Spatial uncertainty and environmental geometry in navigation

Knowing ones place in the world requires integrating sensory inputs with respect to the geometry of the environment. Consistent with this, variations in environmental geometry, such as the shape and size of an enclosure, have profound effects on navigational behavior1-4 and its neural underpinning in grid cells5-7. Here, we show that these effects arise as a consequence of a single, unifying principle: to navigate efficiently, the brain must maintain and update the uncertainty about ones location. We develop an image-computable Bayesian ideal observer model of navigation, continually combining noisy visual and self-motion inputs, and a neural encoding model representing the spatial uncertainty computed by the ideal observer. Critically, we find that a key determinant of spatial uncertainty is the dimensionality reduction inherent in the retinal projection of the environment. Mathematical analysis and simulations show that spatial uncertainty accounts for a diverse range of sometimes paradoxical distortions in human homing behavior across trapezoidal, stretched, and compressed environments. Moreover, the neural encoding of this uncertainty accounts for observed changes in grid field size, anisotropy, rescaling, and boundary-dependent tethering under analogous geometric manipulations. Our results show that spatial uncertainty arising unavoidably during navigation is key to understanding navigational behavior and its neural underpinnings.