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Generic stabilizability for time-delayed feedback control.

ABSTRACT: Time-delayed feedback control is one of the most successful methods to discover dynamically unstable features of a dynamical system in an experiment. This approach feeds back only terms that depend on the difference between the current output and the output from a fixed time T ago. Thus, any periodic orbit of period T in the feedback-controlled system is also a periodic orbit of the uncontrolled system, independent of any modelling assumptions. It has been an open problem whether this approach can be successful in general, that is, under genericity conditions similar to those in linear control theory (controllability), or if there are fundamental restrictions to time-delayed feedback control. We show that, in principle, there are no restrictions. This paper proves the following: for every periodic orbit satisfying a genericity condition slightly stronger than classical linear controllability, one can find control gains that stabilize this orbit with extended time-delayed feedback control. While the paper's techniques are based on linear stability analysis, they exploit the specific properties of linearizations near autonomous periodic orbits in nonlinear systems, and are, thus, mostly relevant for the analysis of nonlinear experiments.

SUBMITTER: Sieber J

PROVIDER: S-EPMC4893174 | biostudies-literature | 2016 May

REPOSITORIES: biostudies-literature

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Generic stabilizability for time-delayed feedback control.

Sieber J J

Proceedings. Mathematical, physical, and engineering sciences 20160501 2189

Time-delayed feedback control is one of the most successful methods to discover dynamically unstable features of a dynamical system in an experiment. This approach feeds back only terms that depend on the difference between the current output and the output from a fixed time <i>T</i> ago. Thus, any periodic orbit of period <i>T</i> in the feedback-controlled system is also a periodic orbit of the uncontrolled system, independent of any modelling assumptions. It has been an open problem whether t ...[more]

PMID: 27279763

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Project description:Demand-controlled deep brain stimulation (DBS) appears to be a promising approach for the treatment of Parkinson's disease (PD) as revealed by computational, pre-clinical and clinical studies. Stimulation delivery is adapted to brain activity, for example, to the amount of neuronal activity considered to be abnormal. Such a closed-loop stimulation setup might help to reduce the amount of stimulation current, thereby maintaining therapeutic efficacy. In the context of the development of stimulation techniques that aim to restore desynchronized neuronal activity on a long-term basis, specific closed-loop stimulation protocols were designed computationally. These feedback techniques, e.g., pulsatile linear delayed feedback (LDF) or pulsatile nonlinear delayed feedback (NDF), were computationally developed to counteract abnormal neuronal synchronization characteristic for PD and other neurological disorders. By design, these techniques are intrinsically demand-controlled methods, where the amplitude of the stimulation signal is reduced when the desired desynchronized regime is reached. We here introduce a novel demand-controlled stimulation method, pulsatile multisite linear delayed feedback (MLDF), by employing MLDF to modulate the pulse amplitude of high-frequency (HF) DBS, in this way aiming at a specific, MLDF-related desynchronizing impact, while maintaining safety requirements with the charge-balanced HF DBS. Previously, MLDF was computationally developed for the control of spatio-temporal synchronized patterns and cluster states in neuronal populations. Here, in a physiologically motivated model network comprising neurons from subthalamic nucleus (STN) and external globus pallidus (GPe), we compare pulsatile MLDF to pulsatile LDF for the case where the smooth feedback signals are used to modulate the amplitude of charge-balanced HF DBS and suggest a modification of pulsatile MLDF which enables a pronounced desynchronizing impact. Our results may contribute to further clinical development of closed-loop DBS techniques.

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Generic stabilizability for time-delayed feedback control.

Publications

Generic stabilizability for time-delayed feedback control.

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