We studied the effects of using fractional order proportional, integral, and derivative (PID) controllers in a closed-loop mathematical model of deep brain stimulation. The objective of the controller was to dampen oscillations from a neural network model of Parkinson's disease. We varied intrinsic parameters, such as the gain of the controller, and extrinsic variables, such as the excitability of the network. We found that in most cases, fractional order components increased the robustness of the model multi-fold to changes in the gains of the controller. Similarly, the controller could be set to a fixed set of gains and remain stable to a much larger range, than for the classical PID case, of changes in synaptic weights that otherwise would cause oscillatory activity. The increase in robustness is a consequence of the properties of fractional order derivatives that provide an intrinsic memory trace of past activity, which works as a negative feedback system. Fractional order PID controllers could provide a platform to develop stand-alone closed-loop deep brain stimulation systems.

我们研究了在深部脑刺激闭环数学模型中使用分数阶比例、积分和微分 (PID) 控制器的影响。控制器的目标是抑制帕金森病神经网络模型的振荡。我们改变了内在参数，例如控制器的增益和外在变量，例如网络的可兴奋性。我们发现，在大多数情况下，分数阶分量会增加模型对控制器增益变化的稳健性。类似地，控制器可以设置为一组固定的增益，并在比经典 PID 情况大得多的突触权重变化范围内保持稳定，否则会导致振荡活动。鲁棒性的增加是分数阶导数特性的结果，它提供了过去活动的内在记忆痕迹，作为负反馈系统。分数阶 PID 控制器可以提供一个平台来开发独立的闭环深部脑刺激系统。