Clinical use of DBS in Parkinson’s disease

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Table of contents

List of Figures
List of Acronyms
Resume en francais
1. Introduction
1.1 Aim and structure of this thesis
1.2 Neurons and action potentials
1.2.1 Action potential
1.2.2 Measurement of electrical activity of the brain
1.3 Basal ganglia
1.4 Parkinson’s disease
1.5 Beta oscillations in basal ganglia
1.5.1 Parkinsonian beta oscillations
1.5.2 Origin of the pathological beta oscillations
1.6 Deep brain stimulation
1.6.1 Clinical use of DBS in Parkinson’s disease
1.6.2 Closed-loop stimulation
1.7 Neural activity modelling
1.7.1 Firing rate models
1.7.2 Firing rate model of the STN-GPe loop
1.8 Analysis and control of nonlinear time-delay systems
1.8.1 Notation and comparison functions
1.8.2 Stability and Lyapunov direct method
1.8.3 Systems with output
1.8.4 Systems with input. Input-to-output and input-to-state stability
2. Stability of the ring rate model of STN{GPe loop with proportional feedback
2.1 Global exponential stability of globally Lipschitz systems
2.1.1 Global exponential stability
2.1.2 Lyapunov-Krasovskii approach for global exponential stability
2.1.3 GES LKF characterization
2.2 Stability of the ring rate model of STN{GPe under proportional stimulation
2.2.1 Model description and extension
2.2.2 High-gain proportional stabilization
2.2.3 Issues with the simple proportional controller
2.3 Proofs
2.3.1 Proof of Theorem 8
2.3.2 Proof of Proposition 10
3. Counter example to a sucient condition for uniform asymptotic partial stability
3.1 Adaptive proportional controller for the ring rate model of STN{GPe loop
3.1.1 Simple adaptive controller
3.1.2 Adaptive controller with -modication
3.2 Partial stability
3.3 Link between uniform asymptotic y-stability and IOS
3.4 Importance of uniformity in IOS analysis
3.5 Counterexample to a sucient condition for uniform asymptotic y-stability
3.5.1 Disproved sucient condition
3.5.2 Counterexample
3.6 Proofs
3.6.1 Proof of Lemma
3.6.2 Proof of Proposition
4. Adaptive stabilization with -modication of time delay nonlinear systems applied to the ring rate model of STN-GPe
4.1 Sigma modication for globally Lipschitz time-delay systems
4.1.1 Sigma modication
4.1.2 Stability in the mean
4.1.3 Stability in the mean of time-delay globally Lipschitz systems
4.1.4 Construction of a strict Lyapunov-Krasovskii functional with linear bounds
4.2 Application to the ring rate model of STN{GPe
4.2.1 Stability in the mean of the ring rate model
4.2.2 Numerical simulations
4.2.2.1 Eect of and on controller performance
4.2.2.2 Equilibrium estimation with a low-pass lter
4.2.2.3 Adaptation to changing parameters
4.3 Proofs
4.3.1 Proof of Theorem 27
4.3.2 Proof of Lemma 28
4.3.3 Proof of Lemma 29
4.3.4 Proof of Proposition 30
5. Frequency-selective quenching of endogenous and exogenous oscillations
5.1 Delayed neural elds model of the STN{GPe loop
5.2 Frequency response of the ring rate model of STN{GPe loop
5.3 Frequency-selective adaptive controller
6. Conclusions, issues and perspectives
6.1 Contributions and discussion
6.1.1 Chapter 2
6.1.2 Chapter 3
6.1.3 Chapter 4
6.1.4 Chapter 5
6.2 Future work