From head to tail: a neuromechanical model of forward locomotion in Caenorhabditis elegans

Abstract

With 302 neurons and a near-complete reconstruction of the neural and muscle anatomy at the cellular level, Caenorhabditis elegans is an ideal candidate organism to study the neuromechanical basis of behaviour. Yet despite the breadth of knowledge about the neurobiology, anatomy and physics of C. elegans, there are still a number of unanswered questions about one of its most basic and fundamental behaviours: forward locomotion. How the rhythmic pattern is generated and propagated along the body is not yet well understood. We report on the development and analysis of a model of forward locomotion that integrates the neuroanatomy, neurophysiology and body mechanics of the worm. Our model is motivated by experimental analysis of the structure of the ventral cord circuitry and the effect of local body curvature on nearby motoneurons. We developed a neuroanatomically grounded model of the head motoneuron circuit and the ventral nerve cord circuit. We integrated the neural model with an existing biomechanical model of the worm's body, with updated musculature and stretch receptors. Unknown parameters were evolved using an evolutionary algorithm to match the speed of the worm on agar. We performed 100 evolutionary runs and consistently found electrophysiological configurations that reproduced realistic control of forward movement. The ensemble of successful solutions reproduced key experimental observations that they were not designed to fit, including the wavelength and frequency of the propagating wave. Analysis of the ensemble revealed that head motoneurons SMD and RMD are sufficient to drive dorsoventral undulations in the head and neck and that short-range posteriorly directed proprioceptive feedback is sufficient to propagate the wave along the rest of the body.This article is part of a discussion meeting issue 'Connectome to behaviour: modelling C. elegans at cellular resolution'.

Keywords: invertebrate; locomotion; motor control; neuromechanical model; proprioception.

Figure 1.

Neuromechanical model. (a) Physical model of the body adapted from Boyle et al. [10]: (i) Complete model. Lateral elements are coloured according to the muscles they are driven by. Head and neck muscles are driven by the head motoneuron circuit (grey) (see b(i)). The rest of the body wall muscles are driven by a series of six repeating VNC units (blue, orange, green, red, purple and brown) (see b(ii)). (ii) One of 49 individual segments. Cross-sectional rigid rods (black), damped spring lateral elements (red), damped spring diagonal elements (blue). (b) Neuromuscular model. Dorsal and ventral lateral elements from the physical body represented in grey on the top and bottom, respectively. Dorsal and ventral staggered muscle arrangement. Muscle force is distributed across all lateral elements they intersect. (i) Head neuromuscular unit includes SMD (black) and RMD (grey) motoneurons that connect to muscles on each side. SMD-class neurons receive stretch-receptor input from self and posterior region covered by black process. (ii) One of six repeating VNC neuromuscular units, derived from a statistical analysis of the connectome [3]. Each unit includes one dorsal and two ventral B- (blue) and D-class (magenta) motoneurons that connect to muscles on each side. B-class neurons receive stretch-receptor input from anterior region covered by blue process [6]. Circuits include all chemical synapses (arrows), gap junctions (connections with line endings) and neuromuscular junctions.

Figure 2.

Oscillations in the head motoneuron circuit. (a) Kymogram during normal operation: oscillation originates in the head and travels posteriorly. (b) Kymogram with VNC motoneurons silenced: dorsoventral bends persist in head and neck. (c) Traces from stretch receptors, motoneurons and muscles. Green/red traces dorsal/ventral stretch receptors. Black/brown traces SMD/RMD neural activity. Solid/dashed lines represent dorsal/ventral motoneurons. Blue/orange represents muscle activity from the six head and neck dorsal/ventral muscles. Activity is cyclic so four points are chosen in the cycle (i–iv). (d) Postures at the four instances of time selected in (c). Dorsal/ventral head and neck muscles represented in blue/orange. Dorsal/ventral undifferentiated processes providing stretch information represented in green/red. (e) Mechanics of oscillation. Green bar represents amount of stretch/contraction in the dorsal undifferentiated process with respect to resting state (black vertical line). White arrows represent whether the process is stretching or compressing. Blue rectangle represents the dorsal head and neck muscles. Only dorsal muscles and stretch receptors are shown. The circles below represent the motoneurons. Muscles/neurons are filled in with colour when they are contracted/activated and no colour when they are relaxed/inactivated. The shade of grey represents the SMD neuron mid-activation. SMD motoneurons are shown in black and RMD motoneurons are shown in brown. Synapses appear only when they are in use.

Figure 3.

Wave propagation through stretch reception. (a) Traces from the dorsal stretch receptors (green), DB motoneurons (black) and dorsal muscles (blue) in two neighbouring VNC neural units: second (solid) and third (dashed). The activity is cyclic so the same four unique points used for figure 2 were chosen to analyse the wave propagation: i–iv (vertical dashed lines). (b) Worm postures at the four instances of time selected in a. The second VNC neural unit receives dorsal stretch-receptor input from the solid green region and innervates the muscles in the solid blue region. The third VNC neural unit (posterior to the second) receives dorsal stretch-receptor input from the dashed green region and innervates the muscles in the dashed blue region.

Figure 4.

Role of biomechanics in the propagation of the wave and locomotion. (a) Speed of the worm as a result of silencing entire VNC neural units. Colour coding according to the region of the body those neural units affect. Black represents the speed of the model worm under normal conditions. Propagation of the wave does not depend entirely on stretch-receptor feedback and neural activity in general. (b) Example kymogram of movement while two VNC neural units (2 and 3) have been silenced. Despite the lack of neural activity, and the lack of network oscillators in the tail, there are oscillations in the head and tail.

Figure 5.

Operation of the head motoneuron circuit in the ensemble of solutions. Distribution of speed (a) and magnitude of change in neural activity in head motoneurons (b) of all model worms in the ensemble under different conditions: normal locomotion (blue), when head motoneurons are silenced (orange), when VNC motoneurons are silenced (green), when head stretch-receptor feedback is silenced (red).

Figure 6.

Operation of the VNC in the ensemble of solutions. Distribution of speed (a) and magnitude of dorsoventral bends (b) of all model worms in the ensemble under different conditions: normal locomotion (black), when VNC stretch-receptor feedback, interunit gap junctions, B-class and D-class motoneurons are silenced independently (grey), and when an entire neural unit is silenced (coloured according to position along the body). The black dashed lines represent the value expected of a normally moving model worm; the grey dashed line represents the value expected of a non-moving model worm.