Frequency response of lift control in Drosophila

Abstract

The flight control responses of the fruitfly represent a powerful model system to explore neuromotor control mechanisms, whose system level control properties can be suitably characterized with a frequency response analysis. We characterized the lift response dynamics of tethered flying Drosophila in presence of vertically oscillating visual patterns, whose oscillation frequency we varied between 0.1 and 13 Hz. We justified these measurements by showing that the amplitude gain and phase response is invariant to the pattern oscillation amplitude and spatial frequency within a broad dynamic range. We also showed that lift responses are largely linear and time invariant (LTI), a necessary condition for a meaningful analysis of frequency responses and a remarkable characteristic given its nonlinear constituents. The flies responded to increasing oscillation frequencies with a roughly linear decrease in response gain, which dropped to background noise levels at about 6 Hz. The phase lag decreased linearly, consistent with a constant reaction delay of 75 ms. Next, we estimated the free-flight response of the fly to generate a Bode diagram of the lift response. The limitation of lift control to frequencies below 6 Hz is explained with inertial body damping, which becomes dominant at higher frequencies. Our work provides the detailed background and techniques that allow optomotor lift responses of Drosophila to be measured with comparatively simple, affordable and commercially available techniques. The identification of an LTI, pattern velocity dependent, lift control strategy is relevant to the underlying motion computation mechanisms and serves a broader understanding of insects' flight control strategies. The relevance and potential pitfalls of applying system identification techniques in tethered preparations is discussed.

Figure 1.

Measurement set-up. A cylindrical arena was constructed from 60 modular LED panels. Lift forces were measured from flies flying tethered to a MEMS (micro-electro-mechanical systems) force sensor (see inset), which we placed in the middle of the arena. The flies were stimulated using vertically oscillating horizontal sine grating patterns, whose spatial frequency we varied systematically between trials. Oscillation amplitude and frequency were also varied systematically.

Figure 2.

Examples of force data measurements. (a) Tracking of pattern oscillation. a(i) shows the position (solid line) and velocity (dashed line) of a vertically oscillating horizontal sine grating (peak-to-peak oscillation amplitude A = 4.76 × 10⁻² m; oscillation frequency f = 1 Hz; spatial pattern frequency SF = 21.0 m⁻¹, not shown). a(ii) shows the resulting lift forces measured from four flies. As in the following plots, force data were low-pass filtered with a zero-phase, 5th order Butterworth filter with a cutoff frequency of 40 Hz. (b) Dependence on oscillation amplitude. A fly was stimulated using oscillation amplitudes of 2.38, 4.76 and 7.14 × 10⁻³ m (b(i)). The measured lift forces are shown in red, green and blue, respectively (b(ii)). (c,d) Dependence on oscillation frequency. Pattern position (red trace) is plotted together with the measured lift force (in black) for oscillation frequencies (c) 1.6 and (d) 3.1 Hz. SF = 15.8 m⁻¹, A = 2.38 × 10⁻² m in either case.

Figure 3.

Frequency response. Shaded areas show ± s.d. The oscillation frequency f was varied between 0.1 and 13 Hz (a–e) or 0.3 and 13 Hz (f–h). (a) Force amplitude response for oscillation amplitudes A₁ = 2.38 (red) and A₂ = 4.76 10⁻² m (black). SF=10.5 m⁻¹. Each curve was measured from N = 5 flies and n = 5000 sinusoids distributed evenly across the oscillation frequency range. The responses to the different oscillation amplitudes in the f range between 1 and 5 Hz are significantly different from each other (Mann–Whitney U-test, p < 0.05). (b) Force gain for two oscillation amplitudes. Data from (a) were normalized as A_force/A and replotted. (c) Phase lag of the force response to the pattern position based on the same data as in (a,b). Dashed lines show the predicted response phase of a velocity-dependent response with delays 0, 100 and 200 ms. (d) Force gain for two spatial pattern frequencies, SF₁ = 21.0 m⁻¹ (black) and SF₂ = 10.5 m⁻¹ (red). Data measured using pattern oscillation amplitudes A₁ and A₂ were pooled for this analysis. (e) Phase lag for the same data as shown in (d). (f) Force gain of combined data. A between 23.8 × 10⁻³ m and 71.4 × 10⁻³ m; SF between 10.5 and 26.2 m⁻¹, including the data shown in (a–e). N = 12 flies; n = 43 000. (g) Phase lag for the same data as shown in (e). The dotted line shows a linear regression between f = 0.1–5 Hz. (h) R² value of sinusoidal fit for the data shown in (F).

Figure 4.

Amplitude/phase analysis. (a) Example of averaged lift force. Raw lift forces of a fly were averaged over 20 oscillations of the pattern (mean ± s.d.). (b) Analysis of fly's estimated free-flight position (P_est). Double integration of the force shown in (a) yields the fly's pseudo-position (± s.d.). This is compared to the pattern position to obtain the phase lag (Δφ). A_in and A_est represent the oscillation amplitudes of the pattern and fly's pseudo-position, respectively.

Figure 5.

Bode plots. (a) Amplitude gain G, calculated from the ratio of the positions of the fly (estimated from equation (3.4)) and the pattern. Data measured using various amplitudes and spatial frequencies (table 2) were pooled. f was varied between 0.1 and 13 Hz. Measurements (mean ± s.d.) are based on n = 12 flies and n = 43 000 oscillations. (b) Phase delay Δφ. The phase difference between the pattern position and pseudo-position is plotted for the same data referred to in (a).

Figure 6.

Superposition property. Fast Fourier transform (FFT) of lift responses of a singe fly (n=3 measurements) measured in presence of a pattern oscillating at a frequency f₁ =3 Hz, f₂ =3.5 Hz, as well as a pattern oscillating with a combination of the two frequencies (black trace). Small peaks in the FFT at 2.5 Hz and 4 Hz reveal a weak non-linearity.