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Dimensionality and dynamics in the behavior of C. elegans
Greg J Stephens, Bethany Johnson-Kerner, William Bialek, William S Ryu
TL;DR
The paper addresses the challenge of quantitatively describing complex naturalistic animal behavior. It analyzes C. elegans in a low-dimensional shape space and reconstructs part of its dynamics, finding reproducible thermal responses that can be steered by synchronizing stimuli to worm state.
Problem
The paper asks how complex, naturalistic motor behavior can be characterized quantitatively when existing approaches emphasize either controlled choices or behavioral richness.
Method
The authors use video microscopy and eigenworm coordinates to represent worm shapes, reconstruct stochastic equations of motion, and test synchronized thermal stimulation.
Results
Four dimensions account for 95% of shape variance, while nonlinear coupled dynamics produce reproducible thermal responses and permit real-time steering into trajectories with definite chirality.
Takeaways & Limitations
Behavioral coordinates can reveal deterministic components hidden by unmeasured variables, and phase-aligned stimuli can reduce apparent response randomness within this system.
Takeaways & Limitations
The study does not derive an equation of motion for the turning mode a3 alone or a fully three-dimensional dynamics predicting more complex correlations and longer stereotyped sequences.
Abstract
from arXiv · showhide
A major challenge in analyzing animal behavior is to discover some underlying simplicity in complex motor actions. Here we show that the space of shapes adopted by the nematode C. elegans is surprisingly low dimensional, with just four dimensions accounting for 95% of the shape variance, and we partially reconstruct "equations of motion" for the dynamics in this space. These dynamics have multiple attractors, and we find that the worm visits these in a rapid and almost completely deterministic response to weak thermal stimuli. Stimulus-dependent correlations among the different modes suggest that one can generate more reliable behaviors by synchronizing stimuli to the state of the worm in shape space. We confirm this prediction, effectively "steering" the worm in real time.
INTRODUCTION
The study seeks a quantitative account of complex animal behavior that preserves naturalistic richness while identifying underlying simplicity. Using freely moving C. elegans, it finds a low-dimensional motor description with reproducible behavioral states and stimulus-dependent steering.
- Animal behavior research balances controlled choice experiments against descriptions of rich behavior in natural contexts.
- C. elegans provides a model system for quantitatively studying complex spontaneous motor behavior in relatively simple conditions.The worms move freely on agar, where their behavior is modulated by chemical, thermal, and mechanical stimuli.
- A low-dimensional shape space captures the worm’s macroscopic motor behavior and supports reconstructed equations of motion with multiple attractors.These attractors are candidates for a rigorous definition of behavioral states.
- Small temperature changes evoke surprisingly reproducible transitions among behavioral states, while correlations among dimensions enable real-time steering with synchronized stimuli.
EIGENWORMS
The authors represent worm posture intrinsically through body-centered angle profiles and reduce these shapes to reproducible eigenworm modes. Four modes capture 95% of shape variance, including during strong thermal responses, providing a compact coordinate system for motor behavior.
- EIGENWORMS: High-resolution tracking microscopy extracts each freely moving worm’s two-dimensional body shape over long periods.The body centerline is sampled at N = 100 equally spaced points along normalized arclength.
- EIGENWORMS: Rotated tangent-angle profiles θ(s) describe posture intrinsically, avoiding arbitrary dependence on the worm’s coordinate position and orientation.This representation contains the same information as curvature while avoiding the noise introduced by taking two derivatives.
- EIGENWORMS: Over 95% of total angle variance is captured by just four eigenvalues, revealing a strongly low-dimensional shape space.The smooth covariance structure indicates that only a small number of modes are significant.
- EIGENWORMS: The four eigenvectors define eigenworm shapes, whose amplitudes provide coordinates for reconstructing posture and are highly reproducible across individual worms.
- EIGENWORMS: Four modes continue to account for approximately 95% of shape variance during strong heat-evoked responses, despite stimulus-dependent changes in shape distributions.Thus the same eigenworm coordinates describe both spontaneous crawling and thermal responses.
