Planning as Dynamics Relaxation: Hippocampal Recurrent Network Realizes Optimal Goal-Directed Navigation

The mammalian brain, most extensively studied in rodents and bats, solves an enormous variety of non-spatial cognitive tasks using neural circuitry, including grid cells, place cells, and hippocampal indexing, that originally evolved for physical navigation. We formalize the above observation within the local Urysohn width (LUW) framework for continual learning. The central construct is the metric slingshot: a learned embedding $φ: X \to Z$ that maps an arbitrary learning problem into a navigational latent space $Z$ where pre-evolved contraction maps (grid cells) already provide the metric machinery, so that only the topological indexing subproblem must be solved de novo. We prove three results. First, the optimal spacing of multi-scale grid cell modules is a geometric series whose ratio is determined by the $Ω(w \log w)$ sample complexity bound of the LUW framework; for ecologically plausible parameters, optimality yields $r^* \approx 1.4$--$1.7$, matching electrophysiological measurements in rodent medial entorhinal cortex. Second, the slingshot preserves the width hierarchy with a Lipschitz-controlled transfer bound: a contractive embedding into a fine-resolution navigational space reduces the effective number of contexts the learner must discover. Third, the anatomical separation of the ventral (what'') and dorsal (where'') visual streams achieves the structural decoupling required by Metric-Topology Factorization (MTF) by architecture, without gradient-routing mechanisms. We demonstrate that such metric slingshot is applicable to both perception cognition and motor control. Together, these results provide a unified, complexity-theoretic account of navigation in non-spatial domain, grid cell multi-modularity, and hippocampal-neocortical complementary learning as consequences of a single exaptation principle: metric slingshot.

Hippocampal place and time cells encode spatial and temporal aspects of experience. Both have the same neural substrate, but have been modeled as having different functions and mechanistic origins, place cells as continuous attractors, and time cells as leaky integrators. Here, we show that both types emerge from two dynamical regimes of a single recurrent network (RNN) modeling hippocampal CA3 as a predictive autoencoder. The network receives simulated, partially occluded experience vectors" containing spatial patterns (location-specific activity sampled during environmental traversal) and/or temporal patterns (correlated activity pairs separated by void" intervals), and is trained to reconstruct missing input. During spatial navigation, the network generates stable attractor-like place fields. But trained on temporally structured inputs, the network produces sequentially broadened fields, recapitulating time cells. By varying spatio-temporal input patterning, we observe hidden units transition smoothly between time cell-like and place cell-like representations. These results suggest a shared origin, but task-driven difference, between place and time cells.

Neural population geometry shapes downstream computation. Recent empirical findings in neurobiology suggest that a hyperbolic structure underlies population activity in the hippocampus. Here we provide a theoretical framework for this phenomenon. First, we propose a plausible construction of hippocampal tuning curves that statistically induces hyperbolic geometry. Next, we establish a connection between neural decoding and associative memory by demonstrating that the Modern Hopfield Network update rule computes the minimum mean-squared-error (MMSE) estimator. Finally, we introduce a novel associative memory model defined in hyperbolic space that yields significantly larger capacity than leading models. Our results suggest that animals encode spatial information as a latent hyperbolic cognitive map, improving both memory capacity and decoding accuracy.