Converting an allocentric goal into an egocentric steering signal

The yaw, pitch, and roll angles of the ball were sampled at 50 Hz, and aligned to our imaging data files using the ball camera's trigger signal. We shifted the acquired ball-position data backward in time by 30 ms due to our measured latency between the trigger pulse for acquiring a frame and when FicTrac finished processing the image. For behaviour only closed-loop wind experiments—which did not require aligning behavioural and neuronal data—no camera triggers were used and all signals were downsampled to 50 Hz.

For Figs. 1 and 5b and Extended Data Figs. 3, 4, 8 and 9g,j we used a 500-ms boxcar filter to smooth the forward walking velocity or turning velocity signal. For several analyses we excluded timepoints when the fly was standing still, or nearly still, which we defined as any moment when the fly's filtered forward walking velocity was ≤1 mm s⁻¹. The fly's virtual 2D trajectory was computed using the bar position, to estimate the fly's heading, alongside the sideward and forward ball rotations to estimate the fly's translational velocity. In Fig. 1, to visualize the relationship between neuronal phases and the fly's orientation over time, we plotted the position of the bar on the arena (instead of the fly's heading) since the EPG phase tracks the inverse of the fly's heading (which is equivalent to the bar position)⁴. In Fig. 2, we flipped the heading direction x-axis to make it easier to compare with Fig. 1.

To analyse the fly's menotaxis behaviour, we isolated straight segments (which we call menotaxis bouts) of the fly's 2D virtual trajectory using the Ramer–Douglas–Peucker algorithm^62,63 (Extended Data Fig. 2a–e). This algorithm simplifies a set of x, y coordinates by iteratively reducing the number of points in the trace. The parameter ε determines the maximum allowed distance between the simplified and original trajectories. We then computed the fly's displacement L for each segment of the simplified trajectory. For all analyses we used ε = 25 mm and only analysed segments with L > 200 mm. In other words, we analysed menotaxis bouts where the fly displaced itself more than 200 mm (roughly equivalent to 70 body lengths), without deviating from its course by more than 25 mm (roughly 8 body lengths). Aside from bar-jump (that is, virtual rotation) experiments (where we used the pre-jump heading angle as the fly's goal angle) and Extended Data Fig. 10a (see 'LAL imaging analysis'), we defined the fly's goal angle as the mean heading angle during each menotaxis bout. For this calculation, we excluded timepoints when the fly was standing still.

The values chosen for parameters ε and L were conservative, in that they tended to break up portions of the fly's trajectory where one might have considered the fly's goal to have remained unchanged into smaller bouts. We preferred this bias over the risk of potentially lumping two bouts together, where the fly's true goal angles might have been different.

To obtain a continuous estimate of the stability of the flies' heading angle (Extended Data Fig. 3o,p), we computed the flies' mean heading vector length (R) similarly as described previously¹¹. In brief, each heading sample point was treated as a unit vector. Each timepoint was then assigned a value of R by taking the mean of the heading vectors within a 30, 60 or 120 s window centred on that timepoint. For this calculation, timepoints in which flies were standing still were first omitted (since this would increase the value of R for trivial reasons); trajectories were concatenated across the omitted standing events, such that analysis windows were not necessarily analysing a continuous trace in time.

To correct for motion artefacts, we registered two-photon imaging frames using the CaImAn⁶⁴ Python package. We defined ROIs for the left and right side of the LAL, the glomeruli of the bridge and columns of the fan-shaped body using a custom graphical user interface written in Python. ROIs were manually drawn using either the time averaged signal or the local correlation image of each z slice. In the case of the fan-shaped body, we used a semi-automated method to define columns as described previously⁴. In brief, we first defined an ROI including the entire fan-shaped body. This ROI was then subdivided into 16 columns of equal angular size using two lines that defined the lateral edges of the fan-shaped body. For each ROI, we defined ΔF/F₀ as equal to (F − F₀)/F₀, where F is the mean pixel value of an ROI at a single timeframe and F₀ is the mean of the lowest 5% of F values.

We computed the FC2 phase in the fan-shaped body using a population vector average^2,4. We computed the EPG phase in the protocerebral bridge as described previously^4,34. For each timepoint, we treated the glomeruli ΔF/F₀ in the bridge as a vector of length 16 and took the Fourier transform of this vector. The phase of the Fourier spectrum at a period of 8.5 glomeruli was used as the EPG phase.

