Topological spatial coding for rapid generalization in the hippocampal formation
The brain navigates complex environments by combining entorhinal grid codes with hippocampal place codes. Although grid codes effectively represent a current environments geometry, their capacity to generalize across topologically analogous environments with different reward and state structures remains poorly understood. We introduce topology-aware grid coding (TAG), a computational theory that leverages topological invariance to generalize to new environments with the same topological structure but with different geometry. Drawing on the Euler characteristic from algebraic topology, TAG integrates complementary neural codes built on foundational grid bases: place codes serving as 0D vertices for self-localization, boundary codes acting as 1D edges to learn policy-independent grid codes through state prediction errors, and corner codes functioning as 2D faces for identifying topologically significant states. TAG grid codes remain stable under topology-preserving deformations yet discriminate among non-isomorphic structures. TAG develops policy-independent grid codes for novel structures more rapidly and robustly than existing approaches, balancing structure and policy encoding for multi-subgoal navigation without extensive planning. Finally, we show that TAG is compatible with transformer architectures, enabling its integration into scalable neural networks. Together, the TAG theory describes the essential nature of geometric objects to explain how the entorhinal-hippocampal system maps the unique topological structure of spaces.