Controllability and observability analysis of basal ganglia model and feedback linearisation control

Deep brain stimulation (DBS) is a clinical remedy to control tremor in Parkinson's disease. In DBS, one of the two main areas of basal ganglia (BG) is stimulated. This stimulation is produced with no feedback of the tremor and often causes a wide range of unpleasant side effects. Using a feedback signal from tremor, the stimulatory signal can be reduced or terminated to avoid extra stimulation and as a result decrease the side effects. To design a closed-loop controller for the non-linear BG model, a complete study of controllability and observability of this system is presented in this study. This study shows that the BG model is controllable and observable. The authors also propose the idea of stimulating the two BG areas simultaneously. A two-part controller is then designed: a feedback linearisation controller for subthalamic nucleus stimulation and a partial state feedback controller for globus pallidus internal stimulation. The controllers are designed to decrease three indicators: the hand tremor, the level of delivered stimulation signal in disease condition, and the ratio of the level of delivered stimulation signal in health condition to disease condition. Considering these three indicators, the simulation results show satisfactory performance.