)
We interpret h(44,, 7) = [wie, w2i..... War] as the recurrent connection weights from the population to neuron such that:
n
(t)
Σ j=1
hy (x)),, T)
(79)
representing the net recurrent input to neuron i. This corresponds to the normalized transition probability q(z(t)]u,, r), and in open fields (Appendix E), it follows Gaussian profile:
a:(l) exp (-
[[(((t) 1ull²
(80)
The theta phase is defined as:
(da((t)
( (accos((((t))
(,(x(t),T)= = T sgn
(81)
dt
dt Wye is the temporal rate of change of this recurrent input, reflecting movement toward (> 0) or away (< 0) from μr as encoded by the weights. arccos(a (t)) E [0, / is the angle between hxx(t), T) and h(4,77 7). and this angle is embedded into the phase, possibly by sinusoidal inhibitory mechanism [65, 66].
where: da,(t) dt Σn jml
(x(1))))
The phase progresses as:
1. At field entry (a; + 0, da:
> 0): Φ; 3r (270%), late phase.
dt
2. At field center (a; dr
0): Φ; π (180"), mid-phase.
3. At field exit (a; + 0.
da:
0): oi _/4 (90%), carly phase.
dt
28

---

This does not require neuron i to encode μa explicitly; rather, h(u,, r) emerges as distributed representation within recurrent connectivity, learned through experience and computed collectively. with @; storing a; (t) as an angular signal.
H.2 Implications for Navigation
This phase also enables navigation by tessellating the environment with {P:}- Each pa anchors tile, and Φ; signals adjacency: 180 indicates proximity to μ4, while 270" and 90º mark entry and exit. For a goal at Py. navigating toward 9, = π (max 0g) guides movement, with {ór] identifying intermediate 14; to approach Mg. This discretizes continuous navigation into a phase-driven sequence over { complementnng our gradient-based approach with a recurrently-driven strategy. During rest, iterating Φ; at Hz drives replay, with da, reflecting virtual transitions, unifying coding and navigation.
de
By lessellating the environment with { and guiding navigation via {oz). this approach unifies spatial representation, temporal coding, and path planning within the embedding framework, offering an elegant, biologically plausible mechanism rooted in recurrent connectivity and the random walk model.
I
Grid Cells Integration
L.1 Grid Cells as a Conformal, Multi-Scale Basis
While place cells form the primary focus of our model, we establish a connection to grid cells in the medial entorhinal cortex. Grid cells exhibit hexagonal firing patterns across multiple scales [67, 9] and are thought provide universal metric that feeds into the more adaptive place cell system.
Let g(x) be the vector formed by the firing rates of the population of grid cells at position x. The high dimensional g(z) forms an embedding of 2D position x. The grid cell representation g(x) provides an effective basis for learning the transformation matrix (T) to produce place cell embeddings h(x, r) = W(r (T))g(x) that approximate q r). Its effectiveness stems from conformal, isotropic, and multi-scale properties of grid cells, aligning with the hippocampus" 's multi-resolution encoding [22].
While place cell representation ha,, 7) preserves the proximity within the environment, g(x) pre- serves the local distance and has the property of conformal isometry.
Conformal isometry [68-71] is defined as:
(1 + dx) g(x)l = dz|| + o[ dz]])
(82)
where s is a scaling factor, and o( [|xx]]) denotes higher-order terms vanishing as ||dx|| + 0, ensuring local distance and angle preservation and isotropic scaling.
plays the role metric. Grid cell population consists of multiple modules, with each module corresponding to metric mirroring the multi-scale property of place cells.
I.2
Implementation of Grid-to-Place Transformation
The place cell embedding is modeled as hxx,7) W(r)g(x), or more generally h(,, 7) Norm (RR.RULLU(()))) + b(r))) where the transformation matrix W(r) is learned to optimize the proximity-ppeesrvinn property:
(W())g(z), (()) = (())=,7)
(83)
The matrix W (T) serves a dual purpose: (1) it captures the appropriate spatial scale by weighting the contribution of each grid cell module to match the place cell scale t, and (2) adjusts for deviations from isotropy and isometry in grid cells. 9(x) facilitates fast adaptation to the environment in that learning (T) may be more efficient than learning h(x, r) directly.
To further enhance adaptability, we can backpropagate gradients to g(x). This allows the grid cell representation to deform response to environmental constraints, simulating the dynamic remapping observed in biological grid cells [72].
29

