How do hoverflies use their righting reflex?

The righting reflex in animals has been investigated in a long series of experiments on vertebrates and invertebrates. Etienne-Jules Marey was the first author to describe in detail how cats orient themselves as they fall (Marey, 1894). Hoverflies, which are among the main flying pollinators, are able to perform impressive aerobatic feats in complex environments (Wang, 2005; Wystrach and Graham, 2012; Mou et al., 2011); in particular, hovering. Flies can land on all surfaces, including plants and walls, and they can even settle upside down on the ceiling (Liu et al., 2019). Just after take-off, they have to adopt the normal right-side up position in order to be able to fly: this action is called the righting reflex.

A classical approach to studying the righting reflex consists of dropping an animal into free fall with its legs pointing upwards. The animal will tend to rotate its body along its body axis and land on its legs. Some well-known experiments have been performed along these lines on mammalian taxa (Magnus, 1924) such as cats (Kane and Scher, 1969) and rats (Pellis et al., 1991), and more recently on wingless insect species such as stick insects (Zeng et al., 2017), ants (Yanoviak et al., 2010) and aphids (Ribak et al., 2013; Meresman et al., 2014). However, to our knowledge, no free-fall righting experiments of this kind have ever been performed on flies with their legs pointing upwards.

Previous studies in wingless animals have shown that righting reflexes can involve two different mechanisms: either an inertial or an aerodynamic control system. During inertial righting processes, the falling animal orients segments of its body so as to create an instantaneous moment of inertia, which triggers a rotation of the body (Marsden and Ostrowski, 1998) by moving the torso (as in cats and rodents), the legs (as in other mammals, reptiles and amphibians) or the tail (as in lizards and kangaroo rats); whereas aerodynamic control occurs in wingless insects such as aphids, ants and stick insect larvae (see Jusufi et al., 2011). When these animals perform appendicular movements (i.e. with their legs), bilaterally asymmetric forces induce a rotation of the body around its longitudinal axis (Jusufi et al., 2011; Ribak et al., 2013; Zeng et al., 2017).

Fruit flies are able to produce torque from aerodynamic forces by generating asymmetric wing motion (Ristroph et al., 2010; Sane, 2003; Beatus et al., 2015; Ristroph et al., 2013). The present study focused on how the hoverfly controls its head and body-roll orientation, focusing in particular on the aerodynamic processes resulting in righting torque. Beatus et al. (2015) have established that fruit flies can control their body-roll angle by establishing a stroke-amplitude asymmetry. Fruit flies can also generate active roll damping (60×10³ deg s⁻¹; see Beatus et al., 2015) to stabilize their roll movements even when exposed to extreme speed perturbations within the duration of a single wingbeat, namely within 5 ms. As fruit flies and blowflies are endowed with an efficient gaze stabilization reflex, which can be used to compensate for thorax rotations and thus to stabilize the visual world across the retina (Hardcastle and Krapp, 2016; Taylor and Krapp, 2007; Hengstenberg et al., 1986; Egelhaaf, 2002), head/body movements can be expected to occur during the righting process.

Based on previous studies on aerial righting reflexes (Zeng et al., 2017), we developed a new experimental setup with which winged insects can be released from an upside-down position (see Fig. 1 and details in Materials and Methods). The present paper first describes the asymmetric wing movements triggering the body righting process, before addressing the kinematics of the hoverfly's righting reflex, focusing on the head, body and head/body angles. Results related to additional experiments (loaded halteres with glue, lighting from below, head–thorax glued) are presented and discussed. A dynamic model for the righting reflex involving both feedback and feedforward control systems is then described. This model is consistent with a previously published control-block diagram of the fly's performance involving a proportional control of the roll movements based on the roll speed measured by the halteres (Beatus et al., 2015). The halteres, which are two ancestral hindwings beating in anti-phase with the wings, are ‘gyroscopic’ sense organs which quickly measure the fly's rotational speed in space (Pringle and Gray, 1948; Nalbach, 1993; Hengstenberg, 1998; Dickinson and Muijres, 2016). We also describe an additional feedforward mechanism possibly controlling the neck motor system, which accounts for the time lag observed between the insect's head and body during the righting process. As suggested in previous studies and confirmed by an elegant recent model for haltere dynamics (Mohren et al., 2019), the halteres act as body angular rate sensors which are sensitive to both Coriolis and centrifugal forces. In the present model, the halteres are involved in two nested feedback loops controlling the roll rate and the roll angle and in the feedforward system, controlling the head orientation with respect to the thorax.