To overlay the FC2 or EPG phase with the bar position (Fig. 1 and Extended Data Fig. 3), we subtracted from the phase its mean offset from the bar position. This offset was calculated, for each recording, by taking the mean circular difference between the phase angle and bar angle, excluding timepoints when the bar was in the 90° gap at the back of the arena or when the fly was standing still. In Fig. 1m,n and Extended Data Fig. 3c,d, we nulled the FC2 or EPG phase in the baseline period by subtracting its mean position, 1 s prior to the bar jump, from every sample point. In Fig. 1o and Extended Data Fig. 3e, we calculated the mean of this adjusted phase during the last 1 s of the open-loop period after a bar jump. To combine +90° and –90° bar jumps for analysis, the mean phase in the last 1 s of the open-loop period was multiplied by −1 for −90° jumps.

In Fig. 1m–o, we imposed strict requirements for a bar jump trial to be included in the analysis. First, the bar jump needed to occur during a menotaxis bout (see 'Processing of menotaxis behavioural data'). Second, we required that the fly return to its previous heading angle following a bar jump —that is, trials when the mean bar position from 5 to 10 s after the start of the bar jump was within 30° from the mean bar position 5 s before the bar jump. Third, the bar needed to jump to a visible position on the arena (rather than the rear 90°, where we had no LED panels). Finally, we only included trials when the FC2 or EPG population vector average amplitude (PVA) (see Extended Data Fig. 3k for a description of how the PVA is computed) was greater than 0.3. These criteria were sensible, in that they selected for trials where we could be confident that the fly's goals had not drifted and that our neural signal estimates were of high quality. However, they were stringent enough that they led us to analyse only 7% of all trials. In Extended Data Fig. 3c–e, we eliminated the first two of these requirements, leading us to analyse 59% of all trials and the results were generally the same.

We used the corrcc function from the pycircstat python package (https://github.com/circstat/pycircstat↗) to compute the correlation between the EPG or FC2 phase and the bar position. For this calculation we excluded timepoints when the fly was standing still or when the bar was located in the 90° gap.

In Extended Data Fig. 3f,g, rapid changes in the FC2 phase position were detected by finding peaks in the filtered phase velocity (500-ms boxcar filter) using the SciPy⁶⁵ function signal.find_peaks. In addition, we required that the FC2 PVA within 1 s from the peak phase velocity was above 0.15 at all timepoints and that the mean PVA during this time was above 0.25. These criteria helped ensure that genuine changes in the FC2 bump position were detected rather than spurious changes in the FC2 phase due to a poorly estimated phase. To overlay all of the detected changes in FC2 phase position, as well as the flies' heading during these moments (Extended Data Fig. 3g), we aligned traces to the start of the peak in phase velocity. In order to combine traces where the peak in the FC2 phase velocity was either positive or negative, we flipped the FC2 phase for traces where the peak phase velocity was positive.

In Extended Data Fig. 3l–n,p,q, we used three different metrics to quantify the EPG or FC2 activity/bump at every timepoint: the population vector average amplitude (PVA), the mean ΔF/F₀ taken across all column or glomerulus ROIs and the maximum − minimum ΔF/F₀ (column or glomerulus ROI with maximum ΔF/F₀ minus column or glomerulus ROI with the minimum ΔF/F₀). In Extended Data Fig. 3q, for each fly, we binned data points based on the fly's forward walking velocity or turning velocity and computed the mean FC2 activity/bump metric for each bin. Likewise, in Extended Data Fig. 3p, we binned data points based on the fly's instantaneous mean heading vector length (see 'Processing of menotaxis behavioural data') and computed the mean FC2 activity bump metric for each bin. Timepoints in which flies were standing still (that is, when the mean forward walking speed was below 1 mm s⁻¹) were removed from the time series before performing this analysis because the fly's mean heading vector length is undefined during standing events.

