A MATHEMATICAL-MODEL OF ADAPTIVE-BEHAVIOR IN QUADRUPED LOCOMOTION

Locomotion involves repetitive movements and is often executed, uncons ciously and automatically. In order to achieve smooth locomotion, the coordination of the rhythms of all physical parts is important. Neurop hysiological 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 vari able environment, and attempt to elucidate how the dynamics of locomot ion pattern generation are adjusted by the environmental changes. Rece nt experimental results indicate that decerebrate cats have the abilit y to learn new gait patterns in a changed environment. In those experi ments, a decerebrate cat was set on a treadmill consisting of three mo ving belts. This treadmill provides a periodic perturbation to each li mb through variation of the speed of each belt. When the belt for the left forelimb is quickened, the decerebrate cat initially loses interl imb coordination and stability, but gradually recovers them and finall y walks with a new gait. Based on the above biological facts, we propo se a CPG model whose rhythmic pattern adapts to periodic perturbation from the variable environment. First, we design the oscillator interac tions to generate a desired rhythmic pattern. In our model, oscillator interactions are regarded as the forces that generate the desired mot ion pattern. If the desired pattern has already been realized, then th e interactions are equal to zero. However, this rhythmic pattern is no t reproducible when there is an environmental change. Also, if we do n ot adjust the rhythmic dynamics, the oscillator interactions will not be zero. Therefore, in our adaptation rule, we adjust the memorized rh ythmic pattern so as to minimize the oscillator interactions. This rul e can describe the adaptive behavior of decerebrate cats well. Finally , we propose a mathematical framework of an adaptation in rhythmic mot ion. Our framework consists of three types of dynamics: environmental, rhythmic motion, and adaptation dynamics. We conclude that the time s cale of adaptation dynamics should be much larger than that of rhythmi c motion dynamics, and the repetition of rhythmic motions in a stable environment is important for the convergence of adaptation.