Supplementary Figure 1

Visual responses of the recorded LPTCs

(a) The mean±SD (n=3 trials) of the direction-selective (DS) responses (i.e., subtracting the null direction, ND, from the preferred direction, PD) of

the HSE cell shown in Fig. 1c, to a wide-field horizontal (left) or vertical (right) moving grating. The light gray background indicates the stimulus

period. (b) The (time) average DS response amplitude during stimulation (grey dots), and the mean value (black lines). (c) Triple immunostaining for

the GFP signal (green) driven by the R81G07Gal4 line, for the biocytin signal (red) from the recorded HS cell, and for the bruchpilot protein (brp,

blue), staining for synaptic neuropile. The dash line delineates the border of the lobula plate. (d–f) Same as (a–c), but for the VS1 cell shown in Fig.

1d. (g–i) Same as (a–c), but for the HSE cell in a norpA mutant background (blind fly) shown in Fig. 1e. (j) DS visual responses from HS cells in

T4/T5-Gal4 (left) and UAS-Kir (middle) control flies, or from HS cells in flies with T4/T5 cells silenced by the expression of Kir. Shown is the

mean±SEM (N is indicated in each corresponding figure).

Nature Neuroscience: doi:10.1038/nn.4435

Supplementary Figure 2

Classification of tethered behavior on the ball

(a) Swing-stance periods (black or white, respectively) during a walking bout as defined by videography. Note that at steady forward walking (from

0.5 to 4 s), the fly displays tripod gate (see zoom-in, N=6 flies). (b) Examples of a subset of the first ten principal components (PCs) of the side-view

fly video recordings (Fig. 1a,b). Asterisks indicate the PCs whose scores are shown in (c). (c) Time-varying scores for different PCs. A zoom-in into

PC5’s scores reveals a periodic behavior of a principal component associated with leg movement. (d) A wavelet transformation of PC5 scores further

indicates the bandwidth of the signal, matching the mean cycle period obtained from the swing-phase analysis. (e) The PCs scores and their wavelet

transform (c,d) were used to train JAABA (top schematic) to classify walking and other possible behaviors of the fly on the ball (bottom color-coded

image). (f) Example of a 1-minute trial displaying the fly’s forward (grey top trace), and angular (grey bottom trace) velocity signals during walking

(pink background) and non-walking (blue background) segments. For angular velocity, CCW turns are positive sign values. For forward velocity,

negative sign indicates backward walking. Arrowheads show activity segments (i.e., non-walking but non-stationary segments).

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Supplementary Figure 3

Walking-specific responses during locomotion in HS cells

Cross-covariance analysis per cell between the fly’s angular velocity (or forward velocity) and the Vm of HS cells, in walking or activity bouts. The

associated distributions indicate the % of activity time spent in walking (“walking ratio”). Note that the example cells included in Figure 3 are not

presented here.

Nature Neuroscience: doi:10.1038/nn.4435

Supplementary Figure 4

Walking direction and speed sensitivity in HS cells

(a) Change in Vm of left-side HS cells with respect to quiescence (ΔVm, color-coded) as a function of the forward and angular velocities of each

individual fly. (b) Same as in (a), but for right-side HS cells. Scale bar is the same as for the maps in (a). (c) Population ΔVm (color-coded) as a

function of the forward and angular acceleration of the fly. (d) Examples of the Vm of HS cells (orange), the forward (Vf, black), and angular

velocities (Va, gray) of the fly in segments of walking bouts where the fly’s angular velocity did not exceed ±35°/s (gray shadow). (e) Left, weighted

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grand mean±SEM of the average angular velocity of the fly for the selected segments with low angular velocity as a function of the averaged Vf of

the fly. The weight is given as a function of number of segments where the fly showed |Va| < 35°/s (mean = 240, range: 50-499 bouts). Right,

weighted grand mean±SEM of the average Vm as a function of the average Vf of the fly. (f) Mean correlation coefficient between the fly’s Vf and

the Vm of HS cells per cell, and between Vf and Va per cell. Significant correlations are red, non-significant correlations are blue (traces were

shuffled 20 times by concatenated walking bouts, Wilcoxon’s signed-rank test).

Nature Neuroscience: doi:10.1038/nn.4435

Supplementary Figure 5

Movement-related modulations in VS cells

(a) Vm of a VS1 cell triggered at the onset of postural adjustments in walking (left), or activity (right) bouts. n = number of bouts. Lighter traces

show triggered data from individual bouts. (b) Left, the mean Vm of the population of VS cells (mean±SEM) triggered at the onset of walking

(black) or activity (gray) bouts. Right, the mean Vm triggered at the offset of walking (black) or activity (gray) bouts. n = total number of bouts. For

walking onset bouts, N=10 cells. For activity onset analysis, N=6 cells. For walking offset analysis, N=7 cells. For activity offset analysis, N=4 cells.

