Phase response analyses support a relaxation oscillator model of locomotor rhythm generation in Caenorhabditis elegans
- Department of Bioengineering, School of Engineering and Applied Science, University of Pennsylvania, United States
- Department of Neuroscience, Perelman School of Medicine, University of Pennsylvania, United States
Figures
Figure 1
Rhythm generation in C. elegans.
(A) Motor neurons generate neuronal signals to control the activation of body wall muscles (BWM), which generates movement subject to internal and external environmental constraints. Sensory input …
Figure 2 with 1 supplement
Undulatory dynamics of freely moving worms.
(A) Worm undulatory dynamics are quantified by the time-varying curvature along the body. The normalized body coordinate is defined by the fractional distance along the centerline (head = 0, tail = …
Figure 2—figure supplement 1
Phase portrait representations of the oscillatory bending dynamics for various body coordinates.
Figure 3 with 8 supplements
Analysis of phase-dependent inhibitions for head oscillation using transient optogenetic muscle inhibition.
(A) Images of a transgenic worm (Pmyo-3::NpHR) perturbed by a transient optogenetic muscle inhibition in the head during forward locomotion. Green shaded region indicates the 0.1 s laser …
Figure 3—figure supplement 1
Normalized deviation to the normal cycle (the unperturbed oscillation) for the head oscillation of the perturbed worms.
Individual dynamics were grouped into different bins by binning their initial amplitude at . In the figure, each trace represents the collective amplitude dynamics of the corresponding group. …
Figure 3—figure supplement 2
The isochron map overlaid with the vector field for the worm’s head oscillation.
On the isochron map, a point on the normal cycle (black trajectory) and all other points off the normal cycle that share the same color form a manifold representing states having an equal phase …
Figure 3—figure supplement 3
Phase response curve of Pmyo-3::NpHR worms (ATR- control group).
Curve was obtained from 414 trials of transient illuminations on head using 116 worms. Each point represents a single illumination (0.1 s duration, 532 nm wavelength) of one worm. Filled area …
Figure 3—figure supplement 4
Phase response curve of Pmyo-3::NpHR worms perturbed by a 0.055 s optogenetic muscle inhibition during normal locomotion.
Curve was obtained from 150 trials of transient inhibitions of head muscles using 115 worms. Each point represents a single illumination (0.055 s duration, 532 nm wavelength) of one worm. Filled …
Figure 3—figure supplement 5
Phase response curves of Pmyo-3::NpHR worms induced by a 0.1 s optogenetic muscle inhibition, perturbed and measured at various body regions.
Anterior = 0.1-0.3; middle = 0.4-0.6; posterior = 0.6-0.8 along the worm body. (Upper) Schematics illustrating the selected spatial regions for optogenetic inhibition (Green shaded area) and phase …
Figure 3—figure supplement 6
Phase response curve of transgenic worms that express NpHR in all cholinergic neurons (Punc-17::NpHR::ECFP).
Curve was obtained from 270 trials of transient inhibitions of cholinergic neurons in the head region using 135 worms. Each point represents a single illumination (0.055 s duration, 532 nm …
Figure 3—figure supplement 7
Phase response curve of transgenic worms that express Arch in the B-type motor neurons (Pacr-5::Arch-mCherry).
Curve was obtained from 551 trials of transient inhibitions of the B-type motor neurons in the head region using 160 worms. Each point represents a single illumination (0.055 s duration, 532 nm …
Figure 3—figure supplement 8
Phase response curve of transgenic worms that express NpHR in the body wall muscles but lack the GABA receptor for the D-type motor neurons (Pmyo-3::NpHR; unc-49(e407)).
Curve was obtained from 259 trials of transient inhibitions of head muscles using 192 worms. Each point represents a single illumination (0.1 s duration, 532 nm wavelength) of one worm. Filled area …
Figure 4 with 1 supplement
Free-running dynamics of a bidirectional relaxation oscillator model.
(A) Schematic diagram of the relaxation oscillator model. In this model, sensory neurons (SN) detect the total curvature of the body segment as well as the time derivative of the curvature. The …
Figure 4—figure supplement 1
Bell-shaped function for modeling the optogenetic muscle inhibition (Equation A14).
