Imagine: You started from point A, somewhere in the middle of a desert. After several days of travel without a map or knowing your destination, seeing only a few rocks at a time and making countless random turns, you finally arrived at point B. There you were given a heavy load to carry back to point A. Without missing a beat, you went home in a straight line. No instruments, no navigation tools. Can you do that?
Our brains are capable of great intellectual feats, such as multiplication of two by two or setting DVD players, but imagining our exact trajectory in 3D space and finding the shortest way back (the home vector) is not our innate ability. Yet insects do that every moment of their lives. How do they do that? Do they use cartesian or polar coordinates? any coordinates at all? How do they compensate for navigation errors? Do they use visual cues? Do they use maps? What kind of maps are these?
...The ability to use currently perceived terrain features to set a course for an area you cannot presently perceive requires the use of a map. It also implies the ability to do navigational computations utilizing the data on the map. The brain of an insect has computational capabilities analogous to those found in a GPS. http://ruccs.rutgers.edu/faculty/GnG/Insect_Symbol_Processing.pdf
...Integration of walking speed and angular variation along the arthropod’s walking route (path integration) gives a global home vector that enables it to determine distance and direction of its nest at any position and time. Charles Darwin was the first to assume that animals may navigate this way. Apart from path integration, many species are capable of using landmarks to get their bearings. However, before these local vectors can be applied successfully, some information about their position has to be stored. Orientation with the aid of local vectors is error-prone, since landmarks can disappear or change their appearance, and, of course, is out of question when no visible landmarks are nearby. The global vector gets updated on the complete trip, even if the orientation is conducted by using landmarks. Moreover, after a sudden failure of the stored landmarks, insects revert to their global vector for orientation. Even if not used for several days, they keep the global vector stored in their memory.
...the main inputs for navigation are spectral skylight gradient, sun position, and the pattern of polarised skylight. The ants continuously use the sky as a reference to determine their body axis orientation. Insects make use of the fact that light waves with their different wavelengths are not equally distributed over the illuminated sky. Direct orientation with respect to the azimuthal position of the sun (or any other light source) has been found in ants, bees, and spiders. The skylight polarisation pattern represents the most effective and stable means for orientation. Each rotation of the ant’s body axis results in a corresponding change of this orientational angle allowing the ant to measure not only its current body direction relative to the skylight pattern, but also the rate of its angular rotation, independent of whether it is moving or turning on spot. Although the polarised light pattern is changing with elevation of the sun, insects use a stereotypical projection that resembles the skylight pattern at dawn or dusk, respectively, in their memory. http://arxiv.org/PS_cache/q-bio/pdf/0512/0512031.pdf
...Desert ants perform large-scale excursions in surroundings devoid of conspicuous landmarks, from which they return to their nest on a direct, shortcut way. During their outbound path they continually update, with astonishing precision, a `home vector', which at any point indicates the homing direction and the distance to the nest. To assess this home vector, the ants need a source of information about the distances they have travelled, i.e. a kind of odometer, and also about the compass direction of their path segments. The polarisation pattern of the sky is the predominant reference system for the estimation of walking directions. The sensory basis of the ants' odometer, however, is less well understood. Three types of cues: (i) energy expenditure, (ii) self-induced optic-flow and (iii) idiothetic cues, i.e. information derived from the animal's own movements, have been proposed. Desert ants use neither energy expenditure nor optic flow cues for gauging distances. Heavy loads did not affect the measurements of walking distances, and the ants arrived at a fairly exact distance estimate even if all optic flow cues had been excluded. The conclusion from these experiments was that ants rely mainly on idiothetic cues, probably on a kind of step counter. Further experiments with ants that were trained to walk over a linear series of hills, however, indicated that the use of idiothetic information cannot be as simple as activating a step counter or monitoring the output of a central pattern generator. The ants indicated homing distances that corresponded to the outbound run's ground distance, not to the (much longer) distance actually walked over the hills. Ants seem to be able to derive the horizontal projections from the uphill and downhill segments of their path. Hence, the animals must be able to measure the slopes of the terrain and to integrate this information into their process of distance estimation. The results show that it is the ground distance that the ants feed into its path integrator, and suggest that the ants are able to keep an accurate home vector also in hilly terrain. http://jeb.biologists.org/cgi/reprint/208/21/4005.pdf
...The integrator is made of linear arranged neurons. Every neuron inhibits its two neighbours and, once activated, stays activated. If a virtual ant starts from the nest, all but the first neurons are reset. Every time the ant has traveled a certain distance, the activation passes to the next neuron. If the ant is able to register the orientation to an objective direction, it might store the two coordinates of the vector in function of the orientation (dx,dy) in two different integrators.
