Exponential Map Models as an Interpretable Framework for Generating Neural Spatial Representations

A fundamental challenge in neuroscience and AI is understanding how physical space is mapped into neural representations. While artificial neural networks can generate brain-like spatial representations, such as place and grid cells, their “black-box” nature makes it difficult to determine if these representations arise as general solutions or as artifacts of a chosen architecture, objective function, or training protocol. Critically, these models offer no guarantee that learned solutions for core navigational tasks, like path integration (updating position from self-motion), will generalize beyond their training data. To address these challenges, we introduce a first-principles framework based on an exponential map model. Instead of using deep networks or gradient optimization, the presented model uses generator matrices to map physical locations into neural representations through the matrix exponential, creating a transparent framework that allows us to identify several exact algebraic conditions underlying key properties of neural maps. We show that path invariance (ensuring location representations are independent of traversal route) is achieved if the generators commute, while translational invariance (maintaining consistent spatial relationships across locations) demands generators producing orthogonal transformations. We also show that preserving the metric of flat space requires the eigenvalues of the generator matrices to form sets of roots of unity. Finally, we demonstrate that the proposed framework constructs diverse biologically relevant spatial tuning, including place cells, grid cells, and context-dependent remapping. The framework we propose thus offers a transparent, theoretically-grounded alternative to “black-box” models, revealing the exact conditions required for a coherent neural map of space.