https://mayalucia.dev/tags/navigation/ Recent content in Navigation on MayaLucIA Hugo -- 0.156.0 en-us Wed, 25 Feb 2026 12:00:00 +0100 https://mayalucia.dev/modules/mayajiva/path-integration/ Wed, 25 Feb 2026 12:00:00 +0100 https://mayalucia.dev/modules/mayajiva/path-integration/ <p>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 <em>Drosophila</em>, the CPU4 neurons in the central complex are believed to perform this computation.</p> <h2 id="the-cpu4-model">The CPU4 Model</h2> <p>Eight neurons with preferred directions spaced evenly around the circle. Each neuron integrates the component of velocity along its preferred direction:</p> <ul> <li><strong>Input:</strong> heading (from ring attractor) and speed (constant in our model)</li> <li><strong>Accumulation:</strong> half-wave rectified projection of velocity onto preferred direction</li> <li><strong>Memory leak:</strong> optional exponential decay parameter λ that causes old displacements to fade</li> <li><strong>Decoding:</strong> population vector gives the home direction; its magnitude gives the distance</li> </ul> <h2 id="the-memory-leak-trade-off">The Memory Leak Trade-Off</h2> <p>A perfect integrator (λ = 0) remembers everything but accumulates drift errors on long journeys. A leaky integrator (λ &gt; 0) forgets old displacements, creating a &ldquo;horizon&rdquo; 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.</p> https://mayalucia.dev/modules/mayajiva/bug-model/ Wed, 25 Feb 2026 12:00:00 +0100 https://mayalucia.dev/modules/mayajiva/bug-model/ <p>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&ndash;Maruyama integrator of heading and position, with rotational diffusion providing the &ldquo;random walk&rdquo; that makes exploration possible.</p> <h2 id="the-locomotion-law">The Locomotion Law</h2> <p>The bug moves at constant speed along its heading direction. Steering is governed by:</p> <ul> <li><strong>Goal-directed torque</strong> &mdash; derived from the difference between current heading and home direction (from path integrator)</li> <li><strong>Rotational noise</strong> &mdash; Gaussian white noise scaled by a diffusion coefficient</li> <li><strong>Compass input</strong> &mdash; the ring attractor&rsquo;s decoded heading, which itself depends on the quantum compass</li> </ul> <p>The balance between deterministic steering and stochastic exploration determines whether the bug can navigate home. This is quantified by the <strong>Péclet number</strong> &mdash; the ratio of directed transport to diffusion.</p>