Building an allocentric travelling direction signal via vector computation

Abstract

Many behavioural tasks require the manipulation of mathematical vectors, but, outside of computational models^1-7, it is not known how brains perform vector operations. Here we show how the Drosophila central complex, a region implicated in goal-directed navigation^7-10, performs vector arithmetic. First, we describe a neural signal in the fan-shaped body that explicitly tracks the allocentric travelling angle of a fly, that is, the travelling angle in reference to external cues. Past work has identified neurons in Drosophila^8,11-13 and mammals¹⁴ that track the heading angle of an animal referenced to external cues (for example, head direction cells), but this new signal illuminates how the sense of space is properly updated when travelling and heading angles differ (for example, when walking sideways). We then characterize a neuronal circuit that performs an egocentric-to-allocentric (that is, body-centred to world-centred) coordinate transformation and vector addition to compute the allocentric travelling direction. This circuit operates by mapping two-dimensional vectors onto sinusoidal patterns of activity across distinct neuronal populations, with the amplitude of the sinusoid representing the length of the vector and its phase representing the angle of the vector. The principles of this circuit may generalize to other brains and to domains beyond navigation where vector operations or reference-frame transformations are required.

Extended Data Figure 1 |. Characterizing the anatomy and physiology of hΔB cells, showing that sytGCaMP and RGECO1a yield similar EPG phase estimates in the ellipsoid body, quantifying the EPG phase tracking of the closed loop dot, and evidence that the hΔB phase tracks the fly’s traveling direction in walking flies.

a, At least sixteen somas are labeled by the hΔB split-Gal4 line used in this paper. By comparison, the hemibrain connectome (v1.1) reports nineteen hΔB cells. b, GFP expression of the hΔB split Gal4 in the fan-shaped body. c, Same as panel b, but not showing the anti-nc82 neuropil stain. d, hΔB cells from hemibrain connectome v1.1¹⁷. e, Top, hΔB GCaMP7f signal in a tethered, flying fly experiencing optic-flow (in the time window bracketed by the vertical dashed lines) with foci of expansion that simulate the following directions of travel: 180° (backward), −120°, −60°, 0° (forward), 60°, 120°. Bottom, Phase-nulled and averaged hΔB activity patterns in the fan-shaped body, calculated from the above [Ca²⁺] signals in the last 2.5 s of optic flow presentation. Population means with s.e.m. are shown. f, Same as panel e, but with hΔB sytGCaMP7f signal. Note that the single-bump structure in the sytGCaMP7f signal is clear than the structure in the cytoplasmic GCaMP7f signal, which is consistent with sytGCaMP7f biasing GCaMP to axonal compartments of hΔBs. g, Probability distributions of the difference between the EPG phase and the bright dot’s angular position, without and with optic flow. h, Circular standard deviation of the EPG phase – dot position distributions, without and with optic flow. Two-tailed unpaired t-test was performed. i, Correlations between the angular velocities of the EPG phase and the visual landmark position under different conditions. The first two columns use the same data as in panels g and h. The third and fourth columns use data from simultaneous GCaMP7f imaging of EPGs and PFRs in tethered-flying flies with a closed-loop dot. The fifth column use data from GCaMP6m imaging of EPGs in tethered-walking flies with a closed-loop bar. Two-tailed one sample t-tests were performed against zero. P values are 1.7e-3, 1.3e-4, 5.3e-4, 1.2e-5 and 3.6e-4 comparing each column (from left to right) to zero, respectively. The relatively low, but significantly different from zero, r values show that the EPG phase tracks, even if poorly, the rotation of the landmark. The EPG phase measured in walking experiments tracks the closed-loop stimulus better than in tethered flight. See Main Text for possible technical reasons for why one would observe this difference. The fact that EPG-phase tracking of the closed loop dot is better when we co-imaged EPGs and PFRs compared to when we imaged EPGs and hΔBs argues that the flies’ genetic background (and thus how reliably flies perform tethered flight) can also quantitatively impact these measures. j, Angular-velocity correlations of the EPG phase and the visual landmark position under different conditions as a function of the time-lag between the two velocity signals. Same data as in panel i, but data with and without optic flow are lumped together. Correlation is highest at 290 ms, 260 ms and 375 ms for the three panels from left to right, respectively. Thus, we used time lags of 275 ms (mean of 290 and 260) and 375 ms for calculating the correlations in flight and walking experiments in panel i, respectively. k, Probability distribution of the angular position of the dot on the arena. Same data as in panels g and h, but data with and without optic flow are lumped together. We tested the uniformity of the distribution across angles using reduced χ2 test. P value is > 0.995, meaning that we cannot reject the hypothesis that the dot position is not evenly distributed on the arena. l, Circular standard deviation of the EPG phase minus the hΔB phase distributions, without and with optic flow. Same data as in panels g and h. Two-tailed unpaired t-test was performed. P value equals 1.3e-6. m, Correlations between the EPG phase and the hΔB phase. Same data as in panels g, h and l. Two-tailed unpaired t-test was performed. P value equals 3.9e-4. n, Data collected from tethered flies walking on a floating ball in complete darkness are shown in this panel and all subsequent panels in this figure. Sample time series of simultaneously imaged EPG and hΔB Gal4 lines. Top two traces show [Ca²⁺] signals. Third trace shows the phase estimates of the two bumps. Bottom two traces show the forward velocity and sideslip velocity of the fly. Quasi-unidirectional walking bouts are labeled with walking directions indicated. o, Probability distribution of the difference between EPG phase and hΔB phase from time segments where flies were walking in three different general directions (Methods). p, EPG – hΔB phase as a function of the egocentric traveling direction. Gray: individual fly circular means. Black: population circular mean and s.e.m. The sign of EPG – hΔB phase deviations seen here, in walking, are consistent with the signs observed in flight, for the same directions of backward-left and backward-right travel. Watson-Williams multi-sample tests were performed. P values are 1.6e-3 and 2.6e-6 comparing the 1^st and 3^rd columns (from left to right) to the 2^nd column, respectively.

