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Path integration is the ability to maintain an estimate of displacement from a starting point by accumulating self-motion cues. In desert ants and bees, this is the primary homing mechanism. In Drosophila, the CPU4 neurons in the central complex are believed to perform this computation. The CPU4 Model Eight neurons with preferred directions spaced evenly around the circle. Each neuron integrates the component of velocity along its preferred direction: Input: heading (from ring attractor) and speed (constant in our model) Accumulation: half-wave rectified projection of velocity onto preferred direction Memory leak: optional exponential decay parameter λ that causes old displacements to fade Decoding: population vector gives the home direction; its magnitude gives the distance The Memory Leak Trade-Off A perfect integrator (λ = 0) remembers everything but accumulates drift errors on long journeys. A leaky integrator (λ > 0) forgets old displacements, creating a “horizon” beyond which the bug cannot navigate home. This trade-off generates a phase diagram: for each noise level, there is an optimal exploration duration beyond which homing fails. ...

The bug is a Braitenberg-inspired vehicle that navigates using stochastic Langevin dynamics. It has a position, a heading, and a single steering input derived from its compass circuit. The locomotion model is intentionally minimal: an Euler–Maruyama integrator of heading and position, with rotational diffusion providing the “random walk” that makes exploration possible. The Locomotion Law The bug moves at constant speed along its heading direction. Steering is governed by: Goal-directed torque — derived from the difference between current heading and home direction (from path integrator) Rotational noise — Gaussian white noise scaled by a diffusion coefficient Compass input — the ring attractor’s decoded heading, which itself depends on the quantum compass The balance between deterministic steering and stochastic exploration determines whether the bug can navigate home. This is quantified by the Péclet number — the ratio of directed transport to diffusion. ...