To compare the effect of columnar stimulation of FC2 neurons across flies, we nulled the heading angle using the following procedure. For each fly, we computed its mean heading during a stimulation A trial, excluding timepoints when the fly was standing still. We then took the mean heading across all stimulation A trials and subtracted this value from the fly's heading angle in all trials. The histograms in Fig. 2c–f used 10° bins and also excluded timepoints when the fly was standing still. In some trials, flies were standing still throughout the entire trial (1.7% of all trials), which resulted in the trial being discarded for relevant analyses.

For Extended Data Fig. 4c, an ROI was considered inside the stimulation ROI if it had at least one pixel within the boundaries of the stimulation ROI scan path and was otherwise considered outside the stimulation ROI. In Extended Data Fig. 4d, we only analysed ROIs that were outside the stimulation ROI. The change in the column ROI ΔF/F₀ was computed by dividing the mean ΔF/F₀ during the 30 s stimulation period by the mean the ΔF/F₀ during the 5 s before the stimulation. To calculate an ROI's distance from the stimulation site (in number of ROIs), we first defined the stimulation site as the column ROI with the highest fraction of pixels inside the stimulation ROI. For each ROI we then computed its wrapped distance in number of ROIs. For instance, column ROI 2 and column ROI 15 have a (wrapped) distance of three, given that there are 16 columns in our analysis. Since our stimulation ROI could overlap with multiple column ROIs, in Extended Data Fig. 4d, there are no column ROIs with a distance of one.

In Extended Data Fig. 4e, to compute the stimulation location angle, we treated the fraction of pixels of each column ROI that were inside the stimulation ROI (see red colour map in Fig. 2c,d) as an array. Using this array, we computed the stimulation location angle with the same population vector average method used to compute the FC2 phase. We then took the mean difference between the two stimulation phases (A and B) for each fly. To compute the mean FC2 phase position during the stimulation period (Extended Data Fig. 4f–h), we excluded timepoints when the fly was standing still.

In Extended Data Fig. 4i–k, for each fly and stimulation location, we predicted the fly's goal heading by adding the angular difference between the two stimulation locations in the fan-shaped body (as described above) to the fly's mean heading direction during trials of the opposite stimulation location.

In Extended Data Fig. 4i, we grouped trials based on the fly's heading relative to the predicted goal heading 2 s before the stimulation onset. In Extended Data Fig. 4j, we instead grouped trials based on whether the fly was standing still prior the stimulation onset (defined as any trial where the fly's filtered forward walking velocity was below 1 mm s⁻¹ at all timepoints 5 s before the start of the stimulation).

To detect transient increases in LAL asymmetries (Fig. 5b,c), we first smoothed the right – left LAL ΔF/F₀ signal using a Gaussian filter (σ = 200 ms). We then detected peaks in the filtered signal using the SciPy function signal.find_peaks. Peaks were defined as timepoints where the filtered signal was above 0.1 ΔF/F₀ for at least 1 s, spaced from other peaks by at least 3 s, and had a prominence of one. To detect transient decreases in LAL asymmetries, we flipped the right – left LAL ΔF/F₀ signal and then applied the same algorithm. In Fig. 5c, we aligned the fly's turning velocity and the right – left LAL ΔF/F₀ signal to the timepoint of the peak neural signal and upsampled both the fly's turning velocity and the right – left LAL ΔF/F₀ to a common 100 Hz time base. In Extended Data Fig. 10a, we plot the exact same data as in Fig. 5c, but instead show the flies' heading relative to goal (rather than the rate of change of that signal or turning velocity) in reference to the neural peaks. For this analysis only, we defined the fly's goal angle as the mean heading angle in a 60 s window sliding window. We obtained similar results when only looking at peaks that occurred during a menotaxis bout, where we could define the goal in our more standard way.

To plot the LAL activity as a function of the fly' heading relative to its goal angle (Fig. 5d), we only analysed data during menotaxis bouts (see 'Processing of menotaxis behavioural data'). Because there is a ~200 ms delay between a change in the fly's heading and a change in the EPG phase³⁴, we expected the LAL ΔF/F₀ signal to be likewise delayed relative to behaviour. Therefore, in Fig. 5d only, we shifted the LAL ΔF/F₀ signal forward in time by ~218 ms (2 imaging volumes) prior to relating the signal to the fly's behaviour. We believe that this is the most appropriate signal to analyse, but our conclusions are the same if we do not apply this shift. For each fly, we calculated the mean LAL ΔF/F₀ by binning the data based on the fly's heading relative to its goal using 10° bins. Timepoints in which flies were standing still were removed from the time series prior to analysis.