- EIGENWORMS: The first two modes form a quadrature pair whose phase describes a propagating body-bending wave and whose rotation speed predicts crawling speed.The amplitude distribution forms a nearly constant-radius ring, while abrupt phase reversals accompany changes in motion.
WHAT DO THE MODES MEAN?
The first two modes encode crawling as phase rotation, while the third mode captures turning through large-curvature shapes and trajectories.
- WHAT DO THE MODES MEAN?: The first two modes form a quadrature pair whose mixtures represent different phases of a traveling body wave.Their amplitude distribution forms a nearly constant-radius ring.
- WHAT DO THE MODES MEAN?: The phase-rotation speed predicts the worm’s crawling speed.Forward and backward crawling correspond to clockwise and counterclockwise rotation in the a1-a2 plane.
- WHAT DO THE MODES MEAN?: Large a3 amplitudes produce Ω-like body shapes and coincide with high-curvature regions of the worm’s trajectory.The a3 amplitude distribution has longer tails than a Gaussian, and |a3| > 1 marks turning-associated displacements.
- WHAT DO THE MODES MEAN?: The eigenworm coordinates connect posture directly to movement without using the worm’s external position or orientation.Turning is identified through movement toward larger a3 magnitudes rather than predefined discrete turning events.
ATTRACTORS AND BEHAVIORAL STATES
The worm’s crawling phase can be modeled with deterministic and stochastic components, revealing four attractors that organize continuous motion into behavioral states.
- ATTRACTORS AND BEHAVIORAL STATES: The phase dynamics model mean acceleration as a function of phase and phase velocity, with state-dependent noise representing unobserved influences.The noise strength may vary with the system state, analogous to a position-dependent temperature.
- ATTRACTORS AND BEHAVIORAL STATES: The noise correlation time is short, separating deterministic average motion from rapidly fluctuating jitter and transitions between modes.The deterministic component captures prolonged constant oscillation in the a1-a2 plane.
- ATTRACTORS AND BEHAVIORAL STATES: Asymmetric attracting basins mean that some initially forward-moving states converge to reversal or pause attractors.This defines behavioral states by long-term dynamics rather than instantaneous phase velocity alone.
- ATTRACTORS AND BEHAVIORAL STATES: Four deterministic attractors correspond to forward crawling, backward crawling, and two stationary pause states.Different initial phase and phase-velocity conditions converge to one of these possibilities at late times.
PAUSE STATES AND REPRODUCIBILITY
Brief, weak thermal pulses drive forward-crawling worms rapidly and reproducibly toward a pause state, making the response detectable in single trials.
- PAUSE STATES AND REPRODUCIBILITY: Brief 75 ms temperature changes of approximately 0.1°C elicit a rapid response while remaining below the pain-avoidance threshold.The stimuli were delivered using infrared-laser pulses to worms crawling forward at stimulation onset.
- PAUSE STATES AND REPRODUCIBILITY: Within one second, phase velocities become concentrated near zero, corresponding to the pause states in the reconstructed dynamics.The initial velocities span a wide range of positive values because all worms were moving forward before stimulation.
- PAUSE STATES AND REPRODUCIBILITY: Pause-state arrival is stereotyped across both trials and worms.This reproducibility contrasts with a response description based only on probabilistic turning or reversing.
- PAUSE STATES AND REPRODUCIBILITY: A single measurement of phase velocity after the pulse detects the small temperature change with approximately 75% correct accuracy in single trials.The result uses the worm’s response as a psychophysical signal.
COUPLING THE MODES AND STEERING THE WORM
Although the shape modes are linearly uncorrelated instantaneously, their dynamics couple after thermal stimulation, allowing phase-synchronized pulses to steer the worm’s orientation.
- COUPLING THE MODES AND STEERING THE WORM: Thermal stimulation creates a strong, stimulus-dependent anticorrelation between phase in the a1-a2 plane and the turning mode a3.This coupling differs from normal crawling and makes the turning response depend on the worm’s phase at stimulation.