Extended Data Figure 2 |. PFN_d and PFN_v activity bumps in the bridge are phase aligned with the EPG heading signal.

a, Sample trace, in tethered flight without optic flow, of simultaneously imaged GCaMP6m in EPGs and jRGECO1a in PFN_ds reveals that the activity bumps of these two cell classes are phase aligned in the bridge. b, Probability distribution of the EPG - PFN_d phase in tethered flight without optic flow. In this panel and throughout, the single fly data are in light gray and the population mean is in black. c-d, Same as panel a-b, but for GCaMP6m in EPGs and jRGECO1a in PFN_vs. e, Top three rows, sample trace of simultaneously imaged GCaMP6m in EPGs and jRGECO1a in PFN_vs in a tethered, flying fly experiencing optic flow (in the time window bracketed by the vertical dashed lines) with foci of expansion that simulate the following directions of travel: −120°, 0° (forward), 120°. Bottom, circular-mean phase difference between EPGs and PFN_vs. f, Probability distribution of the EPG - PFN_vs phase under three optic flow conditions. g, Circular mean of the EPG – PFN phase ad s.e.m. under different visual stimulus conditions. Watson-Williams multi-sample tests, P>0.66 when comparing any experimental group with 0°. Note that we only collected a full EPG-PFN, dual-imaging data set with optic flow (moving dots) with PFN_vs because, for reasons that are not fully clear, the jRGECO1a signal was too weak in PFN_ds to properly estimate the PFN_d phase outside of the context of stationary dots (i.e., during optic flow). When imaging PFN_ds with a split-Gal4 driver and with GCaMP rather than with jRGECO1a (e.g., Fig. 3j-l), the signal is much brighter.

Extended Data Figure 3 |. Δ7 cells are poised to help create sinusoidally shaped activity bumps in PFN_ds, PFN_vs and PFRs in the protocerebral bridge.

Connectivity data are based on those in neuPrint, hemibrain:v1.1. a, Two Δ7 cells from neuPrint reveal a graded increase and decrease in dendritic density across the bridge. b, Synapse-number matrix for detected synapses from EPG cells to Δ7 cells in the protocerebral bridge. Each row represents one Δ7 cell. c, Same data as in panel b, but plotting each Δ7 cell separately. d, Phase-nulled EPG-to-Δ7 synapse # across the glomeruli of the bridge, averaged across all 42 Δ7 cells, based on the data in panel c. The anatomical input strength from EPGs to Δ7s is sinusoidally modulated across the bridge. e, Transforming the EPG activity pattern across the bridge (blue) into a predicted Δ7 activity pattern (green, bottom row) based on the synaptic density profile in panel c (schematized in the middle). We first calculated the dot product between the EPG activity vector and each Δ7 cell’s EPG-to-Δ7 synapse-number vector (panel c). Then, for each glomerulus, we averaged the dot-product-output for all of the Δ7s that have axonal terminals in that glomerulus, thus creating the predicted activity value for that glomerulus. (The size of each green square here schematizes the # of synapses from EPGs to the Δ7 cell of that type in that column; the intensity of each Δ7 row indicates the expected output strength of each Δ7 cell type, after being driven by the EPG signal above.) We plot the inverted, predicted activity output from Δ7s in the bottom row (green) because Δ7s are glutamatergic and glutamatergic neurons in the Drosophila central nervous system typically inhibit their postsynaptic targets (via Glu-Cl channels). After inverting the Δ7 activity one can then imagine simply averaging the Δ7 predicted-activity row with the EPG activity––with some relative weighting for the Δ7 and EPG curves––to generate the net drive to the many downstream neurons that receive both EPG and Δ7 input, like PFNs. Note that the EPG activity bumps are slightly narrower than the sinusoidal fits whereas the Δ7 activity bumps are slightly wider than the sinusoidal fits. f, Same as panel e, but using the phase-nulled, averaged EPG GCaMP activity pattern from a previous study. Note although the EPG bump is narrower in these data from walking flies than in panel e from flying flies, the shape of the predicted Δ7 output remains similar. g, Same as panel e, but starting with (imagined) EPG activity where there is only one active glomerulus on each side of the bridge. Note that the shape of the predicted Δ7 output remains similar to that in panels e-f. h, Measured, phase-nulled activity profiles from PFN_ds, PFN_vs and PFRs. Thin lines: individual flies. Thick lines: population average. All three activity patterns conform well to their sinusoidal fits (gray dashed lines) (see Methods for goodness of fit). We hypothesize that the sinusoidal activity patterns in bridge columnar cells like PFN_ds, PFN_vs, PFRs arises from the combined impact of EPG and Δ7 input. In other words, we posit that Δ7s “sinusoidalize” the EPG bumps in the bridge –– that is, they function to broaden and smoothen the EPG input to the bridge, to create two sinusoidally shaped bumps in their recipient cells, with these bumps often functioning as explicit, 2D vector signals in the fan-shaped body.