To detect spikes, we first filtered the membrane voltage (V_m) trace with a Butterworth bandpass filter. We then detected peaks in the filtered V_m trace above a specified threshold, spaced by >5 ms, using the SciPy function signal.find_peaks. Although this criterion means we could not detect spike rates above 200 Hz, the activity levels of all our cells stayed well below this upper limit. Different cut-off frequencies and thresholds were hand selected for each cell so as to yield spike times that matched what one would expect from visual inspection of the data. To remove spikes from the V_m trace—for analyses of the membrane voltage in Fig. 3c,d and Extended Data Figs. 6d, 7c,d and 9b,c—we discarded V_m samples within 10 ms of a spike by converting those samples to empty entries (that is, the not-a-number (NaN) data type).

When analysing electrophysiological data in comparison to the fly's heading or goal angle (Fig. 3c–f and Extended Data Figs. 6–9), we downsampled the cell's V_m or spike rate to the ball camera frame rate (50 Hz) by either averaging the spike-rate or the spike-removed V_m in the time interval between two camera triggers. In Fig. 3b we plotted the spike rate using a 1-s boxcar filter.

To generate the tuning curves in Fig. 3c,d, we binned the electrophysiological time series data according to the fly's heading, using 15° bins. We then calculated the mean spike rate and the spike-removed V_m for each bin. To estimate a cell's preferred heading angle, we fit the spike-removed V_m tuning curve with a cosine function, with the offset, amplitude and phase of the cosine (the phase is the resulting preferred angle) as fitting parameters. In performing this fit, we excluded timepoints when the bar was located in the 90° gap at the back of the arena because the EPG system is expected to track the fly's heading less faithfully during these moments^2,34,39. We used V_m rather than spike rate for estimating the cell's preferred heading angle because V_m was much less modulated by the fly's goal angle than the spike rate (Extended Data Fig. 7), and thus it was less likely to lead to goal-modulation-related biases in our estimate of the preferred heading angle.

For Fig. 3e, f and Extended Data Figs. 7, 8, we only analysed data from timepoints that contributed to a menotaxis bout (see 'Processing of menotaxis behavioural data'). For each bout, we computed a relative goal angle by subtracting the cell's preferred heading angle from the fly's goal angle. Likewise, for each timepoint, we computed a relative heading by subtracting the cell's preferred heading angle from the fly's current heading angle. We then calculated the mean firing rate (or spike-removed V_m) binned by the fly's relative goal angle using 45° bins (columns in Fig. 3f) and by the fly's relative heading angle, also using 45° bins (x-axis in Fig. 3f). To generate tuning curves (except Extended Data Fig. 8), we removed timepoints when the fly was standing still for the time series, before analysis. By contrast, for Extended Data Fig. 8c–e we only included timepoints when the filtered forward walking velocity of the fly was between −0.5 mm s⁻¹ and 0.5 mm s⁻¹ and the fly's turning velocity was between –5° s⁻¹ and 5° s⁻¹ (that is, the fly was standing still and not turning in place rapidly).

The data contributing to the tuning curves in Fig. 3f were binned according to the heading and goal angles relative to the electrophysiologically preferred heading angle of the cell being studied, which was always made to equal zero. We refer to these relative heading and goal angles as H′ and G′ and we expressed the PFL3 activity in the single-cell model as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(\cos \left({H}^{{\prime} }\right)+d\cos ({G}^{{\prime} }-{G}_{\text{pref}}+{H}_{\text{pref}}))$$\end{document}f(cosH′+dcos(G′−Gpref+Hpref)), with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(x)=a\log (1+\exp (b(x+c)))$$\end{document}f(x)=alog(1+exp(b(x+c))). This form for f, which is called a softplus function, was suggested by examining the shifted spike-rate versus V_m curves in Extended Data Fig. 9c (see below). We then fit the parameters G_pref – H_pref, d, a, b and c by minimizing the squared difference between f and the data. The same value of G_pref – H_pref was used for each cell. The optimal parameters were G_pref – H_pref = −48°, d = 0.63, a = 29.23 Hz, b = 2.17, c = −0.7. The connectomic analysis discussed in the next section indicates that the difference between the preferred heading and goal angles, G_pref – H_pref, is expected to be −67.5°, on average. Several technical and biological reasons could account for the difference between the expected and fitted values. For example, a misestimation the cell's preferred heading direction (see 'Tuning curves') could cause the measured G_pref – H_pref to be smaller than its average anatomical value. In the full model, described in 'Full PFL3 model', we used the angles from the connectome analysis.

