Path integration (PI) is a method of navigation widely used in the animal kingdom. Equivalent to the sailing technique know as dead reckoning, an animal needs only a compass cue and odometer to keep track of its location during a foraging excursion, and subsequently find its way back home. This method therefore works in the absense of landmarks. In the ant species I have modelled, Cataglyphis fortis (C. fortis), the animal's eye has a specialise dorsal rim area which allows it to perceive the pattern of sky light polarisation, which it can use as an accurate compass cue. The animal appears to use step counting as its main odometric cue.
PI requires that these two sources of information (compass and odometer) are continuously processed and integrated during the animal's movements to maintain a running estimate of its location relative to the starting point. This running estimate is know as a home vector (HV), which must somehow be stored in memory throughout any excursion. C. fortis forages over rather flat, featureless salt pans, and tends to relie heavily of PI. The HV is subject to the accumulation of errors, such that when the ant attempts to return to the place it started from (the entrance to its underground nest) it will probably to in slightly the wrong place. The solve this problem C. fortis performs a 'systematic search' pattern, where it performs a series of looping runs centred around its best estimate of where the nest should be, until it locates the nest or perishes from dehydration.
The ant performs PI using only its tiny salt-grain sized brain consisting of perhaps a few tens of thousands of neurons. It is possible to design a PI system using a simplified model of the information processing capabilities of neurons (see the models of Hartmann and Wehner, and Wittmann and Schwegler, 1995, Biological Cybernetics).
My approach has been to use a genetic algorithm (GA) to construct network models of PI via a process of artificial evolution. This (to my knowledge) is the first successful use of this technique in PI modelling. The model produced is considerably smaller than the hand built models, and reproduces two important features of path integration in C. fortis, namely a search behaviour and the curious homing errors seen after certain experimentally imposed outward journey shapes.
The model uses my own modified version of the standard Continuous Time Recurrent Neural Network (CTRNN), referred to here as ModCTRNN. See the CiTRuS page for more details.
The sensors are shown at the top. Two beacon sensors (B_L and B_R) allow the agent to navigate from the nest to a series of light sources placed at random locations in the arena, using a standard Braitenberg architecture. Each light goes out once the agent reaches it, and the next one activates immediately. Hence the outward journey can be controlled by a purely reactive control system. This is not intended to closely model the ants foraging behaviour. Two compass sensors (C_L and C_R) output the cosine and sine of the agent's current compass heading. The speed sensor (S) output is a value proportional to the agent's forward speed. The agent moves only forwards (not sideways) in keeping with the foraging ant. The forward speed is set by the forward motor neuron (F), the left and right turning tendencies are set by the left and right rotation motor neurons (R_L and R_R).
ModCTRNN allows weights to be modified by other weights, turning them into variables in their own right (see CiTRuS page). Weights W_L4 and W_R4 act to copy the speed sensor output to weights W_L3 and W_R3. W_L3 and W_R3 multiply this value by the output of compass sensors C_L and C_R respectively, and input the value to contralateral (opposite side) weights W_R2 and W_L2 respectively, which integrate their inputs. Mathematically, the integral over time of an object's speed multiplied by the cosine and sine of its heading gives the x and y coordinate respectively. Weights W_L2 and W_R2 are therefore the agent's HV. To turn the agent back towards its home we can (using Mittelstaedt's bicomponent PI model) multiply W_L2 by C_L and W_R2 by C_R and take the differece between the two values as the turning signal. This is what the network does, to a first approximation. Homing begins automatically once the last light has been reached since the light sensor outputs drop to zero and no longer supress the homing signal being fed into the rotation motor neurons.
Here the agent starts at the central circle (the nest) and visits the three beacons in an anticlockwise direction. After the last beacon it returns home in a straight line, just like C. fortis. In this case it found the nest first time, so no searching behaviour is seen.
The GA has basically reinvented Mittelstaedt's bicomponent model of PI, (and no I didn't design it to do this, I read Mittelstaedt's paper AFTER the network had evolved) but added some bells and whistles. Specifically the weights storing the HV act as leaky integrators, meaning they gradually lose the information stored in them. Instead of tuning the integrators to be less leaky, evolution has discovered that it can leave them leaky, but scale down the new incoming information to match the amount of leakage since the beginning of the journey. Therefore the stored (x,y) values are effectively placed on an exponentially decaying coordinate system.
The network also displays search behaviour once the agent returns to the nest. Rather than create a separate search pattern generator, evolution has slightly adjusted the homing mechanism to produce efficient searching behaviour as a continuation of normal homing. The search density has a bell shape similar to that seen in the ants.
After performing L shaped outward journeys, a simplified version of the network can be made to reproduce the homing errors seem in the ants by adjusting a single parameter controlling integrator leakiness. This is the first time that a version of Mittelstaedt's model has been show to be capable of reproducing these errors. The diagrams show the angle between the two outward legs of an L shaped journey (x axis) versus the homing direction adopted by the ant or agent (y axis). The dots show data from C. fortis, the upper line shows the fit of my model to the data, the lower lines shows the geometrically correct homing direction.
Paper submitted for review 12/12/2004. RJ Vickerstaff and EA DiPaolo. Many thanks to Tom Collett for suggesting this project to me, and for many helpful comments during its progress.
Sussex Insect Navigation Group