- COUPLING THE MODES AND STEERING THE WORM: Asynchronous pulses produced an average orientation change of 0.01 rad/s, compared with 0.10 rad/s at positive phase and −0.12 rad/s at negative phase.These measurements were collected during a four-minute steering example.
- COUPLING THE MODES AND STEERING THE WORM: Stimulus timing strongly affected steering reliability: 13 of 20 worms differed under stimulated versus nonsimulated conditions, versus 1 of 20 with randomized phase.The trajectory response agrees with the early-time mode correlations.
- COUPLING THE MODES AND STEERING THE WORM: Phase-conditioned thermal pulses predictably bias turning direction because opposite target phases produce opposite a3 biases.The thermal pulse itself has no handedness, so unsynchronized pulses should not systematically favor left or right turns.
DISCUSSION
C. elegans motor behavior is captured by a low-dimensional shape space whose coupled dynamics link continuous posture changes to discrete behavioral states. Phase-aligned thermal stimuli can make responses more deterministic, while a complete dynamical account of turning remains unresolved.
- DISCUSSION: Four eigenworms provide a low-dimensional description whose continuously varying amplitudes combine and generalize discrete behavioral templates.The representation is intrinsic and invariant to center-of-mass position, yet motion in shape space predicts center-of-mass movement.
- DISCUSSION: Single motor actions coordinate multiple degrees of freedom because mode dynamics are nonlinear and coupled.Forward and backward motion correspond to opposite phase velocities, while turns involve coordinated excursions involving a3 and the wriggling modes a1 and a2.
- DISCUSSION: A complete three-dimensional dynamical model remains unavailable because strong coupling prevents isolating an equation of motion for the turning mode a3.The authors identify predicting longer stereotyped sequences such as pirouettes as a further challenge.
- DISCUSSION: Phase-aligned thermal stimuli steer worms into trajectories with definite chirality by exploiting nonlinear correlations among behavioral variables.Although the temperature stimulus is scalar, its correlation with body shape gives the response a definite spatial structure.
- DISCUSSION: The approach may reveal deterministic components of sensory-motor responses in other model organisms and help characterize behavior alongside molecular and circuit mechanisms.The authors frame this as a step toward reducing the imbalance between probing behavioral mechanisms and quantitatively characterizing behavior.
METHODS
The study combines high-resolution worm tracking, eigenworm-based shape analysis, dynamical modeling, and thermal-stimulation experiments to characterize and manipulate C. elegans behavior.
- Data acquisition: Tracking microscopy recorded freely moving worms and used image processing with a motorized stage to maintain observations in the field of view.Images were acquired at up to 32 Hz, while center-of-mass tracking and stage control operated at 4 Hz.
- Shape representation: Worm postures were represented by 100 backbone angles after spline fitting, arclength discretization, and removal of overall rotation.The shape covariance matrix was computed from nine freely crawling worms sampled at 4 Hz for 30 minutes.
- Dimensionality reduction: Eigenworms were obtained as covariance-matrix eigenvectors, and worm phase was calculated from the normalized first two mode amplitudes.The phase was defined as φ = tan−1 (−a2/a1).
- Dynamical modeling: The equations of motion used 32-Hz shape data, polynomial filtering, and directly learned functions F(φ, ω) and σ(φ, ω).F was estimated as the conditional mean acceleration ⟨ω̇|ω, φ⟩ using a functional expansion, with parameters selected by minimizing held-out error.
- Thermal perturbation analysis: Thermal responses were elicited with laser pulses whose beam size, power, duration, and temperature increase were controlled experimentally.Stimulus-dependent correlations were analyzed by comparing post-stimulus and around-stimulus residual correlation matrices using singular value decomposition.
- Thermal steering: Real-time steering computed eigenworms and phase at 8 Hz, firing the laser within a prescribed phase interval during forward motion.Turn effects were quantified from center-of-mass orientation changes, excluding large turns and reversals using compactness-based detection.