Neuronal Network Models of Phase Separation Between Limb CPGs of Digging Sand Crabs

Abstract

Ordinary differential equations are used to model a peculiar motor behaviour in the anomuran decapod crustacean Emerita analoga. Little is known about the neural circuitry that permits E. analoga to control the phase relationships between movements of the fourth legs and pair of uropods as it digs into sand, so mathematical models might aid in identifying features of the neural structures involved. The geometric arrangement of segmental ganglia controlling the movements of each limb provides an intuitive framework for modelling. Specifically, due to the rhythmic nature of movement, the network controlling the fourth legs and uropods is viewed as three coupled identical oscillators, one dedicated to the control of each fourth leg and one for the pair of uropods, which always move in bilateral synchrony. Systems of Morris–Lecar equations describe the voltage and ion channel dynamics of neurons. Each central pattern generator for a limb is first modelled as a single neuron and then, more realistically as a multi-neuron oscillator. This process results in high-dimensional systems of equations that are difficult to analyse. In either case, reduction to phase equations by averaging yields a two-dimensional system of equations where variables describe only each oscillator’s phase along its limit cycle. The behaviour observed in the reduced equations approximates that of the original system. Results suggest that the phase response function in the two dimensional system, together with minimal input from asymmetric bilateral coupling parameters, is sufficient to account for the observed behaviour.