Extended Data Figure 4 |. LNO1 and SpsP cells have [Ca²⁺] responses that are strongly tuned to the fly’s egocentric translation direction––in both walking and flying flies––with responses suggesting that these cells provide sign-inverting input to PFN_v and PFN_d cells, respectively.

Connectivity data and cell-type names are based on those in neuPrint, hemibrain:v1.1. a, LNO1s are a class of cells (two total neurons per side, four per brain) that receive extensive synaptic input outside the central complex and provide extensive synaptic input to PFN_vs in the noduli, with each PFN_v cell on average receive 131 synapses from LNO1s. b, Mean GCaMP signals in PFN_vs and LNO1s in the nodulus as a function of the simulated traveling direction of the fly (via open-loop optic flow). Dotted rectangle indicates a repeated-data column, in this panel and throughout. c, Single-fly (colored circles) and population means ± s.e.m. (black bars) of the average signal in the final 2.5 s of the optic flow epoch. Sinusoidal fits shown in this panel (Methods), and throughout. d, Each SpsP cell (two total neurons per side, four per brain) receives extensive synaptic input outside the central complex and provides extensive synaptic input to PFN_ds on one side of the protocerebral bridge, with each PFN_d cell on average receive 56 synapses from SpsPs. e, Same as panel b, but mean GCaMP signals in PFN_ds and SpsPs in the bridge as a function of the simulated traveling direction of the fly (via open-loop optic flow). A closed-loop bright dot was not present on the LED display when collecting the PFN_d data. f, Same as panel c, but averaging the bridge signal in panel e. g, Same as panel b, but analyzing the PFN_d signal in the noduli. A closed-loop bright dot was not present on the LED display. h, Same as panel c, but averaging the nodulus signal in panel g. i, The optic-flow-simulated egocentric traveling angle at which the activity of each cell type is strongest is depicted with a line at the associated angle. Note that the left-vs.-right angular differences measured in the noduli are smaller, and closer to 90°, than the left-vs.-right angular differences measured in the bridge. This difference might be a purposeful shift in optic-flow tuning related to the use of orthogonal and non-orthogonal PFN axes under different behavioral contexts (see Supplemental Text) and/or originate from differences in how SpsPs in the bridge and LNO1s in the noduli balance optic-flow with proprioceptive/efference-copy inputs to generate their signals. j, Data collected from tethered flies walking on a floating ball in complete darkness are shown in this panel and all subsequent panels in this figure. Mean PFN_v GCaMP signals in the bridge as a function of the fly’s forward speed. k, Right-minus-left PFN_v GCaMP signals in the bridge as a function of the fly’s sideslip speed. l-m, Same as panel j and k, but analyzing LNO1 signals in the nodulus. n-o, Same as panel j and k, but analyzing PFN_d signals in the bridge. p-q, Same as panel j and k, but analyzing SpsP signals in the bridge. In panel b, e, g, j-q, thin lines represent single-fly means and thick lines represent population means. Note that PFN_vs and LNO1s have sign-inverted responses, and that PFN_ds and SpsPs have sign-inverted responses. The response signs to optic-flow simulating the fly’s body translating forward and leftward (rightward) in flight are the same as the signs of responses to the fly walking forward and side-slipping leftward (rightward) when walking. Thus, these data are consistent with all these neurons being sensitive to the fly’s egocentric translation direction, as assessed via optic flow (dominantly) in flight, and via proprioception or efference-copy (dominantly) in walking.

Extended Data Figure 5 |. Multiple, functionally relevant ways of indexing angles across the protocerebral bridge.

