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Capturing the continuous complexity of behaviour in Caenorhabditis elegans

Nature Physics volume 17, pages 275–283 (2021)Cite this article

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Abstract

Animal behaviour is often quantified through subjective, incomplete variables that mask essential dynamics. Here, we develop a maximally predictive behavioural-state space from multivariate measurements, in which the full instantaneous state is smoothly unfolded as a combination of short-time posture sequences. In the off-food behaviour of the roundworm Caenorhabditis elegans, we discover a low-dimensional state space dominated by three sets of cyclic trajectories corresponding to the worm’s basic stereotyped motifs: forward, backward and turning locomotion. We find similar results in the on-food behaviour of foraging worms and npr-1 mutants. In contrast to this broad stereotypy, we find variability in the presence of locally unstable dynamics with signatures of deterministic chaos: a collection of unstable periodic orbits together with a positive maximal Lyapunov exponent. The full Lyapunov spectrum is symmetric with positive, chaotic exponents driving variability balanced by negative, dissipative exponents driving stereotypy. The symmetry is indicative of damped–driven Hamiltonian dynamics underlying the worm’s movement control.

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Fig. 1: State-space reconstruction.
Fig. 2: Reconstruction parameters for maximum predictability.
Fig. 3: The low-dimensional state space of C. elegans locomotion.
Fig. 4: Chaotic state-space dynamics reveal a deterministic component of behavioural variability.
Fig. 5: A symmetric Lyapunov spectrum indicates damped–driven Hamiltonian dynamics.

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Data availability

Posture-mode time series for all worms analysed here are publicly available: https://bitbucket.org/tosifahamed/behavioral-state-space. Original image data for the foraging and escape-response datasets were analysed previously37 and are available from the Dryad Digital Repository: https://doi.org/10.5061/dryad.t0m6p. Data for N2 worms on food and npr-1 mutants were collected from an open-access dataset84 and analysed to solve for coiled postures11. All other data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

Code availability

Code for all analysis reported here was written in MATLAB83 and is publicly available: https://bitbucket.org/tosifahamed/behavioral-state-space.

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Acknowledgements

We thank D. Jordan, I. Etheredge and A. Celani for comments. L. Hebert (OIST Graduate University) developed the custom machine-learning solution for pose estimation of worms in on-food conditions. We would also like to express our gratitude to I. Maruyama for his support during the project. This work was supported by OIST Graduate University (T.A., G.J.S.), a programme grant from the Netherlands Organization for Scientific Research (A.C.C., G.J.S.), Vrije Universiteit Amsterdam (G.J.S.) and the Japan Society for the Promotion of Science (T.A.).

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Authors and Affiliations

  1. Biological Physics Theory Unit, OIST Graduate University, Okinawa, Japan

    Tosif Ahamed & Greg J. Stephens

  2. Department of Physics and Astronomy, Vrije Universiteit Amsterdam, Amsterdam, The Netherlands

    Antonio C. Costa & Greg J. Stephens

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  1. Tosif Ahamed
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T.A., A.C.C. and G.J.S. designed the research, performed the research, analysed the data and wrote the paper.

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Correspondence to Greg J. Stephens.

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Extended data

Extended Data Fig. 1 The predictability Tpred as a function of K and m for N = 12 individual worms.

a, Tpred as a functon of K. b, Tpred as a function of m. We find similar curves across worms, despite the differences in their detailed dynamics. Note that while the distance metric in the SVD space (b) and the space of delays (a) is different, which could result in inconsistencies in the estimation of Tpred, we find only minor differences between the maximum Tpred in the two cases (gray bar in B). Inset shows the normalized singular value spectrum, which does not have a clear cutoff for any worm. c, Expanded inset in B showing the normalized singular value spectrum. d, State space captures a commonly observed sequence where long reversals transition to forward via a deep body bends seen here as a large excitation in Xt2 as the reversal ends. Here we see that the blue (backward) and red (forward) bundles are smoothly connected via a large transient along the turning mode \({X}_{{t}_{2}}\) (data from the example worm in Fig. 3b–e).

Extended Data Fig. 2 Dominant state space modes are stable across different embedding dimensions, with distinct groups independently capturing forward, reversal and turning behaviors.

a–c, Behavioral modes estimated for the worm in Fig. 3 for dimensions, m = 6 (a), m = 7 (b), and m = 8 (c); embedding window is set to K = 12 frames. The modes retain their interpretability across dimensions. In a 6-dimensional embedding, there are two forward, two backward and two turning modes. In 7 dimensions one of the turning modes further splits into an omega-turn like mode (\({\Gamma }_{{t}_{3}}\)) and a delta-turn like mode (\({\Gamma }_{{t}_{1}}\)), while \({\Gamma }_{{t}_{2}}\) changes little. Furthermore, the reversal modes are more separable in 7 dimensions. The 8-dimensional state space retains the forward, reversal and turning dynamics along with an additional and subtle head-bending.

Extended Data Fig. 3 The ensemble embedding across all N = 12 worms is constructed from their concatenated posture time series and characterized by \(K^*=10\) and \(m^*=7\).

a–b, Tpred as a function of K and m. We set \(K^*=10\), approximately when Tpred begins to decrease, and show Tpred(m) at this \(K^*\). We show the resulting modes for m = 6 and the gray bar denotes \(T_{pred}(K = K^*)\). c, m = 7 (d), and m = 8 (e), and these are qualitatively similar to those obtained from our representative worm of Extended Data Fig. 2. The additional modes present for embeddings greater than \(m^*=7\) offer only minor improvements in predictability.

