Research - Neurolocomotion in C. elegans

Using mathematical models to understand neurolocomotion and gait adaptation in the nematode C. elegans.


A central question of neuroethology is how the body and nervous system of an organism interact with its environment to produce complex behaviors like locomotion. This involves the synthesis of both neural and bio-mechanical dynamics in a closed environment-brain-body-behavior loop. The nematode C. elegans is an ideal organism for studying these closed-loop interactions, because of its well-described nervous system of only 302 neurons and dependence on sensory feedback for its undulatory forward gait.

Gait Adaptation

While the motor circuit responsible for forward locomotion has been determined by White et al. in 1986, the exact mechanisms by which neural and mechanical feedback produce coordinated locomotion are not well-understood. Through experiments [Berri et al. (2009); Sznitman et al. (2010); Fang-Yen et al. (2010)] and computational models [Boyle et al. (2012)], C. elegans has been shown to adapt its undulatory gait to fluid environments of different viscosities (see figure at the right), but how it adapts its gait is largely unknown. I developed a mathematical model of the essential elements of the C. elegans forward locomotion system and uncovered the mechanisms underlying gait adaptation. This mechanistic understanding sheds light on the general principles of how a nervous system interacts and coordinates with the outside world.

Neuromechanical Model

The neuromechanical model I developed describes the motor circuit, post-synaptic body wall muscles, and the resulting body shapes of C. elegans. The model is a chain of coupled neuromechanical oscillators, a schematic of which is shown at the left. The modular structure of this model allows us to disentangle the individual nature of the neuromechanical oscillators from the coordinating effects of the passive mechanical body and the active intermodular connections.

Phase Model

Each neuromechanical oscillator module consists of a repeated neural subcircuit, the post-synaptic body wall muscles, the resulting body shape of the post- synaptic body segment, and local proprioception (a neural mechanism that translates physical body in- formation into a signal). Using the theory of weakly coupled oscillators, we reduced the model down to a phase model to separate the oscillator dynamics from the coupling effects. The phase model describes the dynamics of each oscillator by a single equation for its phase and the interactions between each pair of oscillators as a function that depends only on their relative phases.

Gait Adaptation Mechanism

Gait adaptation results from the tendency for the mechanical coupling of the body to pull the oscillators into anti-phase. The figure at the left shows each coupling function of the system and the stable phase differences that they define. Mechanical coupling works against intermodular neural coupling (proprioceptive and gap-junction coupling), which set the oscillators into a phase-wave close to synchrony, resulting in a long wavelength in the low-viscosity environment. Increasing the external fluid viscosity increases the mechanical coupling strength and pulls the oscillators closer to anti-phase, thus shortening the wavelength and providing a mechanism for gait adaptation.

Related Publications

Neuromechanical Mechanisms of Locomotion in C. elegans.

Carter Johnson. Dissertation, UC Davis Applied Mathematics Dept. 2020.

Neuromechanical Mechanisms of Gait Adaptation in C. Elegans.

Carter Johnson, Timothy Lewis, and Robert Guy. Submitted to SIADS June 2020.