Signatures of proprioceptive control in Caenorhabditis elegans locomotion

Abstract

Animal neuromechanics describes the coordinated self-propelled movement of a body, subject to the combined effects of internal neural control and mechanical forces. Here we use a computational model to identify effects of neural and mechanical modulation on undulatory forward locomotion of Caenorhabditis elegans, with a focus on proprioceptively driven neural control. We reveal a fundamental relationship between body elasticity and environmental drag in determining the dynamics of the body and demonstrate the manifestation of this relationship in the context of proprioceptively driven control. By considering characteristics unique to proprioceptive neurons, we predict the signatures of internal gait modulation that contrast with the known signatures of externally or biomechanically modulated gait. We further show that proprioceptive feedback can suppress neuromechanical phase lags during undulatory locomotion, contrasting with well studied advancing phase lags that have long been a signature of centrally generated, feed-forward control.This article is part of a discussion meeting issue 'Connectome to behaviour: modelling C. elegans at cellular resolution'.

Keywords: microswimmers; nematodes; neural control; proprioception; undulatory locomotion.

Figure 1.

A continuum neuromechanical model reproduces the swim–crawl transition. (a) Model schematic describing (i) geometry and (ii) neural circuit. (b) Frequency–wavelength relationship (i) plotted for a wide set of drag coefficients, K = K_ν/K_τ, as shown in (ii), including Newtonian environments (red circles). Parameters used for sample kymograms (black circles) span the swim–crawl transition. (c) Curvature kymograms (showing the curvature along the body along the vertical axis from head, at u = 0, to tail, u = 1, and in time) in environments from water (top left) through intermediate fluids (top right to bottom left) to agar (bottom right).

Figure 2.

The effect of environmental drag on (a) undulation frequency, (b) wavelength and (c) the speed of the nematode in Newtonian environments (K = 1.5). Simulation results spanning E-K_τ space are plotted against the non-dimensional parameter e. Outliers represent simulations exhibiting uncoordinated locomotion.

Figure 3.

Internal modulation of proprioceptive pathways gives rise to an inverse frequency–wavelength relationship. Internal neural modulation of (a) proprioceptive threshold and (b) proprioceptive range, for K = 7.5 (circles), K = 12 (squares), K = 18 (diamonds), K = 30 (triangles) and K = 45 (stars), with E = 100 kPa.

Figure 4.

Neuromechanical phase lag along the body, simulated in agar, over the same range of Young's modulus shown in figure 2. (a) Under feed-forward control, lag is negligible for E ≳ 0.5 MPa but increases approximately linearly for lower E. Note that for a constant undulation period T_f the phase lag ϕ and time lag Δ are related by ϕ = 2πT_f/Δ. (b) Under proprioceptive control, the lag is negligible in the anterior half of the body and increases dramatically towards the tail (u = 1).

Figure 5.

Zero contours of the muscle torque, β(u, t) (blue), and body curvatures, κ(u, t) (red), obtained from simulations in an agar-like medium and E = 100 kPa. Both positive and negative zero-crossings are included. (a) Under a model of feed-forward (CPG) control, the latency between activation and body bend grows linearly along the body. (b) Under a model of proprioceptive control with a diminishing proprioceptive range in the posterior half of the body, the curvature and muscle activation are tightly coupled in the anterior half; the increasing phase lag towards the tail arises from an accelerated neuromuscular wave speed of muscle torque in the posterior half of the body. (c) Kymograms of the body curvature corresponding to (b). Black and magenta lines show peak negative and positive values of muscle activation β(u, t), along the body, respectively.