Fitting our model to the mean turning curves in Fig. 3f accounted for 95% of the variance in these data. In addition, we used our model, with the above parameter values, to predict the time series of the firing rates of individual PFL3 neurons during menotaxis over 20 ms intervals. The model accounted for ~30% of the variance in these unaveraged data. The relatively low amount of variance explained is unlikely the result of tuning to either forward or angular velocity because averaged data that depended only on heading explained 93% and 92% of the variance of the heading/forward velocity and heading/angular velocity data shown in Extended Data Fig. 8a,b. An analysis of spike count variability showed approximately Poisson spike-count variability, and this is a likely source of the extra variance in the unaveraged data.

For the fits shown in Extended Data Fig. 6d, the data were fit to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A\cos (H-{H}_{\text{pref}})+{V}_{0}$$\end{document}Acos(H−Hpref)+V0, with A, H_pref and V₀ as fitting parameters, by minimizing the squared difference between this expression and the data points.

Extended Data Fig. 9b shows the relationship between the spike-rate and V_m (spikes removed) obtained from our whole-cell recordings. To generate this plot, we used the data shown in Fig. 3 and Extended Data Fig. 7 (that is, we included timepoints when the fly was performing menotaxis and not standing still). We binned the data according to the fly's goal angle relative to the cell's preferred heading angle (using the same 45° bins as in Fig. 3 and Extended Data Fig. 7) and also according to each cell's V_m (4 mV bins). We used a cut-off of −46 mV, since at more depolarized membrane potentials spikes were not as well estimated and might have been missed. To include right PFL3 neurons in this analysis, we flipped the goal heading relative to the cells' preferred heading values of right PFL3 cells prior to averaging across all cells.

To generate Extended Data Fig. 9c, in which the curves from Extended Data Fig. 9b are aligned, we shifted the curves for different goal directions along the horizontal (V_m) axis by amounts determined to minimize the squared difference between the spike rates in each bin across the different goal directions and a common function of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(x)=\alpha \log (1+\exp (\beta ({V}_{{\rm{m}}}^{{\prime} }))$$\end{document}f(x)=αlog(1+exp(β(Vm′)), where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${V}_{{\rm{m}}}^{{\prime} }$$\end{document}Vm′ is the shifted membrane potential (black curve in Extended Data Fig. 9c). In other words, we computed the shifts that made the spike-rate curves for different goal directions maximally align. The resulting voltage shifts are plotted in Extended Data Fig. 9d. The parameters α and β of this fit are distinct from the parameters for the fits in Fig. 3f, and it is the parameters of the latter fit that are used to build the full model.

