A Theory of Spatial Continuous Attractors in Hopfield Energy Landscapes

Spatial cognition requires stable neural manifolds of orientation and location that track self-motion and sensory cues. Continuous attractors provide a natural dynamical principle, but learning such mechanisms that form and control manifolds with theoretical guarantees remains difficult. We propose Spatial Energy Attractor Learning (SEAL), a theory of neurodynamics learning for spatial cognition, where continuous spatial representations arise as learned low-energy manifolds in Hopfield energy landscapes through population-level Fourier energy learning. For head-direction and grid-cell systems, Fourier energy landscapes yield ring and torus attractors with learned normal attraction and input-driven tangent transport. We prove that Fourier construction is a special case of a general Hopfield-compatible energy extension whose autonomous dynamics preserve energy descent and whose expressive energy families approximate target attractor energies and restoring fields. Experiments on HD and grid-cell populations show that the learned energies restore perturbed states to ring and torus manifolds, stabilize velocity-driven moving bumps across multiple input regimes, and induce continuous-attractor interactive structures with local excitation and surround inhibition. Together, these results provide a stable, learnable, and biologically interpretable energy-based account of continuous spatial representation.