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

TL;DR

The paper models hippocampal place cells as a population of non-negative position embeddings $h(x,\\tau)$ derived from the spectral decomposition of multi-step symmetric random-walk transition kernels, with $\\langle h(x,\\tau), h(y,\\tau)\\rangle = q(y|x,\\tau)$. The time-scale parameter $\\sqrt{\\tau}$ defines a multi-scale, Euclideanized cognitive map, where emergent sparsity arises from non-negativity and orthogonality constraints, naturally explaining localized place fields. Global spatial relationships are built efficiently through matrix squaring ($P_{2\\tau}=P_\\tau^2$), enabling trap-free gradient-based path planning with adaptive scale selection, and theta phase is linked to the angle structure within the population embeddings. Experiments in open-field and obstacle-rich mazes demonstrate accurate reproduction of transition kernels, robust navigation across scales, and topology-driven remapping, supporting a biologically plausible, scalable framework that connects diffusion theory, cognitive maps, and neural population coding.

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.

Figures (4)

• Figure 1: 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 $q(y|x, \tau)$ at multiple scales with gradient vector fields. $y$ is fixed at the center of the environment. (C) Learned activation patterns of $h(x, \tau)$ at different scales across randomly chosen cells, exhibiting Gaussian-like firing fields centered at specific locations within the open environment.
• Figure 2: Place Cells in Complex Maze Environments. (A) Path planning through obstacle-containing environments. (B) Topologically-informed transition kernels $q(y|x, \tau)$ with gradient fields. Target $y$ is marked as a red point. (C) Randomly sampled place cell profiles at multiple spatial scales. (D) Remapping with environmental modification.
• Figure 3: Successful navigation across eight randomly sampled start-goal configurations in the four-room environment.
• Figure 4: Remapping and path adaptation in response to added obstacles in the four-room environment.

Theorems & Definitions (11)

• proof : Proof sketch of Horn's theorem
• Theorem 1: Gaussian Embeddings in Open Fields
• proof
• Lemma 2: Non-negativity $\,+$ orthogonality $\Rightarrow$ disjoint support
• proof
• Proposition 3: Clique structure and sparsity bound
• proof
• Corollary 4: Scale-dependent localization
• proof
• Theorem 5: Optimal Scale Selection in an Open Field