Episyrphus balteatus (De Geer 1776) pupae were purchased from Katz Biotech AG, Baruth, Germany. They were fed with pollen and honey ad libitum. To facilitate the manipulation, a drop of iron glue (composed of 45% beeswax, 45% rosin and 10% iron dust) was placed on the thorax. Small cylindrical magnets (Supermagnete, Gottmadingen, Germany; diameter 2 mm, length 1 mm, mass 23 mg) magnetized with the iron glue were used to make manipulation of the insects with ceramic clamps easier and gentler. Flies were released upside-down from a suction-based custom-built device. Its surface consisted of a downward-facing Fluon-coated plastic angled ridge, which minimized the occurrence of leg adhesion to the substrate. Suction was applied via a capillary glass tube (diameter ∼1 mm) introduced into a small hole in the middle of the plastic ridge. The device was attached to the top of the box, the horizontal position of which was monitored with a spirit level. The capillary tube was connected to a vacuum source, and the air pressure was valve controlled. In each trial, the insect to be tested was placed on a Fluon-coated carrier surface with its mesosternum (i.e. the mid-thoracic ventral surface located near the centre of mass) connected via suction to the opening of the capillary tube. Magnets were detached after transferring the insects to the device, before they were released. We always checked before each experiment whether the hoverflies equipped with the iron glue could fly in the breeding cages, in order to ensure that their flight ability was not affected by the glue. A total number of 13 falls involving 4 males and 6 females were recorded.

Aerial righting performances were recorded with a high-speed video camera (Phantom Miro M11O) at a rate of 1600 frames s⁻¹ (1280×800 pixels). The macro lens used (Nikon Micro-Nikkor AF-S DX Micro 40 mm f/2.8 G) gave a good compromise between the size of the fly, the resolution and the visual field (the fly-to-camera distance was about 20 cm). The hoverfly was placed upside-down with a custom-built device inspired by Zeng et al. (2017), and released by turning off the vacuum source. The experimental arena was covered with a white diffuser (PMMA WH02, 3 mm thick) and illuminated from above by a halogen light (Kaiser Studiolight H; 5.6×10⁻¹³ W m⁻²). An optic fibre ring light (Schott, KL 1500) connected to a halogen light source via an optic fibre (Xenophot HLX, OSRAM) was placed around the camera lens to enhance the quality and the brightness of the video. The camera was triggered automatically as soon as the insect entered its field of view (Fig. 1A).

As shown in Fig. 1D, the wing positions were recorded from two pairs of natural markers (at the wing junction points and wing tips). In the head, body and head/body study, the positions of the hoverfly's body were indicated by three pairs of markers placed at the two pairs of legs (front legs and hindlegs) and the wing junction points, and a single pair of markers placed at the top and bottom of the head was used to indicate the position of the head. All the trackers moving over a uniform background were recorded using the Tracker Video Analysis and Modeling Tool (^©2018 Douglas Brown; see Fig. 1B,D). The stroke amplitude of the left (φ_L) and right wings (φ_R) relative to the body's orientation were defined as the angle between the wing angle (wing junction and wing tip) and the body roll angle (at the junction between the two wings). The stroke plane angle could not be measured because of the orientation of the camera, but we were only interested here in the relative difference between the left and right wingbeat amplitudes. The head and body roll orientation relative to the vertical were estimated in this way during aerial righting (Fig. 1C). To check the body roll orientation values (where θ_TR denotes thorax roll), we calculated the mean position of the two pairs of legs and the orientation of the wing junction angle. The leg and wing vectors are orthogonal to the dorsoventral body orientation. To determine the head orientation (where θ_HR denotes head roll), the two head tracking points (top and bottom) were used (Fig. 1B,D). In the normal right-side up position (legs pointing downwards), the head and body were taken to have a 0 deg inclination, while in the upside-down starting position (legs pointing upwards), the head and body were taken to have a 180 deg inclination (Fig. 1A,B).

