Path integration in complex number space.

Desert ants and foraging rodents return home along surprisingly direct paths after meandering outward journeys. Traditional path integration models explain this through cumulative vector addition, yet struggle to account for the neurobiological mechanisms underlying this computation. Here we show that animal navigation emerges naturally when trajectories are represented in complex number space. We propose that head direction (HD) cells provide allocentric reference frame rotations, bilateral brainstem neurons (Chx10) control motion magnitudes within each frame, and left-right alternating theta sweeps in entorhinal-hippocampal maps continuously sample these bilateral activation states. Frame rotations at directional changes map to multiplication by unit complex numbers, converting egocentric motor commands into allocentric position tracking. This framework offers a neurobiologically grounded mathematics of navigation.