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

arXiv: https://arxiv.org/abs/2609.13219 · HTML: https://mlsanity.com/paper/2609.13219/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 1 of its 103 references that are in the map.
- First posted on arXiv 30 Aug 2026.

## Builds on (most cited first, 1 of 1)

- [Reinforcement Learning and Control as Probabilistic Inference: Tutorial and Review](https://mlsanity.com/paper/1805.00909/) (2018)

## Abstract (arXiv)

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.