The head/body angle θ_HB was obtained as the difference between the absolute head roll angle and the absolute thorax roll angle: θ_HB=θ_HR−θ_TR. Cluster analyses of normalized wingbeat amplitude differences (see Fig. 2C) were performed with R (www.R-project.org) while the other analyses were performed with MATLAB (R2018a, MathWorks, Natick, MA, USA). The evolution with time of all the angles measured and the roll angle velocities were calculated by applying a Savitzky–Golay filter (order 2, window: 35). As the body always rights before the complete reorientation of the head (see Fig. 3), it was assumed here that the righting reflex started with the first wingbeat and ended when θ_HB was equal to zero (i.e. when the head was realigned with the body). We therefore focused on the hoverfly's body kinematics during this brief part of the fall.

In order to test the dynamic model for the righting reflex, we performed some additional experiments. In previous experiments (Nalbach, 1993; Nalbach and Hengstenberg, 1994), it has been shown that flies with the halteres removed are unable to stay aloft in flight or start their wingbeats. To confirm the role of the halteres, we loaded the halteres with a small drop of glue. We deposited a 0.2 mg drop of glue (50% beeswax, 50% rosin; equivalent to 1% of the mass of E. balteatus) onto the tip of each haltere under visual control using an eyepiece-less stereo microscope (Mantis Elite by Vision Engineering) with a magnification of 15. We also tested the hoverfly's response to a strong change in the lighting conditions by illuminating hoverflies from below with a white halogen light (Kaiser Studiolight H, irradiance of 1.76×10⁻¹¹ W m⁻²). Finally, we tested whether the righting reflex was driven by head rotation by gluing the head to the thorax, as detailed above for the halteres but with a 0.6 mg drop of glue. Body and head rotations were recorded using a high-speed video camera (Phantom VEO E310) at a rate of 3600 frames s⁻¹ (resolution: 1280×800 pixels). A total of 11 falls (from 1 male and 2 females) were recorded in the loaded halteres condition, 13 falls (from 3 males and 1 female) in the altered lighting conditions (illumination from below) and 6 falls (from 2 females) in the head–thorax glued condition. Mean additional mass estimates were obtained by loading a piece of paper with 10 drops of glue and calculating the extra mass as the total mass divided by 10.

In the loaded halteres condition, the falling flies could be viewed from the side. Projected distance calculations were therefore performed in order to estimate the head and body angles (see Fig. S1; Viollet and Zeil, 2013).

Statistical comparisons were performed with R (www.R-project.org) on the experimental variables between the normal condition and the two additional conditions – illumination from below and loaded halters – with a post hoc Kruskal–Dunn test and on one variable – time to onset of the righting reflex – between six samples of six trials of the normal condition and the head–thorax glued condition with a Wilcoxon test.

After being released upside-down, E. balteatus hoverflies are able to right themselves starting from the first wingbeat by generating a 180 deg roll rotation via an asymmetric stroke amplitude between the right and left wings, defined by the positional angles φ­_R and φ­_L, respectively, as shown in Fig. 2A. Two phases can be observed during the righting process: the first phase where, in the case of a clockwise rotation, φ­_L is larger than φ­_R, followed by the second phase, where φ­_R becomes larger than φ­_L (see Fig. 2A,B). The stroke amplitudes of the two wings recorded during trial 7 are shown in Fig. 2A. Fig. 2B shows the occurrence of a smooth transition between the peak-to-peak stroke amplitudes of the left and right wings versus each wingbeat period. It can therefore be concluded that the body rotation results from the difference in stroke amplitude. At the first wingbeat, active torque roll is triggered by the difference between the right and left wing stroke amplitudes, but after five wingbeats, the sign of the difference is reversed, resulting in an active counter-torque roll, stabilizing the fly's attitude on the roll axis. As shown in Fig. 2C, where the thorax angle (θ_TR) is plotted versus the normalized wingbeat amplitude difference, the presence of two clusters is clear. The reversal of the sign of the normalized wingbeat amplitude difference observed during the righting manoeuvre suggests that the torque roll movement is suddenly reversed during the righting process, in line with the occurrence of an aggressive manoeuvre.

