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. 2011 May 3;108(18):7286-9.
doi: 10.1073/pnas.1007868108. Epub 2011 Apr 18.

Emergence of long timescales and stereotyped behaviors in Caenorhabditis elegans

Greg J Stephens  1 , Matthew Bueno de Mesquita, William S Ryu, William Bialek
Affiliations

Emergence of long timescales and stereotyped behaviors in Caenorhabditis elegans

Greg J Stephens et al. Proc Natl Acad Sci U S A. .

Abstract

Animal behaviors often are decomposable into discrete, stereotyped elements, well separated in time. In one model, such behaviors are triggered by specific commands; in the extreme case, the discreteness of behavior is traced to the discreteness of action potentials in the individual command neurons. Here, we use the crawling behavior of the nematode Caenorhabditis elegans to demonstrate the opposite view, in which discreteness, stereotypy, and long timescales emerge from the collective dynamics of the behavior itself. In previous work, we found that as C. elegans crawls, its body moves through a "shape space" in which four dimensions capture approximately 95% of the variance in body shape. Here we show that stochastic dynamics within this shape space predicts transitions between attractors corresponding to abrupt reversals in crawling direction. With no free parameters, our inferred stochastic dynamical system generates reversal timescales and stereotyped trajectories in close agreement with experimental observations. We use the stochastic dynamics to show that the noise amplitude decreases systematically with increasing time away from food, resulting in longer bouts of forward crawling and suggesting that worms can use noise to modify their locomotory behavior.

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Conflict of interest statement

The authors declare no conflict of interest.

Figures

Fig. 1.
Fig. 1.
Reversals in shape space correspond to reversals in the crawling direction. (A) Tracking video microscopy gives both the x-y trajectory of the worm as it crawls on an agar plate, and the shape of the worm’s body at high resolution. (B) Shape is described by the tangent angle θ vs. arc length s, in intrinsic coordinates such that ∫dsθ(s) = 0. (C) We decompose θ(s) into four dominant modes. (D) The joint probability density of the first two modes. Amplitudes along the first two modes oscillate, with nearly constant amplitude but time varying phase ϕ = tan-1(a2/a1); here the amplitudes are normalized so that formula image. (E) The phase trajectories exhibit abrupt reversals, moments when ω ≡ dϕ/dt change sign. The red cross marks the onset of a body wave reversal and the green and magenta dots mark times prior to and during a reversal. These same times are also marked in A demonstrating that phase reversals correspond to reversals in the crawling direction.
Fig. 2.
Fig. 2.
The Langevin model for the phase dynamics, Eqs. 3 and 4, reveals discrete attractors and noise-induced transitions between them. (A) The deterministic component of the force F(ω,ϕ), in units cycles/s2. The black lines are attracting limit cycles corresponding to forward and backward crawling, and the white dashed lines mark boundaries for our analysis of trajectories that start within the forward attractor. (B) The noise strength σ(ω,ϕ), in units cycles/s3/2. (C) A sample of the trajectories resulting from Eqs. 3 and 4, illustrating transitions between attractors at positive and negative ω, corresponding to forward and backward crawling.
Fig. 3.
Fig. 3.
Forward crawling survival times are well captured by noise-induced transitions in the model phase dynamics. (A) The distribution of survival times measured from worm data. We measure the probability that a worm’s trajectory, which is in the neighborhood of the forward attractor at time t, has not crossed to negative phase velocity by time t + τ. The decay is exponential, with a mean time 〈τ〉 = 16.3 ± 0.3 s. (B) The predicted mean time 〈τ〉theory as a function of the noise level. We scale the strength of the noise σ2 by a factor 1/β and solve Eqs. 3 and 4 for many noise realizations. The noise at β = 1 corresponds to the strength derived from actual worm motion and the average survival time at the measured noise level (〈τ〉theory = 15.7 ± 1.3 s) is in close agreement with worm data. In the low-noise limit (β≫1) we find 1/〈τ〉theory ∝ exp(-βE) (blue line), analogous to the Arrhenius temperature dependence of chemical reaction rates. Inset shows the region near β = 1 and the red point marks the measured 1/〈τ〉data. The red error bar denotes the bootstrap error in the noise strength, β = 1 ± 0.05.
Fig. 4.
Fig. 4.
The emergence of stereotyped behaviors in worm and model phase dynamics. (A) The conditional density ρ(ϕ|t) constructed from an ensemble of N = 469 worm trajectories aligned to exit the forward attractor at t = 0 via a path with ϕ(0) < 0. Color scale is for ln[ρ·(1 rad)]. (B) The same density generated from simulations of the stochastic model, Eqs. 3 and 4.
Fig. 5.
Fig. 5.
The changes in the stochastic dynamical system, Eqs. 3 and 4, as a consequence of increasing time away from food (A, B) and with genetic perturbations (C, D). (A) The mean deterministic force F(ϕ,ω) along escape trajectories derived from wild-type worms in early (black), middle (red), and late (blue) epochs. Units are cycles/s2 and differences among the three 700 s epochs are relatively small. Within each epoch we fit the stochastic model and generate N = 104 trajectories with initial conditions in the forward crawling attractor. As before, we evolve each trajectory until a phase reversal. The escape trajectories are aligned to the moment of the reversal (t = 0) and errors denote standard errors in the mean. (B) The mean noise amplitude σ2(ϕ,ω) along escape trajectories in early, middle and late epochs. Units are cycles2/s3. The mean noise amplitude systematically decreases resulting in longer times within the forward crawling state. (C, D) The mean deterministic force and noise amplitude derived from a goa-1 mutant during the early 700 s epoch. For comparison we also show the wild-type (N2) dynamics from the same early epoch. Befitting the general nature of the goa-1 gene, the mutant dynamics reveal substantial changes to both the deterministic force and the noise.
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