---

This formulation provides computational account of how place field embeddings adapt to environ- mental constraints based on flexible metric provided by grid cells. The transformation matrix (T)). combined with the learnable parameters of gar), effectively functions as computational cognitive map, translating the multi-scale grid metric into place representations that reflect both the appropriate spatial scale and deviations from isometry and isotropy.
J Implementation Details
All the models were trained on single NVIDIA A6000 GPU for 2,000 iterations with a learning rate of 0.001. For fine-tuning, we use 50 iterations with 0.0005 learning rate. All models contain 500 cells. For batch size, we learn all combinations of different locations in the field for each iteration. The training time less than 5 minutes.
We evaluate planning using two metrics: (1) Success, whether the agent reaches within unit of the goal, and (2) Success weighted by inverse Path Length (SPL.) [28]. For an agent's path [xi,...; with initial geodesic distance de for episode i (as computed by baseline algorithms), we compute:
T
4
Σ r=2
-1
(84)
Then SPL for episode i is defined as:
Success; d:
SPL,
(85)
We report SPL as the average of SPL; across all episodes. In Table 1, we treat Bug algorithm with oracle the baseline and in Table method serves as the baseline.
K
Additional Experiment Results
K.1
Additional Path Planning Results in Four-Room Environment
Figure 3: Successful navigation across eight randomly sampled start-goal configurations in the four-room environment.
To further demonstrate the robustness of our path planning framework, here we provide additional navigation trials in the four-room environment with randomly sampled start and goal locations.
30

---

Figure 3 shows eight scenarios spanning diverse spatial configurations. In all cases, our gradient- based path planning with adaptive scale selection successfully generated smooth, obstacle-avoiding trajectories that reached the goal. These results, combined with the quantitative evaluation in Table showing 100% success rate across 50 random trials, demonstrate that our model achieves reliable navigation across diverse start-goal configurations even in complex multi-room environments.
K.2
Remapping
We evaluated our model's ability adapt bidirectional environmental modifications the four- room environment (our most complex tested scenario). In addition to the obstacle removal experiment shown in Figure 2 (D), we conducted the complementary scenario: initially, rooms were connected allowing direct northward navigation; we then added an obstacle blocking this direct pathway and fine-tuned the model with 100 iterations at learning rate 0.01. The agent successfully adapted, discovering alternative routes through adjacent rooms to reach the target (see Figure 4). Together, these experiments demonstrate opology-dependent remapping - well-documented phenomenon where place fields reorganize in response to changes in spatial connectivity. By both removing obstacles (creating novel shortcuts) and adding obstacles (blocking direct paths), we show that minimal fine-tuning enables the position embeddings adapt to environmental changes that alter topological connectivity, whether exploiting new connections or navigating around new barriers. This bidirectional adaptability aligns with hippocampal responses environmental modifications.
2048
1024
Time Scale (T)
128
Blocked pathway
64
18
Start
Target
Decision Point
Figure 4: Remapping and path adaptation in response added obstacles in the four-room environ- ment.
31