Righting was therefore studied here from the first wingbeat to the end of the righting process (see Figs 3 and 4), defined as the moment when θ_HB (head/body angle) became equal to 0 deg. As can be seen from Fig. 3, the amplitude and duration of the righting process were fairly variable. It was observed in some cases that hoverflies kept on rotating after reaching 0 deg (i.e. −90 deg in trial 1; see Fig. 3), while in other cases, they stopped righting before reaching an angle of 0 deg (i.e. 25 deg in trial 12; see Fig. 3). Despite the disparities observed among the responses, the time elapsing before the first wingbeat (median value 23.75 ms), the time taken by the hoverflies to reorient themselves (median value 48.8 ms), the number of wingbeats (median value 6 wingbeats) and the distance travelled during the righting process (median value 39.6 mm) did not vary conspicuously (Fig. 5).

As observed in Fig. 3, a time lag occurred systematically between θ_TR (thorax roll angle) and θ_HR (head roll angle): the body initiated the righting response before the head. The maximum value of this time lag was determined by subtracting the time at the minimum value of the head roll angular speed (Ω_HR) from the time at the minimum value of the body roll angular speed (Ω_TR) (see Fig. 4). As shown in Figs 3 and 5E, the maximum time lag ranged from 9 ms to 28 ms.

To account for the time lag observed between θ_TR and θ_HR, it seemed worth analysing the dynamics of θ_HB. The time lag between θ_TR and θ_HR resulted from changes in θ_HB (Fig. 3). θ_TR rotated first, while θ_HR remained constant, resulting in an increase in θ_HB from a null value to a maximum value, and θ_HR then started to rotate and to catch up with θ_TR, resulting in a decrease in θ_HB from the maximum value down to null. Disparities were also observed between the 13 trials in the maximum value of θ_HB, which ranged from less than 50 deg (in trials 4 and 13) to more than 100 deg (in trials 7 and 11; see Fig. 3 and Fig. 5F).

Fast 180 deg roll rotations performed by hoverflies during the righting process resulted from angular speeds Ω_HR and Ω_TR as fast as −10×10³ deg s⁻¹ (see trial 2 in Fig. 4). Thanks to the low inertia (I_roll of 9.76e−12 kg m⁻²) and the fast dynamics (48.8 ms; see Fig. 5), a hoverfly can produce fast body angular speeds when required to achieve a complete righting process. The time elapsing between Ω_TR and Ω_HR is also clearly visible in Fig. 4: Ω_TR started to increase before Ω_HR. Two phases can be observed, depending on the value of Ω_HB. The hoverfly's righting states and phases are shown in Fig. 4, where it can be seen that the head rotated first anti-clockwise to compensate for the rotation of the thorax, and then clockwise in order to become realigned with the body.

Previous studies have shown that θ_HB is stabilized and compensates for the changes in θ_TR (Hengstenberg, 1993; Hardcastle and Krapp, 2016; Sandeman and Markl, 1980). The best way of compensating quickly for these changes is surely to measure them directly and to use this measurement to compensate for them. We adopted the hypothesis that the time elapsing between θ_TR and θ_HR might result from the gaze stabilization reflex.

For the three additional experiments shown in Fig. 6 and Figs S2–S4 (illumination from below, loaded halteres and head–thorax glued), hoverflies were always able to right themselves. When illuminated from below, hoverflies righted themselves in a similar short period of time (median value 61.11 ms) with a shorter time lag (median value 8.1 ms, significant difference) and a smaller θ_HB (median value 44.19 deg, significant difference) than for the illumination from above condition. When halteres were loaded with glue, flies righted themselves faster (median value 29.72 ms, significant difference) with a smaller time lag (median value 6.1 ms, extremely significant difference) and a similar θ_HB (median value 58.21 deg). When the head and thorax were glued together, no significant differences were found in terms of duration of the righting reflex (median value 44.72 ms).

