Adaptive global synchrony of inferior olive neurons

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.