Connectivity data and cell-type names are based on those in neuPrint, hemibrain:v1.1. a, The previously described mapping between EPG dendritic locations in the ellipsoid body and axonal-terminal locations in the bridge. Numbers ordered based on the location of each EPG cell in the ellipsoid body. b, EPG cells divide the ellipsoid body into 16 wedges, each 22.5° wide. Each glomerulus in the bridge inherits its angle, in our analysis here, based on the EPG projection pattern shown in panel a. The angles of the outer two bridge glomeruli––which do not receive standard EPG input, but only EPGt input¹⁷––were inferred to have angles equal to the middle two glomeruli (0° and 22.5°, respectively) based on how other cell types (e.g., PENs) innervate the bridge, as discussed in past work. This angular assignment maintains a 45° step size between adjacent glomeruli on each side of the bridge, which seems natural due to symmetry considerations. (Note that EPGt cells map from the ellipsoid body to the outer two glomeruli of the bridge with a small angular offset compared to the pattern set up by the EPGs that target the central 16 glomeruli––as reported by other studies––a caveat that slightly complicates our angular assignments; however, EPGt cells receive extensive axonal input in the bridge that has the potential to align their output signals with the rest of the bridge system.) Glomeruli are numbered 1 to 18 from left to right, to aid the comparisons made below. c, Two Δ7 cells from neuPrint (and past work) reveal that the axonal terminals of each Δ7 cell are 8-glomeruli apart (#5→#13 for cell A and #2→#10→#18 for cell B). This anatomy argues that any two glomeruli 8 apart, such as #5 and #13, will experience Δ7 output of equal strength. Compelling physiological evidence for this statement is available in the [Ca²⁺] signals of the PEN2 (equivalently, PEN_b) columnar cell class in the bridge, which is a strong anatomical recipient of Δ7 synapses and shows [Ca²⁺] activity across the bridge––clearly dissociable from the activity in EPGs––with consistently equal signal strength at glomeruli spaced 8 apart, perfectly following the Δ7 anatomical prediction (orange trace in Fig. 3d and data points in Extended Data Fig. 2i from ref. 15). Note that in the EPG indexing, shown in panel b, glomeruli #5 and #13, as examples, have angular indices that are not identical, but differ by 22.5°. d, Angles assigned to each bridge glomerulus based on the Δ7 axonal anatomy. Because the Δ7 output anatomy requires that any two glomeruli 8 apart, across the whole bridge, have the same angular index assignment, this results in a situation where all neighboring glomeruli have angular assignments that are separated by 45°. Note that almost all neighboring glomeruli are separated by 45° in the EPG mapping as well, except that, critically, in the EPG mapping the middle two glomeruli are separated by only 22.5°. This discontinuity is not evident in the Δ7 output. To create an angular indexing of the bridge for Δ7s that accommodates the anatomical constraints just described––i.e., one that incorporates an additional 22.5° in the bridge representation of angular space and thus “erases” the EPG discontinuity––we shifted the angular index for each glomerulus on the left bridge leftward by 11.25° relative to the EPG indexing and we shifted the angular index for each glomerulus on the right bridge rightward by 11.25° relative to the EPG indexing. e, The EPG indexing in panels a-b predicts that EPG activity in the left bridge (#2→#9) will be left-shifted by 22.5° compared to EPG activity in the right bridge (#10→#17). Indeed, when we overlapped the left- and right-bridge EPG signals we found the two curves are detectably offset from each other. f, To quantify the data from panel e, for each imaging frame in which the fly was flying, we calculated the phase of the EPG bump in the left and right bridge separately (via a population-vector average) and took the difference of these two angles (black bars: population mean and s.e.m.). We then averaged this angular difference across all analyzed frames for the same fly. For EPGs, this angular difference should be −22.5° if it follows the EPG indexing in panel b and it should be 0° if the activity follows the Δ7 indexing in panel d. Across a population of 9 flies, we found the angular difference is close to −22.5°, but shifted toward 0° by 4.6°, consistent with the fact that the EPG signal itself receives strong anatomical input from the Δ7s and thus could be modulated in its shape to follow the Δ7 indexing, in principle. It seems that the Δ7 feedback to EPGs reshapes its signal, but incompletely. g-h, Same as panel e, but analyzing the PFN_d and PFN_v activity in the bridge. Because PFNs only innervate the outer 8 glomeruli in each side of the bridge (unlike EPGs, which innervate the inner 8), we compared glomeruli #1→#8 in the left bridge overlapped with glomeruli #11→#18 in the right bridge here (the middle two glomeruli contain no signal for PFNs). i, Same as panel f, but analyzing the PFN_d and PFN_v activity in the bridge. Black bars: population mean and s.e.m. Note that because PFN_ds and PFN_vs innervate (and thus we can only analyze) the outer 8 glomeruli of the bridge, the angular difference in phase estimates between the left- and right-bridge activity should be +67.5° if it follows the EPG indexing (panel b) and +90° if it follows the Δ7 indexing (panel d). We found that the average angular difference in both PFN_ds and PFN_vs is intermediate between +67.5° and +90°, consistent with PFNs receiving functional inputs from both EPGs and Δ7s. We use the angular offsets measured in this panel as the basis for slightly adjusting the PFN_d and PFN_v angular indices in the bridge to an intermediate value between the EPG and Δ7 indexing options, described above. We believe that this approach represents the most careful way to combine the known anatomy and physiology to determine the azimuthal angle that each PFN cell signals with its activity in driving the hΔB neurites in the fan-shaped body, which we analyze in the next figure. j, Angles assigned to each bridge glomerulus for PFN_ds, based on the EPG indices from panel b and the physiologically determined adjustment required, based on the measurements in panel i. k, Same as panel j, but for PFN_vs.

Extended Data Figure 6 |. Computing the angular shift implemented by the PFN-to-hΔB connections.