For the full population model, the response of each PFL3 cell is expressed as\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r=f(\cos (H-{H}_{\text{pref}})+d\cos (G-{G}_{\text{pref}}))$$\end{document}r=f(cos(H−Hpref)+dcos(G−Gpref))with the 12 left and 12 right PFL3 cells all modelled with the same function f and parameters d, a, b and c described in the section on fitting the PFL3 tuning curves. The values of the preferred angles, however, differ across the cells, and their values were obtained from the connectome^12,13 (Fig. 4b and Extended Data Fig. 5g). For the preferred goal angles, we used the values G_pref = −(15°, 45°, 75°, 105°, 135°, 165°, −165°, −135°, −105°, −75°, −45°, −15°) for both the left and right PFL3s. For the preferred heading angles, we began by assigning angles to the 18 glomeruli across both sides of the protocerebral bridge, from left to right: −22.5°, 22.5°, 67.5°, 112.5°, 157.5°, −157.5°, −112.5°, −67.5°, −22.5°, 22.5°, 67.5°, 112.5°, 157.5°, −157.5°, −112.5°, −67.5°, −22.5°, 22.5° (Extended Data Fig. 5e). These angles were projected down to the fan-shaped body using the wiring diagram shown in Fig. 4b. There are 18 bridge angles but only 14 of them are used for these projections because the left two outermost glomeruli and the right two outermost glomeruli (first and last two entries in the above list) are not innervated by PFL3 cells. Individual PFL3 cells innervate with their dendrites either one or two of the innermost 14 bridge glomeruli. For the PFL3 cells that innervate two bridge glomeruli, we used the angle corresponding to the innermost of the innervated pair (see Extended Data Fig. 5f,g). The resulting preferred heading angles for the PFL3 population are therefore as follows: for the right PFL3 cells (that is, the PFL3 cells projecting to the right LAL), H_pref = −(67.5°, 112.5°, 157.5°, 157.5°, −157.5°, −112.5°, −112.5°, −67.5°, −22.5°, −22.5°, 22.5°, 67.5°) and, for the left PFL3 cells (that is, the PFL3 cells projecting to the left LAL), H_pref = −(−67.5°, −22.5°, 22.5°, 22.5°, 67.5°, 112.5°, 112.5°, 157.5°, −157.5°, −157.5°, −112.5°, −67.5°). The overall minus sign in these two lists of angles (and in the expression above for the preferred goal direction angles) reflects the fact that angles extracted from the connectome, which are given inside the parentheses, are referenced to the ellipsoid body, whereas the preferred angles listed here are referenced to heading angles, and the EPG bump and heading angles differ by a minus sign. These preferred angles determine the directions of the vectors shown within the fan-shaped body compartments in Fig. 4b, with angles measured positive anticlockwise and the zero-angle pointing directly downward.

In the analysis described in the previous paragraph, we used glomerular angles implied by the Δ7 innervation of the protocerebral bridge (Extended Data Fig. 5e–g). Alternatively, we could have used glomerular angles based on the innervation of EPG neurons (Extended Data Fig. 5c). We opted to use the Δ7 scheme because Δ7 neurons provide the majority of PFL3 neurons' synaptic input in the protocerebral bridge^12,13 (Extended Data Fig. 5a).

We also assumed that the PFL3 cells form twelve functional columns in the fan-shaped body due to anatomical considerations (Extended Data Fig. 5k). PFL3 neurons can, alternatively, be viewed as forming nine columns¹². The model was also tested assuming nine columns (in this case, the preferred goal angles used were -(0°, 45°, 90°, 135°, 180°, −135°, −90°, −45°, 0°)), and qualitatively similar results were obtained. Note that in this 9-column angle assignment, the first and last columns represent the same angle, which would mean that the entire left/right extent of the fan-shaped body would encode more than 360º, a feature that we do not favour and which contributed to our using the 12-column model.

To simulate the effect of silencing subsets of PFL3 neurons (Extended Data Fig. 11h), we used the model described above, but set the response of randomly selected PFL3 cells to zero for all heading and goal angles. For each number of PFL3 cells silenced (from 0 to 24), we took the circular averaged error between flies' intended goal, G, and the zero heading from their PFL3 turning curve across 5,000 simulations. We added noise to the goal direction input of the model, with an amplitude chosen to make the model with no silenced neurons match the performance of PFL3 > TNT_inactive control flies.

