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

A foundational 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 those observed in 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 the exponential map. Bypassing gradient-based optimization entirely, we use generator matrices to map physical locations to neural population vectors. Within the proposed formalism, essential navigational capabilities emerge analytically as intrinsic geometric properties. Exact, trajectory-invariant path integration naturally dictates commuting generators. Furthermore, skew-symmetric generators inherently yield equinorm representations with translationally invariant similarities. Translational similarity invariance is critical for stable egocentric navigation in open fields and empirical models typically enforce them via explicit regularization or architectural constraints. Extending our geometric perspective, we demonstrate that preserving the metric of flat space restricts the spatial wavevectors (derived from the generator eigenvalues) to form sets of roots of unity. The proposed framework supports diverse, biologically relevant spatial tuning profiles, including periodic grid fields, localized place fields, and context-dependent remapping. Beyond explaining mechanisms that underpin recent deep learning models of spatially-tuned cells, our work provides a formal connection between continuous attractor dynamics and oscillatory interference models. Finally, we show how the framework natively supports goal-oriented, multi-map navigation by identifying superpositions of representations alongside remapping as hyperdimensional computing operations. By grounding spatial representations in strict algebraic laws, this work offers a transparent alternative to empirical representation learning, revealing the exact conditions required for a coherent neural map of space.