The manifold structure of limb coordination in walking Drosophila
- Yale University, United States
Figures
Figure 1 with 2 supplements
Measurements of body and limb kinematics in freely-walking Drosophila.
(A) Schematic of experimental setup. Fruit flies walk in a circular arena while illuminated from above and tracked from below using a high-speed camera (150 fps). (B) Data transformations from …
Figure 1—figure supplement 1
Linear regression accurately identifies footfall positions.
(A) Euclidean error distance (mm) between predicted and actual footfall positions as a function of number of principal components (PCs) included in model, estimated using 10-fold cross-validation. …
Figure 1—figure supplement 2
Statistics of centroid kinematic behavior in freely-walking flies.
Error patches show 95% confidence intervals obtained from bootstrap distributions over experiments (N = 8 videos; see Materials and methods). (A) Sample centroid trajectories before and after …
Figure 2 with 2 supplements
Drosophila use a two-cycle limb coordination pattern across all walking speeds.
(A) Canonical hexapod gaits. In all gaits, limb swings (black) propagate ipsilaterally posterior to anterior on each side of the fly. Tripod gait is defined by three limbs simultaneously swinging at …
Figure 2—figure supplement 1
Estimating limb phases.
Analytic signal decomposition of an example limb position time series. Limb movements in the direction parallel to the fly’s body axis were decomposed into instantaneous measures of phase and …
Figure 2—figure supplement 2
A two-cycle coordination pattern is used across all walking speeds.
(A) Joint distribution of midlimb phase and forward walking speed conditioned on the number of feet in stance phase reveals two peaks per limb cycle for 5, 4, and 3-foot down conditions across all …
Figure 3 with 3 supplements
Relative phase measurements reveal a continuum of coordination patterns across all walking speeds with contralateral limbs in antiphase.
(A) Diagram of pairwise limb relative phases, with ipsilateral pairs in blue and contralateral pairs in red. (L1-R1 = Left forelimb – Right forelimb, L2-R2 = Left midlimb – Right midlimb, L3-R3 = Lef…
Figure 3—figure supplement 1
Synthetic canonical gaits differ in relative phasing from the coordination patterns used by free-walking Drosophila.
(A) Distributions of limb relative phases for synthetic tripod, left tetrapod, right tetrapod, and wave gaits. (B) Joint probability distribution of L2-R2 and L3-L1 relative phases for synthetic …
Figure 3—figure supplement 2
The conditional distributions of pairwise limb relative phases are unimodal at all forward walking speeds.
Probability density functions of each of the nine pairings of limb relative phase shown in Figure 3A–D, conditioned on forward walking speed.
Figure 3—figure supplement 3
Additional measurements of limb phases.
(A) Joint probability distribution of L2-R2 and L3-L1 relative phases for bottom, middle, and top thirds of forward velocity distribution: slow walking (0–10.2 mm/s), medium walking (10.2–19 mm/s), …
Figure 4 with 5 supplements
Dimensionality reduction reveals the manifold structure of limb coordination patterns.
(A) UMAP embedding of limb coordinate time series colored by the mean frequency of forward walking. Frequency (correlated with forward walking velocity) maps to the height along the vase-shaped …
Figure 4—figure supplement 1
Principal component analysis of limb kinematic data.
(A) The covariance matrix of the segments of standardized limb kinematic data used to generate the UMAP embedding in Figure 4. This matrix is approximately a block-symmetric-Toeplitz matrix. (B) The …
Figure 4—figure supplement 2
Representation of UMAP embedding in cylindrical coordinates.
(A) The UMAP embedding of limb kinematic data shown in Figure 4 was converted into cylindrical coordinates as , , . Here, the distribution of the resulting UMAP phase angle is shown. (B) Joint …
Figure 4—figure supplement 3
Contralateral antiphase is preserved at all phases of the global oscillator.
The UMAP embedding of limb kinematic data shown in Figure 4, colored by the instantaneous phases of the left and right midlimbs at the center point of each segment.
Figure 4—figure supplement 4
Changing the segment duration dilates the axial extent of the UMAP manifold while maintaining the same structure.
(A) UMAP embedding of 100 ms segments of limb positional data, colored by mean stepping frequency (see Figure 4A). (B) As in (A), but for 400 ms segments. (C) As in (A), but colored by the position …
Figure 4—figure supplement 5
The manifold structure of synthetic canonical gaits differs qualitatively from that of free-walking Drosophila.
(A) UMAP embedding (see Materials and methods) of synthetic tripod (red), left tetrapod (blue), right tetrapod (green), and wave gaits (orange) reveals separate manifolds for each canonical gait …
Figure 5 with 1 supplement
A single-parameter model with speed-independent coupling predicts a continuum of inter-limb coordination patterns.
