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

TL;DR

This paper models hippocampal place cells as multi-scale spatial embeddings derived from random walk kernels, providing a unified framework for understanding spatial representation, navigation, and temporal coding in the brain.

Contribution

It introduces a novel spectral decomposition approach to model place cell activity as non-negative embeddings from multi-step transition kernels, linking spatial and temporal coding.

Findings

Embeddings encode multi-scale spatial relationships.

The model explains localized firing fields without explicit constraints.

Intrinsic theta phase relates to embedding angles.

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.