Egocentric Path Integration Models and their Application to Desert   Arthropods
  Path integration enables desert arthropods to find back to their nest on the shortest track from any position. To perform path integration successfully, speeds and turning angles along the preceding outbound path have to be measured continuously and combined to determine an internal {\em global vector} leading back home at any time. A number of experiments have given an idea how arthropods might use allothetic or idiothetic signals to perceive their orientation and moving speed. We systematically review the four possible model descriptions of mathematically precise path integration, whereby we favour and elaborate the hitherto not used variant of egocentric cartesian coordinates. Its simple and intuitive structure is demonstrated in comparison to the other models. Measuring two speeds, the forward moving speed and the angular turning rate, and implementing them into a linear system of differential equations provides the necessary information during outbound route, reorientation process and return path. In addition, we propose several possible types of systematic errors that can cause deviations from the correct homeward course. Deviations have been observed for several species of desert arthropods in different experiments, but their origin is still under debate. Using our egocentric path integration model we propose simple error indices depending on path geometry that will allow future experiments to rule out or corroborate certain error types. 
Tobias Merkle, Martin Rost and Wolfgang Alt