Fig 1.
The motion detection section of the model for one eye.
A shows the structure of a detector unit spanning two neighbouring ommatidia, with purple and green outputs responding to angular velocity (AV) in the progressive and regressive directions respectively. M denotes multiplication, / denotes division, the ∑ denotes summation over the entire eye. The τ are the time constants of leaky integrators. B shows the horizontal layout of the units and connections for a single eye, and the mapping from visual space to the model photoreceptors, with 0° representing directly in front of the modelled fly. C shows a diagram of the visual environment, which can be rotated. D shows the response of the angular velocity detector to a single 11.5° bar moving across the field of view at different angular velocities. The parameters of the model are (full details can be found in [24]): τ1 = 5ms, τ2 = 15ms, τb = 1ms, τS = 10ms, and control the range and nature of the angular velocity response of the AVDU.
Fig 2.
The components are as follows: Cyan: ring attractor neurons; Green: clockwise motion rotational neuron (with dark blue driver neuron); Red: anti-clockwise rotational neurons (with pale red driver neuron); Pink: Positional inputs. A shows the connectivity within the ring attractor for a single wedge, with local excitatory connections that decrease with distance, and uniform inhibitory connections. B shows the neural circuits for rotating the bump of activity around the ring clockwise and anti-clockwise for a single pair of wedges. The activity is gated by a single driver neuron (centre of the ring) which multiplies the ring activity to produce the output to the next wedge. C shows the positional input to one half of the ring for one neuron, and the angular mapping to the environment. D shows (left to right) the full structure of the ring; the ring being seeded by positional activity (green) to create a bump; the clockwise driver neuron (red) rotating the bump around the ring; the anti-clockwise driver neuron (blue) rotating the bump back to the initial location.
Fig 3.
A Single stimulus. B Dual stimuli. C Panorama of stimuli. These stimuli are based on the angular dimensions of the layouts used by Seelig and Jayaraman [11].
Fig 4.
Ring attractor azimuth estimate accuracy.
A shows the simulated (blue) and estimated (grey) azimuthal rotations when the ring attractor is driven only by the motion of the bar. There is overall good tracking for much of the simulation however there are time periods where the estimate drifts away from the simulated value. B show the same rotations when positional cues drive the ring attractor. There is better tracking, however D shows that the width of the receptive fields leads to a stepped profile for the ring attractor estimate as movement of the bar within a receptive field cannot be detected. C shows the same rotations with combined position and motion driving the ring attractor. There is good tracking, and E shows that the motion signal can compensate for the insensitivity of the motion system within a receptive field. F summarises the results showing that the circular mean and standard deviation (calculated using the Matlab Circular Statistics Toolbox [38]) of the difference between the simulated and estimated rotations is largest with motion only driving the ring attractor and smallest with combined motion and position.
Fig 5.
Receptive field number correlates with tracking accuracy in the ring attractor.
A Greater numbers of receptive fields provide a reduction in the circular standard deviation (calculated using the Matlab Circular Statistics Toolbox [38]) of the error between the actual azimuth and the estimate from the ring attractor. B The correlation between the change in the tracking error and the change in azimuth shows significant positive correlation, which is significantly different for larger RF numbers than for smaller RF numbers. C The correlation between the change in tracking error and the absolute azimuth shows no significant correlation (see main text for details of the test). D An example of the correlation between the change in tracking error and the change in azimuth (for 16 RF, data sampled every 1000 iterations, using the temporal offset calculated to find the mean azimuthal offset). The colour of each point indicates the absolute azimuth, showing no correlation to error change. One example is used as it is representative of all such plots.
Fig 6.
The model can learn to map landmarks to positions on the ring attractor with slow consolidation (β = 0.5) of the plastic weights. In addition the evolved offsets when the ring attractor is seeded to one position are shown.
A The model performance (see E for comparison with the results from Experiment 2) in terms of mean and circular standard deviation (calculated using the Matlab Circular Statistics Toolbox [38]) of the ring attractor direction from the world direction (corresponding to the location of the single stimulus, and remapped from 270° to 360°) is shown at the top, and shows that the model is able to track the motion of the world well in all cases, with an offset to the position of the stimulus on the visual field. Additionally, a polar plot contains an example of the evolution of the weights over the course of the simulation (with time increasing from the centre to the outside) for a single stimulus. In the polar plot each receptive field (RF) is given a colour, with the key around the outside of the ring, and the angular position of each line denotes the position on the ring attractor that each RF maps to. An ordered mapping, with no offset, should therefore be shown by each line lying in the circular segment under the corresponding key item. Instead, we see that the weights remap the 260° world onto approximately 360° of the ring attractor, and the learning is established early in the simulation, although the weights do show changes in the mapping over the course of the simulation. A second Cartesian plot shows the clustering of the RFs into retinotopic maps (see Methods for details), with the size of the marker denoting the number of RFs in a cluster. For the single stimulus it can be seen that a single map evolves, but changes offset over time. B As A, but with the panorama. The polar plots are not helpful for the panorama as there are multiple retinotopic mappings developed. Here there is once again a good performance in tracking the world in most cases, albeit with one run where tracking performance is poor. The example showing the weight clustering shows that multiple retinotopic mappings are developed, however these mappings change over the course of the simulation. C As A, but with dual stimuli. In this case the model exhibits poor performance tracking the world, and the weights remain largely changeable throughout the simulation. D This panel shows the correlations between the changes in azimuthal position and the changes in the offset between the actual azimuth and the azimuth represented on the ring attractor. The single and panorama simulations show a similar correlation to that found with fixed RFs, however the dual stimuli simulations show a negative correlation between the change in position and the change in error.
Fig 7.
The model can learn to map landmarks to positions on the ring attractor with fast consolidation (β = 2.0) of the plastic weights. In addition the evolved offsets when the ring attractor is seeded randomly are shown.
A As Fig 6: A except with β = 2.0. Here the weights finish evolving early on in the simulation and remain fixed after that point. There is only one retinotopic mapping developed. B As Fig 6: B except with β = 2.0. The weights form three retinotopic mappings, which persist for the rest of the simulation. C As Fig 6: C except with β = 2.0. The weights form many weak mappings, with few RFs belonging to any one mapping. D This panel shows ‘transitioning’ between stimuli as found in the experimental data of Seelig and Jayaraman [11]. E This panel shows the correlations between the changes in azimuthal position and the changes in the offset between the actual azimuth and the azimuth represented on the ring attractor. The single and panorama simulations show a similar correlation to that found with fixed RFs, however the dual stimuli simulations show a negative correlation between the change in position and the change in error.