(c) Distribution of the 10% rise time of the Vm (see Methods) relative to the onset of walking bouts. Arrowhead indicates the median value. (d) ΔVm

(color-coded) as a function of the forward and angular velocities of each individual fly. (e) ΔVm (color-coded) is plotted as a function of the forward

and angular velocities of the fly for the population of recorded VS cells, right and left cells were pooled together (see Methods). Scale bar is the

same as for the maps in (d).

Nature Neuroscience: doi:10.1038/nn.4435

Supplementary Figure 6

Walking-specific signals are independent of vision, are not coupled to the movement of antennae or halteres, and do not depend on the

synaptic activity of leg mechanosensory cells.

(a) Walking velocity tuning for the recorded HS cell shown in Fig. 1e. PDmotor and NDmotor are the cell’s preferred and null turning direction of the fly

(see main text). (b) Walking velocity tuning map for the population of HS cells recorded from blind (norpA) flies (N=9 cells). (c) Walking velocity

tuning maps for the population of HS cells recorded from flies with antennae and halteres waxed (N=11 cells, right and left-side HS cells were

pooled, see Methods). (d) Left, walking velocity tuning maps for the population of HS cells recorded from flies with leg mechanosensory cells

expressing tetanus toxin to disrupt their synaptic activity (N=8 cells). Right, walking velocity maps of parental control flies with functional synaptic

activity in leg mechanosensory cells (N=7 cells). (e) Top, paths of freely walking experimental (red, flies with leg mechanosensory cells expressing

tetanus toxin, TNT) or control flies (black). Bottom, zoomed-in view to highlight the straightness of the paths of the walking flies. (f) Probability

distributions (mean±SD) of the straightness of walking paths in experimental (red) and control (black) flies (see Methods). (g) Mean path

straightness for experimental (red) and control (black) flies. For straightness analysis: 13 control flies, mean±SD: 0.87±0.03; 15 experimental flies,

mean±SD: 0.71±0.07, ***,P< 0.0001, Z=4.38, Wilcoxon’s rank-sum test. (h) Probability distributions (mean±SD) of walking speed in experimental

(red) and control (black) flies. For walking velocity analysis: 13 control flies, mean±SD: 11.5±1.9 mm/s; 15 experimental flies, mean±SD: 9.6±1.5

mm/s, P<0.01, Z=2.63, Wilcoxon’s rank-sum test.

Nature Neuroscience: doi:10.1038/nn.4435

Supplementary Figure 7

Correlation between head and body movements during tethered walking in darkness

(a) Top: Schematic of head-tracking set-up. Middle: example frame from a top-view camera. Bottom: example frame from a side-view camera.

Contrast has been enhanced for clarity. (b) Example traces from one trial comparing the head yaw movement and the body’s turn. Top, angular

velocity of the fly. Middle, head yaw angle. Bottom, head yaw velocity. Arrows indicate head-yaw angle offsets. (c) Cross-covariance between

angular velocity and head angle for all walking bouts in darkness for a single fly (n=390 bouts from 126 trials). Thick black trace, mean; shadow, SD.

The magenta line in all cross-covariance plots indicates the bootstrapped 95% confidence level of the analysis (see Methods). Light gray trace,

example from (b). (d) Grand mean cross-covariance. The head precedes the body by 50 ms on average (mean SEM =48.3±24.0 ms, N=9 flies). (e)

Instantaneous head yaw position as a function of the angular and the forward velocities of the fly. (f) Idem as in (c) but the head yaw velocity was

compared with the angular velocity of the fly. (g) Same as in (d) but with the head yaw velocity. (h) Instantaneous head yaw velocity as a function of

the forward and angular velocities of the fly. (i) Cross-covariance analysis between the head pitch angle and the forward velocity of the fly. Grand

mean±SEM (341±143ms, N=6 flies).

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Supplementary Figure 8

Model performance analysis

(a) Predicted velocity tuning maps for different single-component models. (b) Mean cross-correlation coefficients per cell between observed and

predicted Vm for different models (black, matched pairs; red, mismatched pairs; see Methods, lines: mean values, **: P<0.001, Z>3.78, N=19 cells,

Wilcoxon signed-rank test). Right-most column: cross-correlation coefficients between the observed and the predicted yaw head angle. (c)

Performance of the BS and the Va+Vf+BS models in each cell. Indicated are the example cells shown in Fig. 3a, and in (e). (d) Performance of the

Va+Vf+BS and the Va+BS models for each cell. Mean (±SD) correlation coefficients between the predicted and observed HS dynamics (see

Methods). Red and blue: the difference in the magnitude of correlation coefficients between the two models cannot be explained by the reshuffling

procedure (traces were shuffled 20 times by concatenated walking bouts, Wilcoxon’s signed-rank test, P<0.005), whereas the gray pairs can (P