Connectivity data and cell-type names are based on those in neuPrint, hemibrain:v1.1. a, The anatomical angle of each PFN_v cell is indicated based on which glomerulus it inervates in the protocerebral bridge, using the indexing described in Extended Data Figure 5k. b, Same as panel a, but for PFN_ds, using the indexing described in Extended Data Figure 5j. c, Synapse-number matrix for detected synapses from PFN_v cells to hΔB cells in the fan-shaped body. Note that the two stripes in the heatmap represent PFN_vs synapsing onto the dendritic regions of hΔBs. d, Same as panel c, but for synapses from PFN_d cells to hΔB cells. Note that two of the five stripes in the heatmap represent PFN_ds synapsing onto the dendritic regions of hΔBs, whereas the other, brighter, three stripes represent PFN_ds synapsing onto the axons of hΔBs. The average # of synapses that each hΔB compartment (axon vs. dendrite) receives from PFN cells is indicated on the bottom. e, Because hΔBs are postsynaptic to both PFN_vs and PFN_ds that project to the fan-shaped body from both sides of the bridge (panel c-d), each hΔB cell can be assigned an anatomical angle in four potential ways. To calculate the angle for an hΔB cell through its connection with the left-bridge PFN_vs, for example, we averaged the anatomical angles of all the left-bridge PFN_vs that connect to the hΔB cell in question, weighted by the number of synapses from that PFN_v cell to the hΔB cell. f, The anatomical angle of each hΔB cell calculated based on its monosynaptic inputs from left-bridge PFN_vs using the method described in panel e and data in panel c. g, Same as panel f, but calculations were made with right-bridge PFN_v inputs to hΔBs. h, Same as panel f, but calculations were made with left-bridge PFN_d inputs to hΔBs, using only the synapses formed on the axonal terminals of hΔBs. (We test the impact of this assumption––of complete functional dominance of PFN_d axonal synapses to hΔBs––below.) i, Same as panel f, but calculations were made with right-bridge PFN_d inputs to hΔBs, using only axonal synapses. j, For each hΔB cell, we calculated the angular difference between the mean left-bridge PFN_d input and the mean right-bridge PFN_d inputs (i.e., the difference between data points in panels h and i) and we plot a histogram of those values. k-m, same as j for the cell types indicated. n, The anatomically predicted angles for the coordinate axes of the four PFN vectors, as projected to the fan-shaped body and interpreted by hΔB axons and dendrites, calculated by averaging the histogram values in panels j-m, respectively. o, Same as panel n, but including all synapses from PFN_ds to hΔBs, not just the axonal ones as in panel n. We weigh dendritic and axonal synapses by PFN_ds to hΔBs equally in the panel-e calculation. Note that the angles between four coordinate-frame axes do not change very much when also including the dendritic synapses from PFN_ds to hΔBs, likely because they are less numerous than the axonal ones and the impact of the dendritic angles also seem to cancel out in their net effect (compare panels o and n). p, Same as panel n, but using the EPG indexing from Extended Data Figure 5 instead of the adjusted PFN_v and PFN_d indexing. Note that the EPG indexing makes the front angle between the left- and right-bridge PFN_d axes smaller. The same is true for the back angle between the left- and right-bridge PFN_v axes. q, Same as panel n, but using the Δ7 indexing from Extended Data Figure 5 instead of the PFN_v and PFN_d indexing. Note that the Δ7 indexing makes the front and back angles broader than 90°, when used in isolation. This analysis suggests that EPG and Δ7 inputs to PFNs are perfectly weighted to create axes that are orthogonal in our experiments in flying flies and also raise the possibility that orthogonality of this 4-vector system can be dynamically modulated via changing the weights of EPG and Δ7 inputs to PFNs (see Supplemental Text).

Extended Data Figure 7 |. PFR neurons track a variable similar to allocentric traveling direction in walking and flying flies.