Extended Data Fig. 4 Ensemble embedding for different values of K and m.

a–c, Behavioral modes estimated from the ensemble for K = 5, and dimensions, m = 6 (a), m = 7 (b), and m = 8 (c). d–f, Same as above but for K = 15. The modes are qualitatively similar across this variation.

Extended Data Fig. 5 The dominant off-food modes are similar to those of on-food roaming behavior and on-food behavior of npr-1 mutant worms.

We analyze a collection of N = 25 on-food ‘roaming’ N2 worms, N = 25 on-food ‘dwelling’ N2 worms, and N = 7 on-food mutant npr-1 worms from an open access dataset84 (Methods, see also Extended Data Fig. 10a). We show Tpred(m) for (a) on-food N2 roaming worms and (b) on-food npr-1 mutants. c–d, Kymographs of the \(m^*=7\) primary modes from roaming and npr-1 worms coincide, both with each other and with the off-food N2 modes in Fig. 3. The similarity of these embeddings provide new, posture-scale evidence that the NPR-1 mutation overrides the switch to dwelling45. e–f, The combined embedding of roaming-dwelling on-food behavior exhibits an additional ~ 6 modes with small but notable additional Tpred, which was also observed in off-food behavior in the ensemble embedding, Extended Data Fig. 3, and for some individual worms, Extended Data Fig. 1b.

Extended Data Fig. 6 Maximum Lyapunov exponent for different worms and the full Lyapunov spectrum in different embedding dimensions.

a, To quantify the state space divergence we plot the logarithm of the average distance between a trajectory and its nearest neighbors, averaged over several starting reference trajectories. For each worm we find that, after a transient, there is linear region showing exponential divergence. The slope of the linear region provides an estimate of the maximal Lyapunov exponent λmax and the positive exponents are an indication of chaos in worm behavior. b–d, Lyapunov spectra computed from reconstructions of worm behavior in different embedding dimensions. Conjugate pairing of Lyapunov exponents is robustly observed in dimensions 6 and above.

Extended Data Fig. 7 Detecting periodic orbits in the Lorenz System.

a–e, We compare UPO trajectories for the Lorenz system computed from high precision numerical estimates52 (red) with periodic orbits detected using our recurrence based approach (grey) (only 3 UPOs are shown for period 5). The closely-matching trajectories also exhibit agreement between the Floquet exponents estimated from analytical Jacobians (red text) and Local Lyapunov exponents obtained from the estimated Jacobians (grey text). Note that in (e), a fixed point (period-0 orbit) can only be detected by neighboring spiraling trajectories, leading to an overestimation of the exponent. f, Boxplot comparing the entire distribution of Floquet exponents for UPOs up to period 10 (red bars denote the median).

Extended Data Fig. 8 Recurrence Function and Period-1 UPOs.

a, The recurrence function ϵ(r, t) from the same worm in Fig. 3 for 120 frames and 5000 closest recurrences (top). Local minima of this function at times t*, as seen in the average 〈ϵ(r, t)〉r shown below correspond to close recurrences and identify periodic orbits of length t*. The first local minimum is the smallest period pmin, which is 37 frames in this example. For a given value of r, ϵ(r, t*) gives the distance threshold at which we must look to find a periodic orbit of length t*. b, Probability distribution of phase velocities \(\dot{\phi }\) and third eigenworm coefficient a3, which is proportional to mean body curvature, across all period-1 orbits of duration pmin from all worms in the dataset. We see two clusters corresponding to forward and backward locomotion, as well as orbits with a dorsal or ventral bias (for example orbits at bottom right and top left). c, Example period-1 orbits from the same worm in (a) corresponding to forward (top) and backward (bottom) locomotion.

Extended Data Fig. 9 Example E(τ) curves.

a, Error curves are plotted for different embedding dimensions for the Lorenz system state space reconstruction \((K^* = 25)\). b, Error curves in different embedding dimensions for the sample worm in Fig. 3b–e. Better embeddings lead to a lower error curve. The ratio of the area between these curves and the saturation value es to es estimates Tpred. c, A schematic showing the fixed point algorithm for robust estimation of the asymptote es, and the area Δ.

Extended Data Fig. 10 Roaming/dwelling states and further embedding details.

a, Centroid speed and angular speed (averaged in 10 s windows) for the collection of worms used in the N2 on food dataset (red and blue) and the npr-1 dataset (black).We initially collected 150 recordings of N2 worms crawling on food-full plates from an open dataset84. From these, we selected 25 worms with a large fraction of dwelling states (blue) and 25 worms with a large fraction of roaming states (red), defined as in17. We downsample the data to 3 Hz (consistent with17), and average the centroid speed and angular speed in 10 s windows. Roaming and dwelling states are identified by a threshold defined by the line y = x/450 in the plane defined by x, the angular speed, and y, the centroid speed. Points above the line (high speed and low angular speed) are classified as roaming, while points below the line (low speed and high angular speed) are classified as dwelling. Data from npr-1 mutants (black) show predominantly roaming behavior, consistent with previous reports45. b, Example of the one step error, E(1, Nb) curve used to pick the number of nearest neighbors. This was calculated on the same worm as in Fig. 3b–e. c, A transient can be seen For the Lorenz system before the linear regime indicating exponential growth of local finite-sized perturbations (sized ≈ 10−4) begins. d, Transient decreases when perturbations of size ≈ 10−8 are used.

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Ahamed, T., Costa, A.C. & Stephens, G.J. Capturing the continuous complexity of behaviour in Caenorhabditis elegans. Nat. Phys. 17, 275–283 (2021). https://doi.org/10.1038/s41567-020-01036-8

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