A mathematical model of adaptive behavior in quadruped locomotion

Abstract.

Locomotion involves repetitive movements and is often executed unconsciously and automatically. In order to achieve smooth locomotion, the coordination of the rhythms of all physical parts is important. Neurophysiological studies have revealed that basic rhythms are produced in the spinal network called, the central pattern generator (CPG), where some neural oscillators interact to self-organize coordinated rhythms. We present a model of the adaptation of locomotion patterns to a variable environment, and attempt to elucidate how the dynamics of locomotion pattern generation are adjusted by the environmental changes. Recent experimental results indicate that decerebrate cats have the ability to learn new gait patterns in a changed environment. In those experiments, a decerebrate cat was set on a treadmill consisting of three moving belts. This treadmill provides a periodic perturbation to each limb through variation of the speed of each belt. When the belt for the left forelimb is quickened, the decerebrate cat initially loses interlimb coordination and stability, but gradually recovers them and finally walks with a new gait. Based on the above biological facts, we propose a CPG model whose rhythmic pattern adapts to periodic perturbation from the variable environment. First, we design the oscillator interactions to generate a desired rhythmic pattern. In our model, oscillator interactions are regarded as the forces that generate the desired motion pattern. If the desired pattern has already been realized, then the interactions are equal to zero. However, this rhythmic pattern is not reproducible when there is an environmental change. Also, if we do not adjust the rhythmic dynamics, the oscillator interactions will not be zero. Therefore, in our adaptation rule, we adjust the memorized rhythmic pattern so as to minimize the oscillator interactions. This rule can describe the adaptive behavior of decerebrate cats well. Finally, we propose a mathematical framework of an adaptation in rhythmic motion. Our framework consists of three types of dynamics: environmental, rhythmic motion, and adaptation dynamics. We conclude that the time scale of adaptation dynamics should be much larger than that of rhythmic motion dynamics, and the repetition of rhythmic motions in a stable environment is important for the convergence of adaptation.