Stability of the ring rate model of STN{GPe loop with proportional feedback

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Origin of the pathological beta oscillations

The origin of Parkinsonian beta oscillations present in the basal ganglia is still unknown and subject to much debate. The two leading, not necessarily mutually exclusive, theories are endogenous generation with STN-GPe pacemaker and entrainment by external input.
In [Plenz and Kital, 1999], the authors have shown, based on in vitro experiments, that the excitatory neurons of STN, in a closed loop with the inhibitory neurons of GPe form a feedback system that is capable of producing synchronized oscillations. On this basis they have put forward a hypothesis that STN and GPe make up a central pacemaker, responsible for both normal and pathological oscillatory acitivity in the basal ganglia. In [Nevado Holgado et al., 2010], the authors have demonstrated that a ring-rate model of STN{GPe loop is capable of producing pathological beta oscillations, undergoing a transition from healthy to pathological state as the connection strength between the populations increases.
An alternative hypothesis says that the beta oscillations present in the basal ganglia are driven by cortical or striatal input. The cortical entrainment hypothesis is supported by experimental evidence showing that STN neurons exhibit low-frequency oscillatory activity, correlated with slow-wave cortical activity in rat models of PD [Magill et al., 2001], as well as by the fact that cortex is capable of producing beta oscillations [Yamawaki et al., 2008]. Additionally, experimental and computational data shows that striatum, the main input structure of the basal ganglia, is itself also capable of generating beta oscillations [McCarthy et al., 2011].

Clinical use of DBS in Parkinson’s disease

The use of DBS in PD most notably consists of high-frequency stimulation of ventral intermediate nucleus (VIM) of the thalamus for tremor reduction [Benabid et al., 1991; Picillo and Fasano, 2016]; subthalamic nucleus (STN) for improvements in gait, tremor, and bradykinesia [Benabid et al., 2009]; and the internal part of globus pallidus (GPi) for all major motor symptoms of PD [Perlmutter and Mink, 2006] (see Figure 1.7 for illustration).
The electrodes are implanted under local or general anesthesia. Local anesthesia (with patient conscious) allows intraoperative assessment of DBS e cacy and thus increases the chances of a successful surgery. The signal generator is inserted subcutaneously a few days later. In the following weeks the programming of the generator is conducted by a neurologist and usually lasts at least another few days. The frequency is usually set at 130 Hz, the pulse duration at 60 s and the voltage is progressively increased, while checking for improvement in the symptoms as well as for stimulation-induced side e ects, such as dyskinesias (involuntary muscle movements), paresthesias (abnormal sensation of the skin), and muscle contraction. The operating voltage is chosen to maximize the clinical improvement in the symptoms while avoiding the side e ects [Benabid et al., 2009].
The drawback of this method is that once the stimulation parameters are set, the stimulation pattern remains constant. Since the severity of symptoms varies with time on timescales ranging from diurnal rhythms to disease progression over multiple years, this leads to several issues. Insensitivity to changes in severity may lead to overstimulation, not to mention that electrical stimulation of the brain is not contained to targeted area because, due to volume conductance in the brain, electrical stimulation can in uence healthy areas as well. This in uences the patients’ quality of life, as it induces DBS-related side e ects, as well as drains the battery faster, forcing the patient to undergo battery replacement operations more frequently. On the other hand, as the disease progresses, and as the electrode lead is surrounded by scar tissue [Vedam-Mai et al., 2018], the sensitivity to DBS of the stimulated structures decreases, and a stronger stimulation may be necessary.

Firing rate model of the STN-GPe loop

In the paper [Nevado Holgado et al., 2010], the authors employed the approach of Wilson and Cowan to propose a ring-rate model of STN-GPe loop in order to examine conditions that lead to generation of pathological oscillations. Both the structure of the model and the values of the parameters were chosen based on experimental data from various sources. Their main conclusion was that the system will spontaneously produce oscillations when the connections between the populations are strengthened. Their reasoning was based on wealth of experimental data and gave credibility to the theory of endogenous generation of the pathological beta oscillations, supported by in vitro evidence [Plenz and Kital, 1999].
The model is as follows:
1×1(t) = x1(t) + S1(c11x1(t 11) c12x2(t 12) + cCtxu1(t)) (1.3a).
2×2(t) = x2(t) + S2(c21x1(t 21) c22x2(t 22) cStru2(t)): (1.3b).
Activity of STN (in spikes per second) is represented by x1, activity of GPe by x2. The coupling constant cij represents connection strength from population j to population i and ij represents time delay that occurs due to nite velocity of signal propagation. The inputs u1 and u2 (with coupling constants cCtx and cStr) represent cortical and stri-atal inputs to the system, respectively. All the coupling constants are positive, and the sign represents whether neurons in the presynaptic population have excitatory (STN and cortex) or inhibitory (GPe and striatum) e ect on the postsynaptic population.

Stability of the ring rate model of STN{GPe under proportional stimulation

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The results of Section 2.1 allow us to examine stability of the analyzed model of parkin-sonian basal ganglia under proportional feedback control. These results will form a cor-nerstone of the analysis of the stability of this model under adaptive control, proposed in the next chapters. Let us rst recall the ring rate model from Section 1.7.2 and extend it to include control input.

Model description and extension

As recalled in Section 1.7.2, the model described by equation (1.3) has a unique xed equilibrium x for any xed input u . A change of variables ui ui ui lets us eliminate of variables x x x external inputs and puts (1.3) in the form (2.1). Another change[ [ and modi cation of the activation functions Si(x) [Si(x + Si 1(xi )) xi (2.4).
makes the system conform to the requirement f(0) = 0, putting the equilibrium of the system at the origin. Finally, we extend the model with a stimulation signal (t) 2 R that we will use to stabilize the system. The system takes the form 1×1(t) = x1(t) + S1 c11x1(t 11) c12x2(t 12) + (t) (2.5a).

Adaptive proportional controller for the ring rate model of STN GPe loop

A simple solution of the problem of unknown would be to dynamically increase the value of as long as the pathological activity in the system persists. For example, we could use an adaptive controller of the form:
(t) = (t)x1(t) (3.3a).
_(t) = (jx1(t)j); (3.3b).
where : R 0 ! R 0 is a locally Lipschitz function satisfying (r) = 0 if and only if r = 0 and > 0 is an additional tuning parameter, that regulates the increase rate of the gain parameter . With this controller, the proportional gain is increased as long as x1 (representing activity of the STN) is not at the equilibrium. Since the proportional controller (3.2) is e ective for any , this strategy successfully nds a that stabilizes the system, as illustrated in Figure 3.1.
This approach is not without problems, however. With a small time constant , the value of quickly overpasses and continues to grow well beyond this value while the system converges to the equilibrium. And since is nonnegative, t 7!(t) is nondecreas-ing, so there is no way to correct the overestimation of (see Figure 3.2). Additionally, as pointed out in [Ioannou and Kokotovic, 1984], adaptive controllers of this kind su er from parameter drift instability, when the system is subject to exogenous disturbances. This type of instability can induce the adaptive variable to grow inde nitely in the presence of bounded disturbances (see Figure 3.3).

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

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