To predict the PFL3 output signal in Extended Data Fig. 9g,h,j we used our full PFL3 model, as described above. As a goal input to this model, we used our FC2 imaging data during menotaxis. As a heading input to the model, we used a computer-generated (that is, synthetic) EPG and ∆7 heading signal (see below) for each timepoint; we could not use a measured heading input because we did not co-image EPG or ∆7 neurons during the relevant experiments. Before inputting the FC2 imaging data into the model, at every timepoint we first interpolated the ΔF/F₀ array of the 16 imaging ROIs to a 12-ROI array in order to match the 12 columns used in our model. We then normalized the interpolated ΔF/F₀ of each ROI independently such that each ROI's signal ranged from negative one (the minimum value observed in the ROI) to positive one (the maximum value observed in the ROI). The resulting activity in each column ROI was used in place of the term \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cos (G-{G}_{\text{pref}})$$\end{document}cos(G−Gpref) in the equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\left(\cos \left(H-{H}_{\text{pref}}\right)+d\cos \left(G-{G}_{\text{pref}}\right)\right)$$\end{document}fcosH−Hpref+dcosG−Gpref for each PFL3 neuron. To generate the synthetic EPG/∆7 heading signal, we had the phase of the synthetic bumps of activity in the bridge invariably track the angle of the bar on the arena. We time-shifted the phase of the synthetic EPG/∆7 signal forward by ~200 ms in relation to the bar's instantaneous position on the LED display. This latency was chosen so that the synthetic data accorded as closely as possible with past measurements on how the real EPG/∆7 phase lags changes in bar position³⁴ (Extended Data Fig. 3b). Recall that the EPG phase has a variable, fly to fly, offset to the bar position on the LED screen, which means that there will be an arbitrary offset between the FC2 phase in the brain and the expected bar position that a given fly stabilizes on the LED display. To account for this arbitrary offset, we added a fixed offset to the bar position so that its angular position and the FC2 phase aligned on average—which makes sense if one assumes that flies, on average, maintain a heading that is aligned with their goal angle. We used the inverse of this new, offset, bar position over time as the phase of the EPG/∆7 heading signal (that is, the fly's heading) or H, in the expression \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cos (H-{H}_{\text{pref}})$$\end{document}cos(H−Hpref), which determines the heading input into each PFL3 neuron in the model. We could then predict the difference between the left and right population-level PFL3 activity at every timepoint using the same function f and parameters d, a, b and c as in our single-cell model and full PFL3 model. In Extended Data Fig. 9j, we binned data points that fell within a menotaxis bout—excluding timepoints when the fly was standing still—by the fly's heading relative to goal angle and computed the mean predicted R–L signal and the fly's filtered turning velocity for each bin. For this analysis, we first shifted the predicted R–L signal by ~200 ms forward in time in time since our LAL imaging data indicates that this is the latency where the relationship between R–L activity and the fly's heading relative to goal angle is maximal (Extended Data Fig. 10a, see also 'LAL imaging analysis'). We also shifted the fly's turning velocity by ~200 ms earlier in time to account for the expected delay between the processing of internal, navigation-related, information to compute the fly's heading relative to goal error and the execution of a motor command¹¹.

In Fig. 6, for each trial, the allocentric wind direction was computed by taking the mean of the difference between the bar position and the spigot angle, at every timepoint, during the time period when the airflow was on. This value was not necessarily identical to the nominal allocentric wind direction set by our code because of inertial/mechanical latencies associated with the air-delivery spigot needing to physically rotate to deliver air from a new direction. The set point and trial-computed allocentric wind direction could differ by up to 13°.

To generate the histograms in Fig. 6e and Extended Data Fig. 11a, we computed the fly's heading relative to the wind direction by taking the difference between the fly's heading on the ball and the allocentric wind direction, at every timepoint, during either the 30-s period when the wind was on or during the 30-s period starting 5 s after the wind was turned off, referred to as the test period. In the test period, the allocentric wind direction experienced in the most recent wind-on period was used as the alignment point for the histogram. We did not include the 5 s after the wind was turned off since this time period includes a 2-s open-loop 180° bar jump and because flies do not instantly correct for this virtual rotation.

In Fig. 6f and Extended Data Fig. 11b,c, the absolute distance to wind was taken to be absolute value of the flies' heading relative to the wind direction, computed as described above.

To generate the plots in Fig. 6h, for each fly and wind-direction block, we computed the flies' mean heading direction during the test period of the block's second and third trial and plotted this value as a function of the same block's mean allocentric wind direction. The absolute difference between these two values yielded the fly's wind-direction error during the test period, which is plotted, averaged across all six directions, in Fig. 6j. To compute the fly's wind-direction error in in Fig. 6i, we applied the same analysis as described above, but in this case, we used the flies' mean heading direction during the wind period.

In Fig. 6k, for each wind-direction block, a fly was considered to have oriented along the correct direction if its wind-direction error during the test period was less than 30°. In other words, its mean heading direction during the test period needed to be within ±30° from the allocentric wind direction.

The flies' performance index (PI) was defined as the fraction of time that a fly spent oriented toward the 180° hemifield centred on the previously experienced allocentric wind direction minus the fraction of time that a fly spent oriented toward the opposite hemifield (Extended Data Fig. 11d).