>0.02, Wilcoxon‘s signed-rank test). (e) Example of a cell with the lowest three-compartment model performance. Note that the HS cells’ dynamics

are still well described by the three-component model. (f) Top, the predicted velocity-tuning map for the example cell in Figure 5c scaled in mm/s

units in both axes. θ: the angle between the forward velocity axis and the membrane potential change (ΔVm) gradient (see Methods). Bottom,

distribution of θ for each cell; the mean value is indicated in black. The θ value for the observed population velocity map (N=19 cells) is indicated in

red. (g) The observed (black) and predicted (magenta) head yaw angles, estimated from the three-component model fitting the head yaw angle

instead of the Vm. (h) Power spectrum analysis of the observed, the predicted, and the difference between the two for Vm (left), or yaw head angle

(right). Note that the largest difference for the head angle prediction is on the DC component, i.e., the offset of the head position (arrows in (g)).

Nature Neuroscience: doi:10.1038/nn.4435

Supplementary Figure 9

Decoding walking velocities from the bilateral activity of HS cells

(a) Recordings from HS cells on one side of the brain were combined with predicted HS dynamics of the other right side using the walking behavior

of the fly as input for the BS+Va+Vf model (Figure 5). The dynamics of the bilateral HS cells are labeled as modeled + recorded. (b) The mean

angular (left) or forward (middle) velocities of the fly and normalized data points (right) were plotted as a function of the negative (VmLeft-

VmRight) or positive (VmLeft+VmRight) combination of the bilateral dynamics of modeled + recorded cells. (c) Using the bilateral modeled +

recorded dynamics, or the unilateral recorded cell dynamics, we applied a linear decoder (see Methods) to predict the angular velocity (Va) of the

fly. Left, top two plots, example of decoding using bilateral HS cells’ dynamics. Bottom two plots, examples of decoding using the recorded cell

dynamics only, i.e., the unilateral HS cells’ activity. Red trace, estimated; black trace observed. Right, covariance coefficients (Cov Coef) between

the predicted and observed velocities using bilateral or unilateral HS cells’ dynamics. Walking bouts and the cell dynamics were reshuffled 20 times

to obtain a population of Cov Coefs per fly and the mean value was used. Red, Cov Coefs between matched behavior and Vm (bilateral,

mean±SD=0.85±0.06; unilateral, mean±SD=0.48±0.14,); gray, Cov Coefs between mismatched behavior and Vm. The difference in the magnitude of

Cov Coefs between bilateral and unilateral decoding is significant (P<0.0005, Z=3.82, Wilcoxon signed-rank test, N=19 cells). (d) Decoding of the

forward velocity (Vf) of the fly. Same as in (c) Cov Coefs between matched behavior and Vm for bilateral model: mean±SD=0.50±0.13; for the

unilateral model: mean±SD=0.29±0.12. The difference in the magnitude of Cov Coefs between bilateral and unilateral decoding is significant (

Wilcoxon’s signed-rank test P< 0.005, N=19 cells). (e) Transfer functions for the decoding of angular velocity (Va). Gray, individual fly’s filters,

black, mean filter. Note that the magnitude of the mean filters obtained with the recorded data is similar to those obtained with the modeled data. (f)

Same for the decoding of forward velocity (Vf).

Nature Neuroscience: doi:10.1038/nn.4435

Supplementary Figure 10

Visuomotor interactions in HS cells

(a) Top: Velocity-tuning maps under replay visual stimulation per cell. Bottom, visual and angular velocity maps per cell. (b) Walking velocity

tuning map across the population of right-side HS cells under replay trials (N=13). (c) Estimate of the visual modulation of HS cells’ activity under

replay conditions in quiescent segments: the visual stimulus is scaled by the velocity tuning curve of HS cells (see Methods) to obtain an effective

visual stimulus (red trace). This effective stimulus is convolved with a response kernel (see Methods) to estimate the visual-induced activity in HS

cells (blue trace). For comparison, the observed visual responses under identical conditions is shown (black trace). The delay of the kernel was

obtained by cross-correlation analysis between HS cell responses and visual stimuli in quiescent segments. (d) Distribution of θ (Fig. 6) for each cell

for fits with R2>0.7 (top, 9/13 cells, mean±SD=37±9°). Black: mean value; red, θ for the population map shown in Figure 6b. (e) Predictions from

ideal random forest decoders (see Methods) of the sum of the visual velocity (Vv) and the fly‘s angular velocity (Va, red traces), or the difference

between the two (blue traces). The input signals for the decoder were either the estimated Vm dynamics from the visual stimulus and the three-

component walking of the fly (top row), or the estimated Vm dynamics from Vv and the fly’s Va (bottom). c=correlation coefficient between the

predicted and the observed values of the sum or difference between Vv and Va. (f) Cross-correlation coefficients between the predicted and the

observed velocities for each recorded cell, for the two different inputs to the ideal decoders. Bars: the mean values across cells. Color code as in (e).

Nature Neuroscience: doi:10.1038/nn.4435