Lee Keum W, Singh Sahjendra N
Division of Electronic Information and Communication, University of Kwandong, Gangwon, Korea.
Bioinspir Biomim. 2009 Sep;4(3):036003. doi: 10.1088/1748-3182/4/3/036003. Epub 2009 Aug 28.
This paper treats the question of global adaptive synchronization of inferior olive neurons (IONs) based on the immersion and invariance approach. The ION exhibits a variety of orbits as the parameter (termed the bifurcation parameter), which appears in its nonlinear functions, is varied. It is seen that once the bifurcation parameter exceeds a critical value, the stability of the equilibrium point of the ION is lost, and periodic orbits are born. The size and shape of the orbits depend on the value of the bifurcation parameter. It is assumed that bifurcation parameters of the IONs are not known. The orbits of IONs beginning from arbitrary initial conditions are not synchronized. For the synchronization of the IONs, a non-certainty equivalent adaptation law is derived. The control system has a modular structure consisting of an identifier and a control module. Using the Lyapunov approach, it is shown that in the closed-loop system, global synchronization of the neurons with a prescribed relative phase is accomplished, and the estimated bifurcation parameters converge to the true parameters. Unlike the certainty-equivalent adaptive control systems, an interesting feature of the designed control system is that whenever the estimated parameters coincide with the true values, the parameter estimates remain frozen thereafter, and the closed-loop system recovers the performance of the deterministic closed-loop system. Simulation results are presented which show that in the closed-loop system, the synchrony of neurons with prescribed phases is accomplished despite the uncertainties in the bifurcation parameters.
本文基于浸入与不变性方法探讨了下橄榄核神经元（IONs）的全局自适应同步问题。随着出现在其非线性函数中的参数（称为分岔参数）的变化，ION呈现出各种轨道。可以看出，一旦分岔参数超过临界值，ION平衡点的稳定性就会丧失，并且会产生周期轨道。轨道的大小和形状取决于分岔参数的值。假设IONs的分岔参数未知。从任意初始条件开始的ION轨道是不同步的。为了实现IONs的同步，推导了一种非确定性等效自适应律。控制系统具有由一个标识符和一个控制模块组成的模块化结构。使用李雅普诺夫方法表明，在闭环系统中，神经元以规定的相对相位实现了全局同步，并且估计的分岔参数收敛到真实参数。与确定性等效自适应控制系统不同，所设计的控制系统的一个有趣特征是，每当估计参数与真实值一致时，此后参数估计就会保持不变，并且闭环系统恢复确定性闭环系统的性能。给出的仿真结果表明，在闭环系统中，尽管分岔参数存在不确定性，但神经元仍能实现规定相位的同步。