In line with previous studies carried out at our laboratory (Goulard et al., 2015, 2016, 2018a,b), hoverflies were dropped under free-fall conditions, upside-down with their legs touching the top of the box, in order to study aerial righting in flies for the first time. As expected, hoverflies in the upside-down position were found to trigger their wingbeats and to rotate quickly in order to regain the right-side up position within a short lapse of time (48.8 ms; see Fig. 5C). These rotational manoeuvres involved asymmetric wing strokes (active torque; see Fig. 2), resulting in fast thorax roll velocities of the order of several thousands of degrees per second (range: −2×10³ deg s⁻¹ to −12×10³ deg s⁻¹; see Fig. 4).

In our set-up, hoverflies can probably use two main types of cue when they are held upside-down: the dorsal light response (DLR) and proprioceptive cues. The DLR is a reflex that several families of insects use to determine their orientation based on the fact that the brightest part of the environment is presumably located above them (Mittelstaedt, 1950; Hengstenberg, 1993; Goulard et al., 2015, 2018a; Meyer and Bullock, 1977; Schuppe and Hengstenberg, 1993). In a previous experiment, Goulard et al. (2018a) showed that lighting from below drastically affects hoverfly stabilization during free fall, which proves that hoverflies are highly sensitive to light coming from the ground. In the present case, the halogen light projected from below would provide the hoverfly (upside-down) with a vertical reference frame oriented in the appropriate position. Therefore, if the fly's righting process depended only on the DLR, we would observe the occurrence of no righting in this condition, which was never the case. In addition, we observed only a small effect of the lighting conditions on the head movement during the righting process (see Fig. 6B,C, which shows a shorter time lag and a smaller θ_HB), suggesting that the DLR might be involved in the feedforward control of the head as a modulation of its gain (see F_NM(s) box in Fig. 7A). Further experiments are required to better understand the contribution of the DLR to the righting reflex.

Proprioceptive cues which are sensed by the chordotonal organs in the insect's legs (Tuthill and Wilson, 2016). The proprioceptive processes involved in insects' postural reflexes are stimulated by the weight of the legs (Horn and Lang, 1978; Kress and Egelhaaf, 2012; Horn, 1982). Force and load signals act as orientation cues (walking; Büschges et al., 2008; Duysens et al., 2000). Hoverflies are therefore able to quickly estimate the righting posture required to respond to loss of contact between their legs and the substrate (via the tarsal reflex) (see Binns, 1977; Fraenkel, 1932; Pringle, 1938; Dudley, 2002). As suggested in the model presented in Fig. 7, DLR combined with leg proprioception may be used by flies to estimate the angle of the surface on which they are standing, and would therefore enable them to set the value of the input reference signal Ref before taking off from a tilted surface. The input reference signal is an input signal corresponding to the reference value that the closed-loop system tries to reach in steady state. In conclusion, DLR and proprioception are likely to conflict when the lighting originates from below, but further experiments are now required to be able to clearly define their respective contributions to the righting reflex.

In fruit flies, roll torque results from asymmetric wing movements (Ristroph et al., 2010, 2013; Sane, 2003; Beatus et al., 2015), which leads to a difference in the lift generated by each wing. The results in the normal condition as well as in the head–thorax glued condition obtained here tend to confirm that the aerial righting performance of E. balteatus is purely aerodynamic, i.e. hoverflies produce asymmetric flapping movements in order to trigger a torque roll and right themselves (see Fig. 2). As shown in Figs 2 and 7B (sign of the simulated torque roll signal), two phases exist: the first positive and the second negative, resulting in the righting process. The active force used to start the rotation of the body (phase 1) seems to be larger than the force used to stabilize it at the end of the righting (phase 2). Fig. 2C clearly shows two clusters, suggesting that a passive torque roll (Hedrick et al., 2009) related to the friction of the wings also contributes to the stabilization of the fly, leading to this asymmetry in the wingbeat amplitude observed during a complete righting process.