a, Schematics of two example EPG cells, two example PFR cells and two example hΔB cells, which are the anatomically dominant input to PFRs. b, Sample GCaMP7f frames of the EPG bump in the ellipsoid body and the PFR bump in the fan-shaped body. c, Top, EPG (blue) and PFR (purple) GCaMP7f signal in a tethered, flying fly experiencing optic-flow (in the time window bracketed by the vertical dashed lines) with foci of expansion that simulate the following directions of travel: 180° (backward), −120°, −60°, 0° (forward), 60°, 120°, 180° (backward; repeated data). Third row, EPG and PFR phases extracted from the above [Ca²⁺] signals. Fourth row, circular-mean phase difference between EPGs and PFRs. Bottom two rows, average of left-minus-right and left-plus-right wingbeat amplitude. Single fly means: light gray. Population means: black. Dotted rectangle indicates a repeated-data column. d, EPG – PFR phase as a function of the egocentric traveling direction simulated by the optic flow, at three different speeds. Circular means were calculated in the last 2.5 s of optic flow presentation. Gray: individual fly circular means. Black: population circular mean and s.e.m. Dotted rectangle indicates a repeated-data column. (See Methods for how we calculate the optic flow speed.) Note that the data points deviate slightly from the unity line in a manner that means that the PFR phase is slightly shifted away from the traveling direction indicated by the optic flow and toward a frontal heading direction. The hΔB data in Figure 1h does not show this deviation from unity. We performed two-tailed one-sample t-tests against the diagonal line for data points in the ±60° and ±120° columns for the 35 cm/s data from PFRs here and the 35 cm/s data from hΔBs in Figure 1h. For the PFR results on the left of panel d, P values are 4.7e-5, 4.7e-5, 5.4e-4 and 9.1e-3 for the −120°, −60°, +60° and +120° columns, respectively. For the hΔB results in Figure 1h, P values are 0.39, 0.88, 0.058 and 0.44 for the −120°, −60°, +60° and +120° columns, respectively. e, PFR phase as a function of the inferred allocentric traveling direction, calculated by assuming that the EPG phase indicates allocentric heading direction and adding to this angle, at every sample point, the optic-flow angle. Gray: individual fly means. Black: population mean. In panel d and e, data from the middle column (35cm/s) were the same as in panel c. f, Tethered, walking, [Ca²⁺]-imaging setup with a bright blue bar that rotates in closed loop with the fly’s turns. g, Sample time series of simultaneously imaged EPG and PFR bumps in a tethered, walking fly. Top two traces show [Ca²⁺] signals. Third trace shows the phase estimates of the two bumps. Bottom trace shows the forward speed of the fly. h, Probability distributions of the EPG – PFR phase in walking and standing flies. Thin lines: single flies. Thick line: population mean. i, Circular mean of the EPG – PFR phase in walking and standing flies. WatsonWilliams multi-sample tests, P>0.63 when comparing any experimental group with 0°. Gray dots: single fly values. Black bars: population means ± s.e.m. j, Same as panel i, but plotting circular standard deviation. Two-tailed unpaired t-tests were performed. P value equals 0.042. k, Tethered-walking setup where we used a 617 nm LED focused on the center of the fly’s head to optogenetically trigger backward walking via activation of LC16 visual neurons expressing CsChrimson (Methods). l, An example 2D trajectory of optogenetically triggered backward walking. An arrow is shown every ~0.1 seconds. Red arrows indicate backward walking during the red-light pulse; blue arrows indicate the 1.2 s before the red light turned on. m, Left, time series of EPG (blue) and PFR (purple) bumps and phase-estimates from the trajectory in panel l. Right, time series of forward velocity, sideslip velocity and the difference between the PFR and EPG phase in the trajectory shown in panel l. The ΔF/F heatmap range is more compressed here than in other plots because the PFR signal strength typically dips when the fly initiates backward walking (a phenomenon whose mechanism we have not yet explored). Nevertheless, clear moments where the PFR phase separates from the EPG phase are evident, even after the PFR signal strength has recovered, in this sample trace (and in others). n, Time series of the mean forward velocity, mean sideslip velocity and the circular mean of the difference between the PFR and EPG phase during backward walking, grouped by optogenetic trials in which the fly walked to the back left (left panel) or to the back right (right panel). The sign of PFR-EPG phase deviations seen here, in walking, are consistent with the signs observed in flight, for the same directions of backward-left and backward-right travel. Thin lines and gray dots: individual trials. Thick line and black dot: population mean (circular mean for bottom row). o, Circular mean and s.e.m. of the peak EPG – PFR phase during triggered left-backward and right-backward walking bouts (0.6 s to 1.4 s after the dashed lines in panel n). Watson-Williams multi-sample tests were performed and P value equals 1.6e-6.

Extended Data Figure 8 |. Response-tuning in PFN, PFR and hΔB neurons to the translation speed indicated by our optic-flow stimuli.

a, Bottom row: phase-nulled PFN_d GCaMP activity across the bridge, averaged in the final 2.5 s of the optic flow epoch. We show responses to optic flow simulating traveling backward at four different speeds (left) and responses to optic flow simulating forward travel at four different speeds (right). The mapping between bridge [Ca²⁺] signals and data points in the plots in subsequent panels is indicated (arrows) for a few example points, using measurements from left-bridge PFN_ds, as an example. How we calculate the mean and amplitude of each bump is schematized. b, The population-averaged amplitude of the phase-nulled left-bridge PFN_d [Ca²⁺] activity in the final 2.5 s of the optic flow epoch, plotted as a function of the egocentric traveling direction simulated by the optic flow. The translational speed of optic flow increases across the four columns, from left to right. Gray lines: sinusoidal fits. S.e.m. are shown in this panel and throughout. c, Same as panel b, but analyzing the right-bridge PFN_d activity_. d, Same as panel b, but analyzing the left-bridge PFN_v activity_. e, Same as panel b, but analyzing the right-bridge PFN_v activity_. f, Same as panel b, but analyzing the hΔB activity in the fan-shaped body_. g, Same as panel b, but analyzing the PFR activity in the fan-shaped body. h, Same as panel b, but analyzing the hΔB activity in the fan-shaped body in non-flying flies. i, Same as panel f, but with a more zoomed-in y-axis. j, Amplitude of the four sinusoids in panel b to indicate how PFN_d responses, overall, scale with optic-flow translation speed. k-m, Same as panel j, but for the plots and cell type shown to the left. Note that the amplitudes of the PFN sinusoidal activity patterns are not only scaled by the traveling direction angle (panel b-e), but also by traveling speed (panel j-m). These plots make sense as a way to quantify the amplitude of sinusoidally modulated responses, like those of PFNs, but we also show, for completeness, the results of the same analysis for hΔBs and PFRs, where this way of quantifying forward-speed tuning makes less sense. n, Mean of the four sinusoids in panel f to indicate how hΔB responses, overall, scale with optic-flow translation speed. o, Same as panel n, but for the PFR plots shown to the left. p, Same as panel n, but for the hΔB plots shown to the left. Note that response-scaling with speed in hΔBs and PFRs was not consistent across all traveling directions (panels f-h). The fact that the speed tuning of hΔBs remains nonuniform across traveling directions in non-flying flies (panel h) suggests that this nonuniform tuning is not entirely due to an efference copy/proprioceptive signal being mismatched with backward optic-flow directions in tethered flight, though the interpretation of this nonuniform tuning will need to be resolved in future work. q, Same as panel n, but for the hΔB plots shown to the left. r, Same as panel b, but analyzing the mean (rather than the amplitude) of the left-bridge PFN_d [Ca²⁺] activity patterns. Gray lines: same sinusoidal fits from panel b with a vertical offset and a scale factor that is constant across all four speeds. The fact that our amplitude fits from panel b also fit the mean responses shown here well supports the hypothesis that the heading input and the optic-flow input to PFN cells are integrated multiplicatively (see Methods). s-x, Same as panel r, but analyzing the cell type indicated on the left side of the figure, for each row. See Methods for how the optic-flow speed was calculated.

