(A) Raw image in the tracking microscope. (B) The curve through the center of the body. The black circle marks the head. (C) Distances along the curve (arclength s) are measured in normalized units, and we define the tangent (s) and normal (s) to the curve at each point. The tangent points in a direction θ(s), and variations in this angle correspond to the curvature κ(s) = dθ(s)/ds. (D) All images are rotated so that 〈θ〉 = 0; therefore θ (s) provides a description of the worm's shape that is independent of our coordinate system, and intrinsic to the worm itself.

(A) The distribution of amplitudes ρ(a₃), shown on a logarithmic scale. Units are such that , and for comparison we show the Gaussian distribution; note the longer tails in ρ(a₃). (B) Images of worms with values of a₃ in the negative tail (left), the middle (center) and positive tail (right). Large negative and positive amplitudes of a₃ correspond to bends in the dorsal and ventral direction, respectively. (C) A two minute trajectory of the center of mass sampled at 4 Hz. Periods where |a₃|>1 are colored red, illustrating the association between turning and large displacements along this mode.

(A) The mean acceleration of the phase F(ω,φ) in Equation 4. (B) The correlation function of the noise 〈η (t)η(t+τ)〉. The noise correlations are confined to short times relative to the phase velocity itself. (C) Trajectories in the deterministic dynamics. A selection of early-time trajectories is shown in black. At late times these same trajectories collapse to one of four attractors (red): forward and backward crawling and two pause states. (D) Joint density ρ(ω,φ) for worms sampled at 32 Hz. A sample trajectory of a single worm moving forwards, backwards, and pausing, is denoted by black arrows.

(A) The joint probability density of the first two amplitudes, ρ(a₁, a₂), with units such that . The ring structure suggests that these modes form an oscillator with approximately fixed amplitude and varying phase φ = tan^-1(-a₂/a₁). (B) Images of worms with different values of φ show that variation in phase corresponds to propagating a wave of bending along the worm’s body. (C) Dynamics of the phase φ(t) shows long periods of linear growth, corresponding to a steady rotation in the {a₁, a₂} plane, with occasional, abrupt reversals. (D) The joint density ρ(|ν|,|ω|). The phase velocity ω = dφ/dt in shape space predicts worm’s crawling speed.

(A) The covariance matrix of fluctuations in angle C(s, s′). The inhomogeneity along the diagonal shows that the normal modes of the motion are not sinusoidal but the smooth structure of C(s, s′) means that a small number of modes are significant. (B) We find the eigenvalues of C(s, s′) and compute σ²_K, the fraction of the total variance (integrated along the body of the worm) captured by keeping K modes (see Materials and Methods). (C) Associated with each dominant mode is an eigenvector and we refer to these as eigenworms u_μ(s). The population-mean eigenworms (red) are highly reproducible across individual worms (black). (D) The fraction of variance, σ̃²_K, at each point along the body curve captured by keeping K modes (K = 1 to 4, from bottom to top curve). The overall error in reconstruction of the worm body curve decreases as the number of modes increases, but does so inhomogeneously. (E) In response to strong thermal stimuli, reconstructions using the eigenworms of spontaneous crawling continue to account for most of the shape variance. Worm images are recorded at times synchronized to a heat pulse and we display σ²_K aligned with this pulse (red line). (K = 1 to 4, from bottom to top curve).

(A) The distribution of phase velocities ρ_t(ω) in response to a brief thermal stimulus. Within one second, the distribution becomes highly concentrated near ω = 0, corresponding to the pause states identified in Figure 5. (B) Correlations between phase in the {a₁,a₂} plane and a₃, . Shortly after the thermal impulse (t, t′>0) the modes develop a strong anti-correlation which is distinct from normal crawling. (C) Phase dependent thermal response. Worms stimulated during ventral head swings (−2≤φ≤−1) turn dorsally (red) while worms stimulated during dorsal head swings (2≤φ≤π) turn ventrally (blue). When phase is ignored there is no discernible response (grey). Solid lines denote averages while colored bands display standard deviation of the mean. (D) Worm “steering.” A thermal impulse conditioned on the instantaneous phase was delivered automatically and repeatedly, causing an orientation change in the worm's trajectory. In this example lasting 4 minutes, asynchronous impulses produced a time-averaged orientation change 〈〉 = 0.01 rad/s (black), impulses at positive phase produced a trajectory with 〈〉 = 0.10 rad/s (blue), and impulses at negative phase produced 〈〉 = –0.12 rad/s (red). This trajectory response is consistent with the mode correlations seen in Figure 6C. We found 13 out of 20 worms produced statistically different orientation changes under stimulated and non-simulated conditions while only 1 out of 20 worms responded in the same fashion when the phase was randomized (p<0.01, Fisher exact test).