Schr\"odinger Dynamics and Berry Phase of Undulatory Locomotion

Schrödinger Dynamics and Berry Phase of Undulatory Locomotion

Alexander E. Cohen, Alasdair D. Hastewell, Sreeparna Pradhan, Steven W. Flavell, and Jörn Dunkel

Phys. Rev. Lett. 130, 258402 – Published 22 June 2023

Abstract

Spectral mode representations play an essential role in various areas of physics, from quantum mechanics to fluid turbulence, but they are not yet extensively used to characterize and describe the behavioral dynamics of living systems. Here, we show that mode-based linear models inferred from experimental live-imaging data can provide an accurate low-dimensional description of undulatory locomotion in worms, centipedes, robots, and snakes. By incorporating physical symmetries and known biological constraints into the dynamical model, we find that the shape dynamics are generically governed by Schrödinger equations in mode space. The eigenstates of the effective biophysical Hamiltonians and their adiabatic variations enable the efficient classification and differentiation of locomotion behaviors in natural, simulated, and robotic organisms using Grassmann distances and Berry phases. While our analysis focuses on a widely studied class of biophysical locomotion phenomena, the underlying approach generalizes to other physical or living systems that permit a mode representation subject to geometric shape constraints.

Received 2 June 2022
Accepted 30 May 2023

DOI:https://doi.org/10.1103/PhysRevLett.130.258402

Physics Subject Headings (PhySH)

Animal behavior, social interactions & emergent phenomena Biological movement Locomotion

Physics of Living Systems

Authors & Affiliations

Alexander E. Cohen ^1,2, Alasdair D. Hastewell ¹, Sreeparna Pradhan ³, Steven W. Flavell³, and Jörn Dunkel ^1,*

¹Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA
²Department of Chemical Engineering, Massachusetts Institute of Technology, 25 Ames Street, Cambridge, Massachusetts 02142, USA
³Picower Institute for Learning and Memory, Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology, 43 Vassar Street, Cambridge, Massachusetts 02139, USA

^*dunkel@mit.edu

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Vol. 130, Iss. 25 — 23 June 2023

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Figure 1
Chebyshev mode representation enables an efficient and interpretable low-dimensional description of undulatory locomotion across species and model systems. (a) Experimental image of C. elegans worm with center of mass (c.m.) and mean orientation overlayed. (b) Tracked centerline of worm over 100 s. Arrow indicates direction of motion. (c) A small number of Chebyshev polynomials suffices to accurately reconstruct the worm shape (left). Faint colored lines correspond to centerline reconstructions at different polynomial degrees. Reconstruction error (right) decays rapidly as the Chebyshev degree $n$ increases. (d) The zeroth-order Chebyshev coefficients follow closely the worm’s geometric c.m., illustrating the physical interpretability of the Chebyshev mode representation. (e) Similarly, the first-order Chebyshev coefficients represent the tail-to-head worm orientation. (f) The mode-averaged dominant frequency of Chebyshev mode oscillations correlates closely with the locomotion speed of the worm.
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Figure 2
Inferred Schrödinger dynamics replicate stereotypical C. elegans locomotion. (a) Representative real propulsion vector $h_{0}$ and Hamiltonian $H = S + i A$ for a minimal periodic straight-motion model [Eq. (4) with $S = 0$ and equidistant spectrum of $H$ ] fitted to data from a single oscillation period ( $τ = 3.05 s$ ). (b) Kymographs of $x (s, t)$ and $y (s, t)$ coordinate fields for observed data (left) and model prediction (middle) show little deviation (right), confirming that Eq. (4) can accurately capture undulatory shape dynamics of C. elegans. (c) Real-space dynamics predicted by the Schrödinger model (line) is consistent with the observed worm dynamics (circles); see Supplemental Material, Video S1 [51]. Experimental data have been periodically extended for visualization to avoid overlapping body segments. (d) Real-space shape functions [Eq. (5) ] corresponding to the three smallest magnitude eigenvalues, $λ_{k}^{\pm} = \pm k λ$ for $k = 0$ , 1, 2, account for $> 98 %$ of the shape dynamics, enabling a generalizable low-rank description. More complex turning dynamics can be described using time-varying Hamiltonians with unconstrained spectra (Fig. 4 and [51] ).
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Figure 3
Mode-space Hamiltonians provide a compact dynamical description of undulatory motion across different species and model systems. (a) Living and nonliving systems [35, 39] analyzed here and representative straight-motion Hamiltonians $H = i A$ inferred from a single oscillation period. The eigenspaces of the Hamiltonians enable the comparison and classification of undulation dynamics in (d). Scale bars are 8 mm (centipede), 10 cm (snake), 10 cm (toy snake), 0.25 mm (worm model). (b) Inferred Schrödinger model dynamics (line) provide an accurate description of the observed dynamics (circles). Models were fitted on a single period $τ = 0.19 s$ (centipede), 0.33 s (snake), 0.45 s (toy snake), 2.2 s (worm model); see also [51], Video S1. Experimental data have been periodically extended for visualization to avoid overlapping body segments. (c) The dominant shape eigenvectors $v_{1} (s)$ and $w_{1} (s)$ are consistent within each species and capture differences between species. (d) Pairwise Grassmann distances between subspaces spanned by first excited eigenstates of the Hamiltonians (top) and its 2D planar embedding [bottom, constructed by a multidimensional scaling (MDS) ] capture the similarities and differences between undulatory locomotion in organisms, model simulations, and robots. Each point corresponds to a different trajectory.
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Figure 4
Breakdown of adiabaticity during reversal turning behavior of C. elegans. (a) The turning part $S (t)$ of the Hamiltonian $H (t) = S (t) + i A (t)$ becomes switched on at the turn. (b) The turn is signaled by a sudden change in the geometric Berry phase (blue) of the dominant eigenvector [51], and the rms reconstruction error of the adiabatic approximation increases noticeably after the turn (see Supplemental Material [51], Video S2).
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Physical Review Letters

Schrödinger Dynamics and Berry Phase of Undulatory Locomotion

Alexander E. Cohen, Alasdair D. Hastewell, Sreeparna Pradhan, Steven W. Flavell, and Jörn Dunkel

Phys. Rev. Lett. 130, 258402 – Published 22 June 2023

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