Animal path integration: a model of positional uncertainty along tortuous paths.

Exact closed form mathematical solutions are reported which quantify the dynamic uncertainty resulting from path integration (PI) along tortuous paths. Based on a correlated random walk model, the derived results quantify positional estimation error moments with and without a compass, in discrete and continuous time. Consistent with earlier studies on attempted straight-line navigation, using a compass significantly reduces the uncertainty during PI, making purely idiothetic PI biologically implausible except over short distances. Examples are used to illustrate the contributions of angular noise, linear noise and path tortuosity, under different conditions. Linear noise is shown to be relatively more important with a compass while angular noise is more important without. It is shown that increasing path tortuosity decreases positional uncertainty, true for long and short journeys, irrespective of whether a compass is used, or the level of noise. In contrast, reducing angular noise also reduces uncertainty, but only below some critical level of noise. Using canonical equations of PI, it is shown that polar PI using a compass accumulates uncertainty in a manner similar to Cartesian PI without a compass. Issues of data sampling bias and intermittent use of a compass are also considered for PI along tortuous paths.