It is worth noting that we have never observed any righting manoeuvres involving body pitch rotations under normal conditions. Body pitch depends on the fly's ability to shift the centre of mass forwards (or backwards) in order to trigger a nose-up (or nose-down) pitch torque movement (Ristroph et al., 2013). Controlling the wingbeat amplitude differentially is probably much more efficient and direct than using each wingbeat's duty cycle – the ratio between the durations of the upper (upstroke ϕ_Up) and lower (downstroke ϕ_Down) parts of a wingbeat (see Fig. 2A). As explained in the Introduction, vertebrates are known to move segments of their body, legs (reptiles) or tail (lizards) in certain configurations in order to control the instantaneous moment of inertia and the angular momentum. However, most hoverflies are much smaller than vertebrates and they are exposed to more viscous aerodynamic forces (see Jusufi et al., 2011). Aerodynamic interactions are therefore likely to predominate over inertial effects. We never observed the occurrence of any righting movements without any wingbeats being triggered and the head–thorax glued condition showed that righting is not coupled to any head movement. We therefore modelled the roll dynamics in terms of a purely second-order system (a double integrator) receiving torque U_roll as its inputs and delivering, via the moment of inertia I_roll, outputs specifying the appropriate thorax roll speed Ω_TR and thorax roll angle θ_TR. No evidence showing the presence of a neural integrator serving to estimate the roll has been obtained so far, although the existence of this component is biologically plausible as the temporal drift inherent to any vibrating rate gyro such as the halteres (Acar and Shkel, 2009) is likely to be very small during the short time required to perform righting manoeuvres. Once the righting has been accomplished, hoverflies can count on vision in addition to the thorax roll speed to estimate their body roll and compensate for any drift (see the unbiased rate gyro method based on an inertial measurement unit in Wu et al., 2015).

Accurate control of the body attitude (roll) during the righting process is also necessary to be able to reach the steady-state 0 deg position (right-side up) reliably. Flies may use visual motion cues (Meresman et al., 2014; Yanoviak et al., 2010) combined with inertial measurements provided by the halteres (Dickinson et al., 1999; Fraenkel and Pringle, 1938). During flight, the halteres are known to oscillate up and down in antiphase with the wings (see Nalbach, 1993) and are therefore highly receptive to the state of wing activation (see Dickerson et al., 2019; Parween and Pratap, 2015; Deora et al., 2015, 2017; Pratt et al., 2017).

However, because of the extremely short duration of the righting reflex (∼49 ms) and the relatively long processing time required by the motion-processing neurons (about ∼50 ms in the fruit fly; see Warzecha and Egelhaaf, 2000; Frye, 2009), the righting reflex can be assumed to depend only on the angular body speed measurements provided by the halteres. The halteres are known to show fast response times of only ∼5 ms (Pringle and Gray, 1948; Sandeman and Markl, 1980; Taylor and Krapp, 2007; Dickinson and Muijres, 2016; Sherman and Dickinson, 2004, 2003; Liu et al., 2016), and the righting response times recorded here were consistent with those of the halteres. A self-stabilizing reflex has been studied in fruit flies performing cruising flights (Ristroph et al., 2010, 2013; Beatus et al., 2015). Upon undergoing an external disturbance, the fly adjusts its body orientation in order to recover its initial right-side up position within 30 ms. The roll-induced perturbations applied in the later studies amounted to about 45 deg. In addition, the halteres are known to be involved in insects' detection of fast perturbations (Ristroph et al., 2010; Dickinson et al., 1999) and in fruit flies' mediated equilibrium reflexes (Liu et al., 2016; Nalbach, 1993, 1994; Nalbach and Hengstenberg, 1994; Dickinson et al., 1999; Fox and Daniel, 2008; Huston and Krapp, 2009; Frye, 2009).

