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First published online August 18, 2005
Journal of Experimental Biology 208, 3349-3366 (2005)
Published by The Company of Biologists 2005
doi: 10.1242/jeb.01772

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Evolving neural models of path integration

R. J. Vickerstaff^* and E. A. Di Paolo

Centre for Computational Neuroscience and Robotics, School of Life Sciences, University of Sussex, Brighton, BN1 9QG, UK

View larger version (13K): [in a new window]   Fig. 1. The four main classes of possible home vector. Left, geocentric rectangular (x,y) and polar (r,); right, egocentric rectangular (x',y') and polar (r','). N is the nest position, A is the animal's position, the short arrow indicating its orientation. We use the convention of measuring angles as positive anti-clockwise from the x-axis or from the animal's body axis.

View larger version (15K): [in a new window]   Fig. 2. Dynamics of homing using (A) Eqn 3 and (B) Eqn 4. The horizontal axis shows the animal's orientation, ; the vertical axis shows the animal's rate of turn d/dt, where (x,y) is the geocentric rectangular home vector (HV) and (r,) is the geocentric polar HV (see Fig. 1). This example shows the case where (A) x>0, y=2x and, in equivalence, (B) =tan–12=1.107. Both schemes lead to one stable and one unstable equilibrium heading with the same respective values.

View larger version (9K): [in a new window]   Fig. 3. The four compass activation functions used. From the bottom: cosine, positive cosine, piecewise linear approximation of cosine, head direction cell. Plot of sensor output (y-axis) against A–a (x-axis), where A is the agent's current orientation and a is the sensor's preferred direction.

View larger version (19K): [in a new window]   Fig. 4. Experiment 1A CTRNN network solving the constant speed path integration task. BL/R, left/right beacon sensor; CL/R, left/right compass sensor; RL/R, left/right rotation motor neuron; NL/R, interneurons. Arrows are directional weighted links; double-headed arrows indicate two links running in opposite directions. Open circles are neurons involved in the oscillations responsible for generating the various modes of behaviour seen in this agent during its return journey (see Fig. 5).

View larger version (12K): [in a new window]   Fig. 5. Experiment 1A CTRNN agent performing the constant speed path integration task. B1 indicates the single beacon. Beacon and nest are drawn as circles of radius 0.01.

View larger version (24K): [in a new window]   Fig. 6. Experiment 1A CTRNN network dynamics during the return phase of the journey (see Fig. 5). Plots are of neuron firing rates; the y-axis runs from 0 (bottom) to 1 for each. B1 and nest indicate the time of arrival at the beacon and the nest, respectively (the nest is reached at the very end of the trial). NL3/R3 and RL/R are as in Fig. 4.

View larger version (11K): [in a new window]   Fig. 7. Experiment 2B agent (see Fig. 8) performing the variable-speed path integration task. B1,2,3 indicate the order the beacons were presented in. Beacons and nest are drawn as circles of radius 0.01.

View larger version (17K): [in a new window]   Fig. 8. Experiment 2B ModCTRNN network solving the variable-speed path integration task. BL/R, left/right beacon sensor; CL/R, left/right compass sensor; RL/R, left/right rotation motor neuron; S, speed sensor; F, forward motor neuron. Arrows are directional weighted links; double-headed arrows are two links going in opposite directions between the same end points. Lines ending in small squares are links that modify other links. wL1/R1...wL6/R6 are weighted links (synapses). See Table 2 for parameter values.

View larger version (19K): [in a new window]   Fig. 9. Experiment 2B network performing the variable-speed path integration task. Plot shows link weights (y-axis; see Fig. 8) over the whole trial (x-axis; see Fig. 7). B1,2,3 show the approximate time of arrival at each beacon (the nest is reached at the very end of the trial). From 2 time units into the trial until 3 time units, a large change in the values of wL3/R3 reflects noise applied to the forward motor output, as detected by the speed sensor. The second abrupt change in these values reflects the period of enforced captivity of the agent at the last beacon before homing begins.

View larger version (17K): [in a new window]   Fig. 10. Experiment 2B agent showing transient negative phototactic behaviour (loops) near the limit of the `foraging' range selected for during evolution. Here, the agent twice turns away from the beacon (marked B1) before finally reaching it and homing normally.

View larger version (14K): [in a new window]   Fig. 11. The simplified analytic model (SAM) fitted to data (Müller and Wehner, 1988) taken directly from figs 11a–c in Hartmann and Wehner (1995). The ants walked along a channel from the nest, turned through an angle A, walked along a second channel before being released on a test field. The lengths of the channels were (A) 10 m and 5 m, (B) 5 m and 10 m and (C) 10 m and 10 m, respectively. shows the ants' homing direction (diamonds). In each graph, the top line shows the SAM using a time constant of =18.38; the lower line shows the correct homing angle.

View larger version (13K): [in a new window]   Fig. 12. The top line represents y=x, i.e. the correct homing distance. The middle line represents three leaky integrator models fitted to data (diamonds) taken directly from fig. 2 of Sommer and Wehner (2004). The models are Eqn 8 with =98.27, ß=0.0103 (taken from their paper), Eqn 9 with =103.2 and Eqn 10 with =193.6 (fitted using least squared error). The models are indistinguishable. Also shown as the bottom line is Eqn 10 with =18.38, the value obtained fitting the simplified analytic model (SAM) to the L-shaped journey experiments (see text), clearly a poor fit.

View larger version (23K): [in a new window]   Fig. 13. The experiment 2B agent's behaviour if the nest is removed (A) with all sources of noise removed and (B) with 1% sensor noise, 10% rotation motor noise and 70% forward motor noise (as during evolution). A searching behaviour that does not rely on the presence of noise is clearly visible and arises from the coupled dynamics of the agent's network and motion.

View larger version (34K): [in a new window]   Fig. 14. Experiment 2B search density of the agent when the nest has been removed, averaged over the final 35% of 50 independent trials. The agent is returning from a single beacon placed at a distance of 0.75 to the right in all trials. The white dot shows the fictive nest position; the black bar shows the size of the nest (diameter 0.02).

View larger version (23K): [in a new window]   Fig. 15. Distances averaged over 200 trials where the agent has returned from a beacon 0.75 distance units away to the fictive nest position and is performing the search behaviour. Top line: average distance of agent from the nest; linear regression fitted is d=1.5820x10–6t+0.0519. Middle line: average distance of agent from home vector (HV) zero point (see text); model fitted is d=1.0598x10–6t+0.0496. Bottom line: average distance of HV zero point from nest; linear regression model (not plotted) is d=1.0484x10–6t+0.0091. The middle line, which increases during the search behaviour, suggests the agent is `deliberately' searching further from the nest the longer it has been searching.

View larger version (13K): [in a new window]   Fig. 16. The simplified analytic model (SAM) modified to use a piece-wise linear approximation of cosine (see Fig. 3) as the compass response function (upper line) fitted using least sum of squared error to data (diamonds; Müller and Wehner, 1988) taken from fig. 11a in Hartmann and Wehner (1995). The lower line shows the correct homing angle. The value of the time constant obtained is =38.00. The fit is noticeably poorer than that obtained with the cosine-shaped compass sensor response function (see Fig. 11A). The axes are labelled as in Fig. 11.

View larger version (12K): [in a new window]   Fig. 17. Network used to derive the simplified analytic model (SAM), adapted from the network evolved in experiment 2B. See Appendix 2 for explanation.