Fig 1.
Reichardt-Hassenstein detector.
The detector tests whether input at the two locations (top) is correlated in time, with peak response at the time constant τ. M represents multiplication, and—represents subtraction. By taking the difference between the progressive and regressive circuits (in this case the right arm is progressive and the left arm is regressive) the detector gives a response (bottom) from -I to +I, where I is the maximum input to the detector, and a negative value indicates a reverse correlation. The architecture of this detector forms the basis of the retinotopic layers of the model, however the form is modified by the addition of neural dynamics on both arms of the detector. Further details can be found in the text.
Fig 2.
The two versions of the AVDU detector. Left: with dynamic time constants; right: with delays.
Squares indicate LIN units, circles indicate other operations. In the centre we suggest the corresponding regions of the bee visual system for each stage of the detector. In both versions the input is first temporally filtered in the input layer (first square), then is transmitted to two Reichardt-Hassenstein detectors (Fig 1). These either differ in the time constants of the LINs (τ1 and τ2) or by fixed delays (d1 and d2) shown with gray backgrounds. A base time constant of τb = 1ms is used otherwise. A further LIN is used to apply the subtraction, having a time constant of τR = 5ms, and then the division is performed in the final LIN following summation across all detectors in the array. This final LIN has a time constant of τS = 100ms to smooth the output.
Fig 3.
Example of fits to one data set from Ibbotson 2001.
f(x) (response versus angular velocity) for log, linear and exponential fits are shown with their R2 values.
Fig 4.
Comparison of the responses of dynamic time constant variants of the detector.
(A) For a fixed time constant (τ1) and varying long time constant (τ2). (B) for a fixed long time constant (τ2) and varying short time constant (τ1). A best match to log-linear response for the 5/15ms time constant pair is found. All responses shown are for the detector responding to offsets only. The desired log-linear response is shown by the gray line.
Fig 5.
Comparison of the responses of fixed delay variants of the detector.
(A) For a fixed short delay (d1) and varying long delays (d2). (B) For a fixed long delay (d2) and varying short delays (d1). A non-log linear response curve for all delay pairs is shown. All responses shown are for the detector responding to offsets only. The desired log-linear response is shown by the gray line.
Fig 6.
The average response of the full detector is largely invariant to the spatial frequency or contrast of the stimulus.
The three plots show three different methods of the delay in the RHD-LIN detector, and the graphs within show the four different types of input. (A) a log-linear response is found for the onset-only and offset-only response curves. Both the rectified and non-rectified onset and offset response curves deviate from log-linear at high values of AV. (B) and (C) all response curves deviate from log-linear.
Fig 7.
The average responses of the component RHD-LIN detectors for several spatial frequencies compared to experimental data of optomotor neurons.
Experimental data is from Ibbotson 2001 [18]. The responses of the dynamical detectors (top two graphs) show a better match for the experimental data, notably the shared roll-off, than the fixed delay detectors.
Fig 8.
The average responses of the full detector to different spatial frequencies and different contrasts.
The input has a spatial frequency of 19°, and the model F = 0.25. The response shows a clear velocity tuning that is largely invariant to the spatial frequency or contrast of the stimulus, with the exception of very low and high values of AV where there is greater variance.
Fig 9.
Centering performance of the model with F = 0.0 and F = 0.25 both agree with experimental data, while performance with F = 0.5 does not.
Experimental data are from Dyhr et al [3] for real bees. One wall is held at a constant spatial frequency while the other is varied with sinusoidal patterns. Dashed lines indicate the two points where the spatial frequencies of the two walls are equal, one for each of the two lines. The model error bars show the variance of two runs with differing starting positions in the corridor.
Fig 10.
Centering performance of the model agrees with experimental data from real bees for F = 0.0 and 0.25.
Experimental data is from Dyhr et al [3]. One wall was held at a constant spatial frequency while the other is varied, with square wave and sinusoidal patterns.
Fig 11.
Odometry using the flight of the model bee.
The simulated bee was flown in the for 5 seconds with the same stimuli on both walls. The stimuli were square wave gratings with spatial frequencies from 0.1 to 0.8 cycles per degree when observed from the corridor centre. Post-simulation the distance in cm per unit of the summed logged detector output is compared, and shows a consistent estimation of distance from the total summed detector output until 0.6 cycles per degree.
Fig 12.
Layout of the full system showing subregions and AVDU placements.
The input to the full system consists of 32x32 ommatidial locations (blue grid), which are processed by AVDUs in three subregions, left (green), right (red) and centre (orange). AVDUs (yellow circles) exist between the location pairs sharing the edge they are located on. The preferred motion direction of each subregion is shown with an arrow. Note that the 32x32 extent of the locations covers a field of view extending 260 degrees horizontally and 180 degrees vertically.