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Reviews:
[{"review_id": "4jFU0KSh5u", "reviewer": "Reviewer_qzbz", "summary": "This 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.", "questions": "Most 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?", "limitations": "See above", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\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", "was_revised": true, "final_justification": "I apologize for the late response to the authors rebuttal. I had only a few very minor gripes with the clarity of the description, all of which were addressed in the rebuttal, so I will maintain my high initial score, as I believe this paper will be of value to the neurips community.", "rebuttal": "We appreciate your recognition of our theoretical framework's strong motivation, its recovery of key hippocampal features, and the clarity of our figures.\n\n### Weaknesses\n\n> **1. It would be nice if they authors discussed connections to known experimental results in a little bit more depth** \n\nWe 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.\n\n\n>**2. The model is obscured a little bit by notation** \n\nWe 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.\n\n\n### Questions\n\n\n>**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** \n\nWe 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.\n\n\n>**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?** \n\nThis 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.\n\n|            | Open field | S-shape |\n|-----------------|------|----------|\n|  No noise       | 100% | 100% |\n|  $\\sigma^2*n=0.05$  | 98%  | 96%  |\n|  $\\sigma^2*n=0.2$  | 90%  | 90%  |\n\n\n\n>**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.**\n \n \nWe 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.\n\n>**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?**\n\nThis 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.\n\nIf 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.\n\nWith 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.\n\n>**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?**\n\nThe 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.\n\nOur 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.\n\n*Thank you again for your comprehensive review. We will revise our paper by incorporating all your insightful comments and suggestions to strengthen our paper.*"}, {"review_id": "INI6W4RlMW", "reviewer": "Reviewer_wLru", "summary": "The 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.", "questions": "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?\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)?", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths: \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?", "was_revised": true, "final_justification": "I think this is a valuable contribution. The authors have addressed my concerns appropriately and it seems to me that they have done a reasonable job of addressing concerns from other reviewers.", "rebuttal": "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.\n\n### Weaknesses\n\n>**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.**\n\nWe 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.\n\nWe 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.\n\n| Environment | Bug algorithm | A* algorithm |\n|-------------|----------------|-------------------|\n| Open field  | 0.991±0.04    | 0.991±0.04 |\n| U-shape     | 0.919±0.25    | 0.801±0.254 |\n| S-shape     | 1.392±0.98    | 0.915±0.42 |\n| Four-room   | 1.519±1.17    | 0.935±0.029 |\n\n\nThe 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. \n\nWe thank the reviewer again for the suggestion and will add this to our revised manuscript.\n\n\n>**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?**\n\nThe 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.\n\n### Questions\n\n>**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?**\n\nThis 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.\n\n>**2. Empirically, rodent place fields often shrink, elongate or duplicate near walls and obstacles. Can your representation somehow capture such distortions?**\n\nThank 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.\n\n>**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?**\n\nThis 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.\n\n>**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?**\n\nWe 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. \n\n>**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)?**\n\nThis 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.\n\nRegarding 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. \n\n\n[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.\n\n[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.\n\n[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.\n\n*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.*"}, {"review_id": "iVj1ZlUU5v", "reviewer": "Reviewer_xpWW", "summary": "This 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.", "questions": "The 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?", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 2, "strengths_and_weaknesses": "The 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.", "was_revised": true, "final_justification": "The authors have significantly improved the paper and addressed all the concerns raised.", "rebuttal": "We are grateful that you found our framework novel and elegant.\n\n### Weaknesses\n\n>**1. Computational Efficiency and Training Bottleneck**\n\nOur 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.\n\nTo 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.\n\n>**2. $C^0$ Continuity and the Gradient Field**\n\nThe 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.\n\nSpecifically, 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.\n\nTherefore, our planner can still produce smooth, effective trajectories. We will clarify this point in the revised manuscript.\n\n>**3. Inconsistency in Random Walk Definition**\n\nWe 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.\n\n>**4. Intuitive Explanations for Concepts**\n\nWe 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.\n\n>**5. Related Work in Appendix A**\n\nWe 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.\n\n>**6. Offline Process Limitation**\n\nWe 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.\n\n>**7. Lack of Sensory Input Integration**\n\nThe 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. \n\n>**8. Differentiation from Successor Representation (SR)**\n\nWe 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:\n\n1. **Shared Foundation**: Both frameworks root spatial representations in transition matrices.\n2. **SR's Discounted Sum**: SR ($M = \\sum_{k=0}^\\infty (\\gamma T)^k$) represents expected discounted future occupancy, with $\\gamma$ setting the predictive horizon.\n3. **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.\n4. **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.\n\n**Key Distinctions and Novel Contributions**\n1. **Biologically-Constrained Factorization with Emergent Sparsity**: \nOur 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.\nCrucially, 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.\n2. **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.\n\nThe 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.\n\nWe will incorporate proper SR citations and discussion in our revision.\n\n### Questions\n\n>**1. Relationship to Successor Representation (SR)**\n\nAs 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. \n\n>**2. Computational Complexity of Training**\n\nSee 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.\n\n>**3. Biological Plausibility of Mechanisms**\n\nWhile 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.\n\n>**4. Effect of $C^0$ Continuity on Path Quality**\n\nSee 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.\n\nBecause 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.\n\n*Thanks for your detailed feedback. We will incorporate all your insightful comments in our revision.*"}, {"review_id": "e5SxISr0Fl", "reviewer": "Reviewer_GgX7", "summary": "This 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.", "questions": "1. 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.", "limitations": "Yes. The paper has considered both theta-phase procession and grid cell integration as major limitations of this work.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "+ 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\".", "was_revised": true, "final_justification": "I am satisfied with authors' rebuttal and willing to increase my rating by one grade.", "rebuttal": "We thank you for recognizing our model as an interesting and novel contribution to computational neuroscience.\n\n### Weaknesses\n\n>**1. Model Simplicity**\n\n**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. \n\n**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.\n\n**Empirical evidence:**\nOur experiments can demonstrate this capability:\n- The \"diffraction-like patterns\" in Figure 2B show how gradient fields bend around corners and through passages\n- 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\n- The boundary layer effects (Appendix E.4) ensure obstacle-parallel flows for complex navigation.\n\n**Multi-scale robustness:**\nAdaptive 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.\n\n**Additional experiment validation:**\nFollowing your advice, we tested our model on the concave environment from Figure 10 of Muller et al. [1], achieving 100% success rate. \n\n>**2. Additional results**\n\nWe 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.\n\nWe 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).\n\n>**3. Remapping**\n\nWe 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.\n\n### Questions\n>**1. Goal-reaching with obstacles**\n\nThank 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. \n\n>**2. Matrix multiplication**\n\nWe consider matrix squaring ($P_{2t}=P_t^2$) as a primary method for two main reasons, highlighting both computational efficiency and biological plausibility:\n\n* **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).\n* **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.\n\nWe 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\". \n\n>**3. Least resistance**\n\nThank 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].\n\nIn 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.\n\n>**4. Regarding Whittington et al. (2022)[2]** \n\nWe 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:\n\n* **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 $t$.\n* **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.\n* **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.\n* **Path Planning Mechanism:** Our path planning relies on adaptive gradient ascent on these multi-scale transition probability fields, which naturally produce robust navigation properties.\n\n**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.\n\n*Thank you again for your thoughtful review and constructive feedback. We are committed to addressing your concerns and will revise our paper accordingly.*"}]

Author rebuttal:

Meta-review: The paper considers a new model of hippocampal place cells using population embeddings with random walk kernels. They demonstrate the suitability of their method in a path modeling framework across extensive simulations, including remapping, to also showcase biological correspondence with real place cells. Reviewers suggested additional connection to known experimental results, relevant methods like SR, and comparisons to other planners; these were clarified and expanded upon by the authors during rebuttal. Reviewers unanimously recommended acceptance.
Decision: Accept (poster)