The curve models the degree of paralysis due to the optogenetic muscle inhibition as a function of time. Referring to Equation A14, the fractional variable describes the reduced proportion of …
Figure 5 with 1 supplement
Simulations of optogenetic inhibitions in the relaxation oscillator model.
(A) Phase response curves measured from experiments (blue, same as in Figure 3L) and model (orange). Model PRC matches experimental PRC with an MSE ≈ 0.12. (B,C) Simulated dynamics of locomotion …
Figure 5—figure supplement 1
Performance of model oscillators: threshold-switch (column 1), van der Pol (column 2), Rayleigh (column 3), and Stuart-Landau (column 4).
(A-D) Time-varying curvatures of the worm’s head region, measured from experiments (red, 5047 cycles using 116 worms) or produced by models (black). The four models match the experimental curvature …
Figure 6 with 2 supplements
The model predicts phase response curves with respect to single-side muscle inhibitions.
(A) (Upper) a schematic indicating a transient inhibition of body wall muscles of the head on the dorsal side. (Lower) the corresponding PRC measured from experiments (blue, 576 trials using 242 …
Figure 6—figure supplement 1
Paralyzing effect analysis of muscle inhibitions induced by illumination on different sides of the worm’s head segment.
(A) Spectra of paralyzing effects across all phases of illuminations, represented by absolute curvature of the head region. shown on y-axis is the value obtained later in phase (or 0.3 s in …
Figure 6—figure supplement 2
Phase response curves with respect to single-side muscle inhibition, simulated from model oscillators: threshold-switch (column 1), van der Pol (column 2), Rayleigh (column 3), and Stuart-Landau (column 4).
(A-D) PRCs with respect to dorsal-side muscle inhibition, measured from experiments (blue, 576 trials using 242 worms) or produced by models (orange). (E-H) PRCs with respect to ventral-side muscle …
Figure 7
Model reproduces C.elegans gait adaptation to external viscosity.
(A) Dark field images and the corresponding undulatory frequencies and amplitudes of adult worms (left) swimming in NGM buffer of viscosity 1 mPa·s, (right) crawling on agar gel surface. The worm …
Author response image 1
Videos
Video 1
Transient illumination of the anterior region of a freely moving Pmyo-3::NpHR worm.
Green-shaded region indicates timing and location of illumination.
Tables
Key resources table
| Reagent type (species) or resource | Designation | Source or reference | Identifiers | Additional information |
|---|---|---|---|---|
| Strain, strain background (E. coli) | OP50 | CGC | Fang-Yen Lab Strain Collection: OP50 RRID:WB-STRAIN:WBStrain00041971 | OP50 |
| Strain, strain background (C. elegans) | YX148 | Fouad et al., 2018 | Fang-Yen Lab Strain Collection: YX148 | qhIs1[Pmyo-3::NpHR::eCFP; lin-15(+)]; qhIs4[Pacr-2::wCherry] |
| Strain, strain background (C. elegans) | YX119 | Fouad et al., 2018 | Fang-Yen Lab Strain Collection: YX119 | qhIs1[Pmyo-3::NpHR::eCFP; lin-15(+)]; unc-49(e407) |
| Strain, strain background (C. elegans) | YX205 | Leifer et al., 2011 | Fang-Yen Lab Strain Collection: YX205 | hpIs178[Punc-17::NpHR::eCFP; lin-15(+)] |
| Strain, strain background (C. elegans) | WEN001 | Fouad et al., 2018 | Fang-Yen Lab Strain Collection: WEN001 | wenIs001[Pacr-5::Arch::mCherry; lin-15(+)] |
Appendix 1—table 1
Objective functions used in the optimization procedures for alternative models.
| Type | Free locomotion model | Complete model |
|---|---|---|
| van der Pol | ||
| Rayleigh | ||
| Stuart-Landau |
-
Two-step optimization procedure for van der Pol, Rayleigh, and Stuart-Landau oscillators. The first-step optimization determines part of parameters such that individual models generate free locomotion dynamics. The second-step optimization leads to complete models such that models’ perturbed dynamics and phase response curves are produced.
