Place Cells as Proximity-Preserving Embeddings: From Multi-Scale Random Walk to Straight-Forward Path Planning

The hippocampus enables spatial navigation through place cell populations forming cognitive maps. We propose proximity-preserving neural embeddings to encode multi-scale random walk transitions, where the inner product $\langle h(x, t), h(y, t) \rangle = q(y|x, t)$ represents normalized transition probabilities, with $h(x, t)$ as the embedding at location $x$ and $q(y|x, t)$ as the transition probability at scale $\sqrt{t}$. This scale hierarchy mirrors hippocampal dorsoventral organization. The embeddings $h(x, t)$ reduce pairwise spatial proximity into an environmental map, with Euclidean distances preserving proximity information. We use gradient ascent on $q(y|x, t)$ for straight-forward path planning, employing adaptive scale selection for trap-free, smooth trajectories, equivalent to minimizing embedding space distances. Matrix squaring ($P_{2t} = P_t^2$) efficiently builds global transitions from local ones ($P_1$), enabling preplay-like shortcut prediction. Experiments demonstrate localized place fields, multi-scale tuning, adaptability, and remapping, achieving robust navigation in complex environments. Our biologically plausible framework, extensible to theta-phase precession, unifies spatial and temporal coding for scalable navigation.