Extended Data Figure 9 |. The neural circuit described in this paper implements an egocentric-to-allocentric coordinate transformation.

a, Schematic of the computation implemented in the Drosophila central complex. Traveling-direction signals referenced to the body axis (i.e., optic flow signals in SpsP and LNO1s, which indicate the egocentric traveling angle, green) are converted into traveling-angle signals referenced to cues in the world (i.e., the hΔB bump position, which indicates allocentric traveling angle, red). b, Schematic of a very similar computation hypothesized to take place in monkey parietal cortex.

Extended Data Figure 10 |. A traveling-direction signal computed via optic flow is robust to changes in the yaw angle of the fly’s head.

a, Fly flying straight with the head aligned to the body axis. EPG and hΔB signals are aligned in the ellipsoid body and fan-shaped body, respectively. b, Fly flying straight forward with the head rotated 20° to the right. The EPG bump––assuming the EPG bump position tracks the fly’s head (rather than body) direction––will rotate 20° counterclockwise. The hΔB bump, however, will remain pointing in the same allocentric traveling direction because the net effect of the EPG bump rotating 20° in one direction and the ego-motion signal from optic flow (not represented in the diagram) rotating 20° in the opposite direction is that the PFR/hΔB bump stably indicates the same traveling direction throughout.

Figure 1 |. hΔB neurons signal Drosophila’s allocentric traveling direction.

a, Fly brain. b, Two example EPGs and two example hΔBs. Each cell type tiles the central complex c, Imaging neural activity in a flying fly with an LED arena. d, sytGCaMP7f frames of the EPG bump in the ellipsoid body and the hΔB bump in the fan-shaped body. e, Simultaneously recorded sytGCaMP7f signal from EPGs (blue) and hΔBs (red) in a flying fly. Top: [Ca²⁺] signal; bottom: phase estimates and dot position. Gray regions represent the 90° gap in the back of the arena. f, Distributions of EPG – hΔB phase without and with optic flow. Gray: single fly means. Black: population mean. g, Top, EPG (blue) and hΔB (red) sytGCaMP7f signal in a sample flying fly experiencing optic-flow with foci of expansion that simulate the following directions of travel (in the time period delimited by the vertical dashed lines): 180° (backward), −120°, −60°, 0° (forward), 60°, 120°, 180° (backward; repeated). Middle, phases of the sample [Ca²⁺] signals above. Bottom, circular mean of EPG – hΔB phase for a fly population. Gray: single fly means. Black: population mean. Dotted rectangle: repeated-data column. h, EPG – hΔB phase versus the egocentric traveling direction simulated by optic flow. Circular means were calculated in the final 2.5 s of optic-flow presentation. Gray: individual fly means. Black: population mean and s.e.m. Dotted rectangle: repeated-data column. i, hΔB phase versus the inferred allocentric traveling direction, calculated by assuming that the EPG phase indicates allocentric heading and adding to this angle, at every sample point, the optic-flow angle. Same data as in panels g-h. Gray: individual fly means. Black: population mean. (Note flipped x-axis indicating that the hΔB bump tracks the negative of the fly’s traveling direction; Methods).

Figure 2 |. The allocentric traveling direction can be computed by vector rotation and summation, which can be implemented by phasors.

a, The traveling-direction vector (green) for a fly translating at an egocentric traveling angle T_ego, referenced to its head direction (gray line with circle), is projected onto four axes oriented ±45° and ±135° relative to the head, yielding four scalars L_1–4. The +45° projection is shown. The fly’s head direction represents 0° in this egocentric reference frame. Angles are positive clockwise. b, The fly’s allocentric traveling direction, T_allo, can be computed either by rotating the egocentric traveling angle (T_ego) such that it becomes referenced to the external world (i.e., the sun) (top) or, as in the fly circuit, by first referencing the ±45° and ±135° projection axes to the external world (bottom) and then taking the vector sum of the four projection vectors. Egocentric vectors are referenced to the external world by adding H, the fly’s allocentric heading angle, to them. c, Two-dimensional vectors can be represented by sinusoids and adding sinusoids then implements vector addition.

