Spatial Encoding of Translational Optic Flow in Planar Scenes by Elementary Motion Detector Arrays

Elementary Motion Detectors (EMD) are well-established models of visual motion estimation in insects. The response of EMDs are tuned to specific temporal and spatial frequencies of the input stimuli, which matches the behavioural response of insects to wide-field image rotation, called the optomotor response. However, other behaviours, such as speed and position control, cannot be fully accounted for by EMDs because these behaviours are largely unaffected by image properties and appear to be controlled by the ratio between the flight speed and the distance to an object, defined here as relative nearness. We present a method that resolves this inconsistency by extracting an unambiguous estimate of relative nearness from the output of an EMD array. Our method is suitable for estimation of relative nearness in planar scenes such as when flying above the ground or beside large flat objects. We demonstrate closed loop control of the lateral position and forward velocity of a simulated agent flying in a corridor. This finding may explain how insects can measure relative nearness and control their flight despite the frequency tuning of EMDs. Our method also provides engineers with a relative nearness estimation technique that benefits from the low computational cost of EMDs.


S1 Values of parameters for theoretical study
The model was evaluated with N f = 2000, f min = 1, and f max = 1000. The five parameters τ, ∆Φ, d, V , and Φ were linearly sampled according to the values presented in the following table. The speed of the agent V was sampled with 49 different values between 0.1m/s and 0.8m/s. The distance d between the agent and the surface was sampled with 49 different values between 5cm and 20cm. The inter-ommatidial angle ∆Φ was sampled with 28 different values between 0.5°and 5°. The time constant τ was sampled with 28 different values between 1 ms and 10 ms The azimuth angle Φ was sampled with 200 different values between π/24 and π/2. The  Figure S2. Apparent angular period for varying viewing direction Φ and ratio between Λ and d. For Λ << d, the small angle approximation holds true and the apparent angular period follows a squared sinus as is expected in the case of a lateral infinite wall. However when Λ > d, the apparent angular period is shorter on the side (Φ ≈ 90 • ), and longer in front and in the back (Φ → 0 • and Φ → 180 • ). .

Apparent angular period
The apparent angular period λ is defined, for a given linear period Λ, as the angular size occupied by a complete cycle on the retina of the agent as shown on Fig. S2a. It is dependent on the distance to the surface d, the linear period Λ and the viewing direction Φ. By posing x + = d tan(θ ) + Λ 2 , x − = d tan(θ ) − Λ 2 , and θ = π 2 − Φ, we obtain geometrically from Fig. S2: Thus the expression for the apparent angular period is given by equation (S2). The variation of the apparent angular period across the visual field can be seen on Fig. S2b.

Apparent temporal and linear frequency
Linear frequencies, linear periods and angular periods are respectively noted f , Λ and λ , and are respectively expressed in m -1 , m, and radians. The relation between f and Λ is given by equation (S3).
Temporal frequency ω is expressed in radians per second. In the case presented in Fig. S2a, the temporal frequency does not depend on the viewing direction Φ and is only function of the speed of the agent and the linear period as given in equation (S4).

Apparent signal amplitude
In a viewing direction Φ, a sinusoidal pattern is seen with an apparent angular period λ through photoreceptors with Gaussian acceptance windows of standard deviation σ = ∆ρ 2 √ 2 ln(2) . The apparent signal amplitude is the result of the convolution of the input signal with this Gaussian window as in equation (S5).
According to 1 , in most diurnal insect species the ratio between acceptance angle and inter-ommatidial angle is given by equation (S6): From equation (S5) and equation (S6), we can express the apparent signal amplitude as a function of the inter-ommatidial angle and the apparent angular period, as given in equation (S7).
3/10 S3 EMD velocity response curve for a broadband signal Here we reproduced the same study for a single-frequency image, i.e. a sinusoidal grating. The spatial period of the sinusoidal grating is Λ = 5 cm, which is in the range of spatial periods used in the main text. We can see that Ψ no longer encodes relative nearness (h). Also it is noteable that contrary to the broadband case, R max is mostly independent from the distance (c), and Ψ is mostly independent from the flight speed (d). In the single-frequency case, our proposed control method allows the agent to stabilise its lateral position in the center of the corridor. This is expected from the prediction (showed in Supplementary Figure S5) that Ψ decreases with the distance to the walls. (Bottom left): In the single-frequency case, the agent is not able to stabilise its flight speed, it varies erratically between 0.5 m/s and 1.0 m/s. This is expected from the prediction (showed in Supplementary Figure S5) that Ψ is independent from the flight speed. It should be noted that a natural setting would realistically not include single-frequency patterns. Furthermore insects are likely to control flight speed using additional regions of the visual field (like the ventral and dorsal regions) that may contain patterns with richer frequency content.