(A) A single-parameter phase oscillator model to generate metachronal waves (see Materials and methods). The frequency of metachronal waves is determined by a single parameter, stance. Limb …
Figure 5—figure supplement 1
A single-parameter model generates a two-cycle coordination pattern across all walking speeds.
Model-generated joint distribution of midlimb phase and forward walking speed shows two peaks per limb cycle for 5, 4, and 3-foot down conditions.
Figure 6 with 1 supplement
Asymmetric, segment-specific modulations of limb movement underlie turning.
Error patches show 95% confidence intervals of the mean obtained from bootstrap distributions over experiments (N = 8 videos; see Materials and methods). (A) Modulation of swing duration in …
Figure 6—figure supplement 1
Average modulations of limb movement parameters.
Error patches show 95% confidence intervals of the mean obtained from bootstrap distributions over experiments (N = 8 videos; see Materials and methods). (A) Average swing duration in individual …
Figure 7
Spontaneous turns are aligned to preferred phases of the limb oscillator.
Error patches show 95% confidence intervals obtained from bootstrap distributions over experiments (N = 8 videos; see Materials and methods). (A) Probability density function of limb positions in …
Figure 8
Experimental perturbations of walking speed modulate stance duration.
(A) Drosophila normalized forward speed over time in response to 8 ms optogenetic activation of moonwalker neurons (red, n = 82) versus random trigger control (blue, n = 91). (B) Cumulative …
Author response image 1
Replication of Figure 2 with halved swing/stance separation threshold.
Author response image 2
Analysis of phase extraction with high SNR (A and B) and low SNR (C and D).
Author response image 3
Example trajectories with gait index analysis from Mendes et al., 2013.
Author response image 4
t-SNE embedding of limb position data, colored by forward walking speed.
(A) shows experimental limb positional data, while (B) shows data generated using the continuum model presented in Figure 5 of the manuscript.
Videos
Video 1
This movie shows a fly walking in our arena with annotated body and limb features.
Video shows annotations in both the camera and egocentric frame of the fly. Body orientation is indicated with a yellow triangle. Limb positions are red (L1), blue (L2), green (L3), orange (R1), yell…
Video 2
This movie shows a fly in its egocentric frame with annotations for each of the individual limbs.
Limb position variables are shown as time series in the direction parallel and perpendicular to the fly’s major axis. Limb positions are red (L1), blue (L2), green (L3), orange (R1), yellow (R2), …
Video 3
This movie shows a grid of 25 fly trajectories seen in the egocentric frame.
Limb positions are annotated with red (L1), blue (L2), green (L3), orange (R1), yellow (R2), and brown (R3).
Tables
Table 1
Fly strains.
https://doi.org/10.7554/eLife.46409.028| Genotype | Source | Experiment | Figure | |
|---|---|---|---|---|
| Wild type | +; +; + | (Gohl et al., 2011) | Free-walking; Visual stimulus induced slowing | 1–8 |
| Moonwalker > Chrimson | +; +; VT-050660-Gal4/UAS-Chrimson | (Bidaye et al., 2014) | Optogenetic induced slowing | 8 |
Table 2
Phase templates for canonical gait coherence analysis (Collins and Stewart, 1993).
Limb ordering is (L1, L2, L3, R1, R2, R3).
| Canonical gait | Phase template (cycles) |
|---|---|
| Tripod | [0, 1/2, 0, 1/2, 0, 1/2] |
| Left tetrapod | [1/3, 2/3, 0, 0, 1/3, 2/3] |
| Right tetrapod | [2/3, 0, 1/3, 0, 1/3, 2/3] |
| Wave | [1/6, 1/3, 1/2, 2/3, 5/6, 0] |
Table 3
Parameters used for model in Figure 5.
https://doi.org/10.7554/eLife.46409.031| Parameter | Value |
|---|---|
| 1/8 | |
| 40 ms | |
| 40–210 ms |
Table 4
Two-sample Monte Carlo resampling tests using the Kuiper V-statistic against the null hypothesis that the distribution of phases at yaw extrema is indistinguishable from that over all instants (105 permutations, N = 8 videos).
https://doi.org/10.7554/eLife.46409.026| Limb | V-statistic | p-Value | 95% CI for p-value |
|---|---|---|---|
| O1 | 1.08 × 10−1 | <10−5 | [0, 3.68 × 10−5] |
| O2 | 1.67 × 10−1 | <10−5 | [0, 3.68 × 10−5] |
| O3 | 1.77 × 10−1 | <10−5 | [0, 3.68 × 10−5] |
| I1 | 1.07 × 10−1 | <10−5 | [0, 3.68 × 10−5] |
| I2 | 2.33 × 10−1 | <10−5 | [0, 3.68 × 10−5] |
| I3 | 1.16 × 10−1 | <10−5 | [0, 3.68 × 10−5] |
Additional files
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Transparent reporting form
- https://doi.org/10.7554/eLife.46409.032