Figure 3 |. PFN_d and PFN_v cells show physiological and anatomical patterns consistent with them functioning to build the traveling-direction signal in hΔBs.

a-b, Sample PFN and hΔB cells (see Text). c, [Ca²⁺] signals and phase estimates of EPG and PFN_v bumps simultaneously imaged in the bridge of a flying fly. d, Phase-nulled and averaged EPG and PFN_v activity in the bridge. On each frame, the EPG bumps were rotated to the same position and the simultaneously imaged PFN_v signals were rotated by this EPG-defined angle; coherent PFN_v bumps in these plots thus indicates strong phase-alignment to the EPGs. Thin lines: individual flies. Thick lines: population average. Gray dashed lines: sinusoidal fits (see Methods for details and goodness of fit). e, Same as panel d, but for EPGs and PFN_ds. f-i, schematics of how PFNs promote hΔB axonal activity with ±45° and ±135° offsets (see Text) j, Phase-nulled PFN_d activity in the bridge, averaged in the final 2.5 s of each optic-flow epoch. Dotted rectangle: repeated-data column here and throughout. Gray dashed lines: sinusoidal fits. Closed-loop bright dot was not displayed. k, Single-fly (circles) and population means ± s.e.m. (black bars) of the amplitude of PFN_d activity across the left bridge, averaged in the final 2.5 s of the optic flow epoch. Sinusoidal fit: gray line. l, same as panel k, but right-bridge PFN_ds. m-o, same as panels j-l, but PFN_vs. p-r, same as panels j-l, but EPGs.

Figure 4 |. A model of how vector computation builds the traveling-direction signal in hΔB cells.

a, When a fly travels forward, both PFN_d sinusoids have large amplitude and both PFN_v sinusoids have small amplitude, leading the sum of the four PFN vectors, i.e., the hΔB (red) vector, to point forward, or in the heading direction (blue EPG vector). b, When a fly travels backward, both PFN_d sinusoids have small amplitude and both PFN_v sinusoids have large amplitude, leading the sum, i.e., the hΔB (red) vector, to point backward, or opposite the heading direction. c, When a fly travels rightward, the right-bridge PFN_d sinusoid and the left-bridge PFN_v sinusoid have larger amplitude than their counterparts on the opposite side of the bridge, leading the sum, i.e., the hΔB (red) vector, to point rightward. d, Same as panel a––fly moving forward––but after the fly has turned clockwise by 90°. This turn rotates all the vectors (i.e., the reference frame) by 90° inside the brain. e, Data from Figure 1h (black bars) and model (diamonds and circles) (see Text).

Figure 5 |. Neural-activity perturbations induce changes in the traveling-direction signal that are consistent with the vector-sum model.

PFR phase used as proxy for hΔB phase here (see Text). a, Prediction. b, PFR bump in the fan-shaped body of a fly walking with a closed-loop bright bar at 34°C, with EPGs expressing (right) or not expressing (left) shibire^ts. c, PFR phase – bar position distributions. Thin lines represent single flies and the thick line represents the population mean (also in panels g,k,o). d, Circular s.d. of PFR phase – bar position distributions for different genotypes. Gray dots represent single flies and black markers represent population means ± s.e.m (also in panels h,l,p). P<2×10⁻⁴ comparing experimental (4th or 7th columns) with any control group (Unpaired two-tailed t-tests). e, Prediction. f, Simultaneous imaging of EPG and PFR bumps in the context of PFN_vs expressing (right) or not expressing (left) Kir2.1. g, EPG – PFR phase distributions. h, Circular s.d. of PFR – EPG phase distributions for different genotypes. P<4×10⁻⁵ comparing experimental (3rd or 5th columns) with any control group (Unpaired two-tailed t-tests). i, Prediction. j, Simultaneous imaging of PFR and EPG bumps in the context of expected optogenetic activation (right) or no activation (left) of PFN_vs. The two-photon laser exciting GCaMP simultaneously excites GtACR1 in LNO1s, which should silence them and thus excite PFN_vs via sign-inverting synapses in the noduli. k, EPG – PFR phase distributions. l, Circular mean of the PFR – EPG phase distributions for different genotypes. P<4×10⁻⁴ comparing experimental (3rd column) with either control group (Watson-Williams multi-sample test). m, Prediction. n, Simultaneous imaging of PFR and EPG bumps in the context of optogenetic activation of SpsP cells expressing (right) or not expressing (left) csChrimson. An external, red laser excites csChrimson in SpsPs, which should activate them and thus inhibit PFN_ds via sign-inverting synapses in the bridge. o, EPG – PFR phase distributions. p, Circular mean of the PFR – EPG phase distributions for various genotypes. P<7×10⁻⁴ when comparing the experimental group (3rd column) with either control group (Watson-Williams multi-sample test). See Methods for exact p values.