Are Grid Cells an Efficient Encoding of Space?

Since their discovery over 20 years ago, grid cells have attracted considerable theoretical interest. Although it is widely agreed they subserve path integration (the tracking of position from self-motion cues), the precise reason for their distinctive activity remains debated. Previous work has generated grid-like representations by optimising objectives motivated from both the efficient coding of spatial position and path integration, often encouraging properties such as conformal isometry, translation-invariant dynamics or high coding capacity. These studies have identified important conditions under which grid-like codes emerge. However, whether grid cells emerge from directly optimising for efficient coding alone, across a broad class of possible representations, has not been fully addressed and still remains an open question. To address this question, we consider a natural task-level formulation of efficient coding: under biologically plausible constraints, we seek representations that minimise the mean squared error in the presence of noise. Specifically, we consider an encoder that receives noisy displacement updates and whose activity itself is corrupted by noise, deriving conditions under which this path-based problem reduces to the classical efficient-coding problem of encoding a source distribution. To analyse this problem, we employ the Ziv–Zakai lower bound, an information-theoretic tool that captures both local decoding precision and catastrophic errors arising from periodic spatial codes. Along the way, we see that several properties previously found to produce grid cells are theoretically advantageous within our coding framework. Grid-cell-like representations then emerge from the optimisation, providing evidence that grid cells constitute a highly efficient spatial code. More broadly, our results support the idea that the problem of efficient spatial coding is tightly related to the one of path integration, this suggests that task-relevant compression can favour representations that reflect the symmetries and compositional structure of the task.