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

Yuhang He; Junfeng Zuo; Tianhao Chu; Si Wu

q-bio.NCAug 30, 2026

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

Authors: Yuhang He, Junfeng Zuo, Tianhao Chu, Si Wu

Organizations: School of Psychological and Cognitive Sciences, Peking University · IDG/McGovern Institute for Brain Research, Peking University · Peking-Tsinghua Center for Life Sciences, Academy for Advanced Interdisciplinary Studies, Peking University · Center of Quantitative Biology, Peking University ·

Abstract

Neural correlates of spatial cognitive map are well documented, yet exactly how neural circuits perform spatial navigation in complex environments - e.g., reaching a goal while avoiding obstacles - remains largely unclear. Here, we show that a hippocampal network with appropriate recurrent connections can naturally achieve optimal goal-directed navigation via its relaxation dynamics. Specifically, we consider that the recurrent weights between the neurons represent the transition probabilities between spatial locations encoded by neurons; obstacles such as walls and blocked corridors are therefore reflected by the vanishing of connection weights. This connection pattern can be learned in the hippocampus via behavioral-timescale synaptic plasticity (BTSP) while the animal is exploring the environment. When a goal signal is presented, the network dynamics will relax into an activity field representing the goal location. We prove that this field is mathematically equivalent to the desirability field of a Linearly-solvable Markov Decision Process (LMDP), and the local log-gradient of the field indicates the navigation direction. Both theoretical analyses and simulations demonstrate that this recurrent network dynamics-mediated navigation is efficient and robust in environments with complex obstacle layouts. Moreover, only low-rank updates of the network's connection pattern are needed when the environment has local changes. We hope this study offers insight into a general circuit principle for planning in abstract rational maps in the brain beyond spatial navigation.

Figures & tables

Figure 1: The relaxation reproduces the LMDP field, and the readout is amplitude-robust. ( A ) Two-room diagnostic maze; color shows the log desirability (low at start, high at goal). ( B ) Network steady state mapped to the LMDP coordinate $\widehat{\bm{z}}$ , plotted against the LMDP solution $\bm{z}$ . Corrected fit on identity ( $L_{2}=9.5\times 10^{-16}$ ); uncorrected residual $3.2\%$ ; weak nonlinear recall ( $A=0.2$ ) contributes $1.2\%$ . ( C ) Strong recall ( $A=5$ ) confines shape distortion to an $8$ -cell shell around the goal; at a probe outside the shell, $\Delta\theta=0.015^{\circ}$ . ( D ) Shell radius grows as $\log A$ ( $R^{2}=0.94$ ). ( E ) Outside-shell $99$ th-percentile readout error across $\alpha$ and $A$ . The main row $\alpha=0.985$ (black box) stays below $1^{\circ}$ across the full recall range; the red contour marks where the $1^{\circ}$ boundary is crossed in adjacent rows.

Figure 2: Goal-free BTSP solves the bug trap, and a local edit reroutes plans globally. ( A ) Top: connectivity learned by goal-free BTSP. Bottom: field correlation and edge recall as functions of goal-free transitions per neuron; the dashed line marks the displayed setting ( $3.0$ transitions per neuron, recall $69\%$ , precision $1.000$ ). ( B ) Goal-clamped field (color) and population-vector rollout (white) trace a $110$ -step route; first readout (cyan) points $93.8^{\circ}$ off the Euclidean direction (orange dashed), which stalls at the wall. ( C ) Two-door diagnostic. Closing the near door updates only the local crossing connections; the next relaxation reroutes through the far door (white) without further learning. Color: field difference (closed minus open). ( D ) Singular spectrum of field differences across $34$ goals (log scale); the first two modes carry $99.5\%$ of the total energy.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 3: The recurrent source response is a screened Green function and approaches the Laplacian Green shape in the weak-cost limit. ( D ) Spectral reconstruction of the screened Green response using the low-frequency eigenmodes of $\bm{L}=\bm{I}-\bm{S}$ . The black curve shows normalized gains $(\lambda_{n}+m_{\alpha})^{-1}$ , and the gray curve shows the shape error after adding successive modes. ( E ) At $\alpha=0.996$ , the source response of the recurrent network matches the screened Green field; white contours show the recurrent response and color shows the Green field. ( F ) As $m_{\alpha}=(1-\alpha)/\alpha$ decreases, the mean-zero screened Green shape approaches the Laplacian pseudoinverse Green column. The algebraic identity between the recurrent resolvent and the screened Green operator holds at numerical precision throughout the sweep.

Figure 4: Asymmetric BTSP deviations are controlled by a single irreversibility scale. ( A ) The perturbation knob $\eta$ decomposes the effective BTSP memory into a fixed reversible support and a scaled directional residue. The relevant error scale is $\chi=\alpha\|\bm{E}_{\rm asym}\|_{2}/(1-\alpha)$ , which combines raw asymmetry with recurrent amplification. ( B ) Representative fields in the bug-trap maze at $\alpha=0.985$ . The left panel shows the reference $\log\bm{z}_{0}^{g}$ . The middle and right panels show signed deviations $\Delta\log\bm{z}_{\eta}^{g}=\log\bm{z}_{\eta}^{g}-\log\bm{z}_{0}^{g}$ ; red means the asymmetric field is higher than the reference and blue means it is lower. ( C ) Population-vector rollouts from the same three fields. Low $\chi$ leaves the route near the reference route, whereas larger $\chi$ produces a visible detour. ( D ) Relative field error collapses when plotted against $\chi$ across seeds, discounts, and $\eta$ values. Point color encodes $\eta$ , marker shape encodes $\alpha$ , the black curve is the binned median, and the gray band is the interquartile range. ( E ) The behavior-relevant readout-angle error follows the same $\chi$ -ordered trend. ( F ) Raw asymmetric replay degrades at larger $\chi$ , while bilateral replay projects the same raw experience back to the reversible support and keeps route quality close to the reference.

Recurrent Neural Networks

Recurrent neural networks process sequences by updating a persistent hidden state, enabling temporal modeling, memory, and streaming computation.

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The Metric Slingshot: Navigational Reuse as Width-Optimal Structural Decoupling in Continual Learning

Xin Li

Department of Computer Science University at Albany, SUNY Albany, NY 12222, USA

Neural Processes Continual Learning Transfer Learning

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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.

Abstract

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