For the above analyses (except when computing the allocentric wind direction), we first removed from the time series timepoints when the fly was standing still. Five out of the 331 flies in our dataset stood still during the entire test period of both the second and third trial of at least one of the wind-direction blocks. Because this would result in an undefined goal angle for one of the wind-direction blocks, we excluded these flies from all analyses. Of the remaining 326 flies, 5 flies stood still during either the entire second or the entire third trial of a wind-direction block, which resulted in these flies only having one trial analysed for that wind-direction block.

For Fig. 1o, to assess whether the FC2 phase changed during a bar jump, relative to its position immediately prior, we performed a V-test⁶⁶ (Rayleigh test for uniformity where the alternative hypothesis is a known mean angle μ) with μ = 0° (P = 6.65 × 10⁻⁴). To assess whether the EPG phase tracks the bar during a bar jump we performed a V-test with μ = 90° (P = 7.99 × 10⁻³). The same tests applied to Extended Data Fig. 3e yielded μ = 0° (P = 7.69 × 10⁻⁸) for the FC2 phase and μ = 90° (P = 2.49 × 10⁻⁵) for the EPG phase.

For Fig. 2g, to assess whether the difference in flies' mean heading direction for stimulation A and B was within the expected difference based on the stimulation locations in the fan-shaped body, we performed a V-test with μ equal to the angular difference between the two stimulation location angles (Extended Data Fig. 4e). For flies expressing CsChrimson in FC2 neurons, this was μ = −173.4° (P = 1.49 × 10⁻³). For control flies that did not express CsChrimson, this was μ = −164.9° (P = 0.93). The expected difference of both groups is not exactly the same since the stimulation ROIs are defined manually without knowledge of the column ROIs (which are only defined later during the imaging analysis).

For Fig. 5h, to assess whether flies expressing CsChrimson in PFL3 neurons showed a change in ipsilateral turning velocity relative to control flies only expressing jGCaMP7f, we performed a two-sided Welch's t-test (P = 1.93 × 10⁻⁵). To compare flies expressing CsChrimson in PFL1 neurons with control flies we used a two-sided Welch's t-test (P = 0.76).

For Fig. 6j, we performed a two-sided Mann–Whitney U-test to assess whether flies expressing TNT in neurons labelled by PFL3 line 1 (57C10-AD ∩ VT037220-DBD) had a lower error during the test period than control flies expressing TNT_inactive instead. This yielded P = 0.05 for our first experimental replicate (rep. 1) and P = 1.20 × 10⁻⁶ for our second experimental replicate (rep. 2). To combine these two P values, we used Fisher's method, which yielded P = 1.08 × 10⁻⁶. Likewise, in Extended Data Fig. 11e, we applied the same test to assess whether flies expressing shibire^ts in neurons labelled by PFL3 line 1 had a lower error during the test period than control flies in which shibire^ts was driven by an empty split driver line (P = 0.29) and whether flies expressing TNT in neurons labelled by PFL3 line 3 (27E08-AD ∩ VT037220-DBD) had a lower error during the test period than flies expressing TNT_inactive instead (P = 0.05). We also performed a two-sided Mann–Whitney U-test to assess whether PFL3 line 1>TNT flies (rep. 2) had a greater error during the wind-on period than TNT_inactive control flies (P = 3.15 × 10⁻³). In Extended Data Fig. 11g—in which we selected flies whose wind-direction error during the wind-on period was between 12° and 45°—we performed the same test to assess whether PFL3 line 1>TNT flies (rep. 2) had a lower error during the test period than TNT_inactive control flies (P = 6.47 × 10⁻⁴). For Fig. 6k and Extended Data Fig. 11f, we performed two-sided Mann–Whitney U-tests to assess whether flies with PFL3 cells targeted for silencing oriented along fewer correct goal directions during the test period than control flies: PFL3 line 1>TNT versus PFL3 line 1>TNT_inactive (rep. 1: P = 0.04, rep. 2: P = 5.25 × 10⁻⁷ and Fisher's method: P = 3.90 × 10⁻⁷), PFL3 line 3>TNT versus PFL3 line 3>TNT_inactive (P = 0.07) and PFL3 line 1>shibire^ts versus empty split>shibire^ts flies (P = 0.15).

All p-values are reported without correction for multiple comparisons.