Hoverflies are able to rotate their head around the roll axis with respect to their body up to a median angle of 77.7 deg (see Fig. 5F). During the righting phase, we observed a median time lag of 16.3 ms (see Fig. 5E) between the head and body rotations. The body starts to rotate first, and then the head rotates in the direction imposed by the body. Four possibilities can be envisaged for modelling this time lag. (1) As suggested by the results of a previous study on wasps (Viollet and Zeil, 2013), head orientation may be controlled by a feed-forward signal, while a copy of the command signals sent to the wing motoneurons is also sent with the opposite sign to the head position (neck) servo system. If a simple time lag is inserted into the pathway controlling the head orientation, one might wonder what the effects of this time lag might be, and how biologically relevant it is. Adding a time lag would definitely alter the performance of the gaze stabilization reflex, which plays a crucial role in blowflies (for review, see Hardcastle and Krapp, 2016). (2) As suggested by previous studies on gaze stabilization (see Hardcastle and Krapp, 2016; Mittelstaedt, 1950), without any need for halteres, passive inertial stabilization may serve to compensate for body rotations. Assuming the neck to act like a simple spiral spring, the head would first stay behind and then be pulled in the direction of the rotation imposed by the body. However, the difference in mass between the body (21 mg) and head (5 mg), corresponding to a ratio of 4.2 between the head and body moments of inertia (assuming the body to be a cylinder and the head a sphere) is not consistent with the idea that a purely passive process of stabilization might occur during the righting process. In addition, Gilbert and Bauer (1998) established that in the flesh fly, head posture results from both a passive and an active control of the head based on the prosternal organs. It would certainly be of interest to investigate hoverfly head dynamics more closely in future studies. Here, we modelled the neck motor system in the form of a first-order filter [a F_NM(s) transfer function] to account for the head's moment of inertia and the friction of the neck muscles. Values of F_NM(s) were obtained by fitting the model's response to the 13 trials (see Fig. 4). (3) As shown in Fig. 8B (and also in the model in Fig. 7A), we assumed that θ_HB is negatively proportional to Ω_TR (see block −1 in the model in Fig. 7A): the head is driven by the body roll speed. The head therefore changes its rotational direction when the body roll speed decreases. Fig. 8B and Fig. 9 show that the head angle is highly correlated with the peak in the roll angular speed and not correlated with any particular value of the head/body angle. The role of a switch control would be very unclear in terms of its functionality, whereas a direct control of the head based on the body's angular speed makes sense in terms of the gaze stabilization reflex.

(4) As suggested by the authors of previous studies on fast haltere-induced compensatory head movements (Hengstenberg, 1993; Hardcastle and Krapp, 2016; Sandeman and Markl, 1980), the time lag observed here between the head and body responses may result from the gaze stabilization reflex being activated at the very beginning of the wingbeats. As described in our model (see Fig. 7A), the feedforward control of the head depends on the body's angular speed transmitted by the halteres. As in the simulated response (see Fig. 8A,B), the orientation of the head with respect to the body θ_HB is proportional to the body's angular speed Ω_TR. The head therefore adopts the opposite response to that of the body. In the first phase, the head compensates for the body rotation while the body speed is increasing, and in the second phase, the head rotates as dictated by the body roll. As indicated by the dashed line in Fig. 8B, the direction of the head rotation is reversed when the body's angular speed starts to decrease. On the basis of the present model, the existence of a feedforward head control based on the thorax's angular speed suggests that the angular position of the head with respect to the body θ_HB must be strongly correlated with the thorax's angular speed Ω_TR. We therefore analysed the cross-correlation between these two normalized signals (see Fig. 9A). The cross-correlation function applied to the 13 trials showed the existence of a median time lag of 7.5 ms (see Fig. 9D), with a standard deviation as small as 2.5 ms, which suggests the existence of a strong correlation between thorax speed and head orientation, as expected. In addition, the simulated time lag observed between the head and body upon combining a physically plausible closed-loop control of the body roll with a feedforward control of the head orientation accounts accurately for the time lag observed in hoverflies (see Fig. 8). It is worth noting that the two phases in the simulated head/body movement would require a seamless feedforward control of the head orientation based on the body's angular speed transmitted by the halteres.

As discussed by Taylor and Krapp (2007) and recently by Hardcastle and Krapp (2016), gaze stabilization is crucial to steady flight because it reduces motion blur. It would therefore not be surprising if the gaze stabilization reflex was activated at a very early stage during the righting manoeuvres once the wingbeats have been triggered and the halteres are therefore vibrating as well. It is worth noting that gaze stabilization is not detrimental to the fly's ability to start rotating within a very short time. A righting response lasting for about 50 ms might suffice to enable the gaze stabilization reflex to be fully operational during the subsequent cruise flight.

We are most grateful to Julien Diperi for his contribution to building the experimental set-up and to Marc Boyron for developing the electronics: all the research presented in this paper was based on their work. We also want to thank Jessica Blanc for correcting and improving the English in the manuscript.