This model is just a speculation. Nobody really knows how insects do path integration, measure distances, position themselves relative to landmarks, and integrate all this information to go straight home. Furthermore, the path integartion alone is generally insufficient for optimum homing, as there are always obstacles getting in a way. Not only one has to find the shortest path home, one has to stay on this path avoiding the known obstacles and do that in an optimum way, as the load is very heavy. Insects do that, too. One theory suggests that they store panoramic views of all obstacles that they identify. Another suggests that insects use vector maps and follow field lines.
...The global path integration is enhanced by long-term memories of significant sites that insects store in terms of the coordinates of these sites relative to the nest. With these memories insects can plan routes that are steered by path integration to such sites. Quite distinct from global path integration are memories associated with familiar routes. Route memories include stored views of landmarks along the route with, in some cases, local vectors linked to them. Local vectors by encoding the direction and/or distance from one landmark to the next, or from one landmark to a goal, help an insect keep to a defined route. Although local vectors can be recalled by recognising landmarks, the global path integration system is independent of landmark information and that landmarks do not have positional coordinates associated with them. The major function of route landmarks is thus procedural, telling an insect what action to perform next, rather than its location relative to the nest.
more on the snapshot theory
...The snapshot model assumes that a panoramic image of the surroundings of the target position (nest, feeding station) is acquired and stored by the animal. The stored visual snapshot is then used to calculate the direction that the animal has to follow in order to return back to its target position. This is done by employing a matching mechanism that compares the current image with the stored snapshot. According to the ALV model, the animal does not need to store a snapshot: the only piece of information needed is the direction of the Average Landmark (AL) vector which is acquired by a simple summation of unit vectors pointing to landmarks (which are simple visual features, e.g. edges). The target direction is calculated at any position as the vector difference between the AL vectors at the target and the current position. The single vector representation of the target direction suggests that there might be a closer link between the landmark navigation system and the celestial compass navigation system.
http://voronoi.sbp.ri.cmu.edu/~motion/papers/sbp_papers/integrated2/lambrinos_landmark_vector.pdf and, in more detail, http://www.verena-hafner.de/papers/emoa.pdf
How is that done? Here is the conclusion from the recent paper:
...in our model, the arthropod does not need to perform complicated calculations such as applying trigonometric or other non–linear functions, but rather updates two cartesian coordinate values of the relative global vector by computing a simple system of linear differential equations. http://arxiv.org/PS_cache/q-bio/pdf/0512/0512031.pdf
You see how simple it is: an animal the size of a pinhead solves the system of linear differential equations, uses spectroscopy and polarimetry for absolute positioning, counts its steps for hours at a time, uses globally referenced maps, derives and memorizes horizontal and stereotypical projections, and adds multiple vectors -- all without using trigonometric and other nonlinear functions. Nothing fancy. Apart from that, it is doing numerous other, equally improbable, tasks. The ability is built-in; it is not learned. It is passed on.
Now imagine this brain to be the size of yours.
Imagine the ability to untangle the loops and zigzags of one's life, integrate over all thoughts and memories, tally up all losses and wasted hopes, and go straight back to one's beginnings, by following the light that shineth unto the perfect day: the global home vector. Do we have this ability?