Place Cells as Multi-Scale Position Embeddings:
Random Walk Transition Kernels for Path Planning
NeurIPS 2025 (Poster) [Paper link] [Code link]
Minglu Zhao*, Dehong Xu*, Deqian Kong*, Wen-Hao Zhang, Ying Nian Wu
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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.
Experiments
I.
We begin our numerical experiments in a simple open field environment to demonstrate fundamental properties. Within this environment, we learn h(x,τ) embeddings across all lattice points
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. Our results demonstrate that the learned model consistently generates near-optimal trajectories.
Figure 1: Place Cell Representations and Navigation in Open Field Environment.
(A) Goal-directed path planning trajectories with adaptive scale selection
(B) Normalized transition probability kernels q(y|x,τ) at multiple scales with gradient vector fields
(C) Learned activation patterns of h(x,τ) at different scales across randomly chosen cells
II.
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 environments 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.
Figure 2: Place Cells in Complex Maze Environments.
(A) Path planning through obstacle environments.
(B) Transition kernels q(y|x,τ) with gradient fields.
(C) Sampled place cell profiles at multiple spatial scales.
(D) Remapping with environmental modification.
III.
We evaluated our model's ability to adapt to bidirectional environmental modifications in the four-room environment. We conducted two scenarios: obstacle removal and obstacle addition.
The agent successfully adapted, discovering alternative routes through adjacent rooms to reach the target. Together, these experiments demonstrate topology-dependent remapping -- a well-documented phenomenon where place fields reorganize in response to changes in spatial connectivity.
Figure 3: Remapping and path adaptation in response to added obstacles in the four-room environment.
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