Feedback deep brain stimulation for rehabilitation in Parkinson’s disease via unknown input observer

doi:10.1016/j.bspc.2020.101937

Biomedical Signal Processing and Control

Volume 59, May 2020, 101937

https://doi.org/10.1016/j.bspc.2020.101937 Get rights and content

Abstract

In this article, a feedback mechanism is used for Deep Brain Stimulation of Basal Ganglia (BG) in order to control excessive tremor caused by Parkinson’s disease. Due to the nonlinearity in the Basal Ganglia model, there are some limitations in performance of previously used feedback control schemes. We enhance the linear approximation of the model by introducing an unknown input (disturbance) to the model which represents the approximation error. Then we apply an unknown input observer to estimate the system states. The estimated state is used to implement a double loop feedback with back stepping and proportional control laws simultaneously stimulating the Globus pallidus internal and Sub thalamic nucleus sections. Simulations show improvements in estimation and control.

Graphical abstract

Introduction

Millions of people around the world suffer from neurological disorders. One of the most common symptoms is tremor caused by dystonia, multiple sclerosis and the Parkinson’s disease (PD) which ranks the second most prevalent neurodegenerative disorder [1,2]. Several treatments have been suggested for tremor among which Deep brain stimulation (DBS) is known to have compelling effects for PD patients [3]. DBS is a surgical technique in which a neural stimulator is implanted into the patient’s brain. The stimulator sends electrical signals to those brain areas that are responsible for body movement. DBS has been used as a treatment technique for essential tremor, Parkinson’s disease (PD), dystonia, obsessive-compulsive disorder and epilepsy [[4], [5], [6]]. The DBS system consists of three parts: The stimulator, a programmable battery-powered device for producing electric pulses, the lead wire, an insulated wire that connects the electrodes to the stimulator, and the electrodes which deliver electric pulses to the brain tissue. Electrodes can be placed in different areas of Basal Ganglia (BG) such as Sub-thalamic nucleus (STN) and Globus pallidus internal (GPi) sections [7]. DBS can be classified into open-loop (conventional) and closed-loop (feedback or adaptive) techniques. It should be pointed out that despite its effectiveness, DBS has its own shortcomings and there are still questions about it. Especially the closed-loop DBS is in the development stage [8].

In open-loop DBS, the stimulating signal is continually applied to the BG regardless of the tremor situation. In this technique a clinician manually adjusts the stimulation parameters after DBS implantation. Tuning of the stimulation signal is done according to a set of guidelines [9] and may not result in the optimal parameter set. Furthermore, the neural plasticity phenomenon may deteriorate DBS efficiency over time [9]. Also, open-loop DBS uses continuous stimulation which shortens the battery life [10].

These shortcomings motivates the feedback or closed loop DBS [[10], [11], [12], [13], [14]] in which the stimulation signal is tuned automatically based on measured signals, known as biomarkers. Several biomarkers including BG action and local field potentials as well as cortical recordings can be used with wearable sensors, e-Health and m-Health devices [15]. Proper selection of the biomarkers, DBS parameters, stimulation pattern and control algorithms are among future research areas in (feedback) DBS.

In [16] a model is proposed for tremor dynamics. In a following work [17], closed loop (feedback) mechanisms are introduced to control tremor. References [[10], [11], [12], [13]] introduced different control algorithms to improve the system performance. In this paper, we use the double-loop controller [10] because it is deemed to yield improved performance [10,13]. In [[10], [11], [12], [13]], a state space model is developed to model the BG dynamics. An observer is designed to estimate the system states after the model is approximated and linearized. The observer is used in combination with a robust back stepping [10] and feedback linearizing [13] controllers. Authors of [10,13 and 16] approximated the sign function in the tremor dynamics by a saturation function since the latter is continuous and differentiable around the origin enabling model linearization. This approximation is not accurate especially when the two functions have arguments close to zero. To remedy this issue, we will first enhance the approximation by introducing a disturbance signal which quantifies the difference between the saturation function and the sign function. Consequently, we replace the Proportional (P) observer by a Proportional Integral (PI) observer which enables estimation of the state vector regardless of the unknown input [[18], [19], [20], [21], [22], [23], [24]].

This paper is organized as follows: In the next section preliminaries are discussed. The third section includes simulation results and the fourth section concludes the paper.

Section snippets

Modelling the tremor in PD

Different models including neuron, neuron behavioural and behavioural models have been used to model tremor dynamics [10]. Behavioural models may use transfer functions [12,16] or state space models [[10], [11], [12], [13]]. In this paper we use the state space representation of [10,13] which is based on the transfer function model given in [16]. The model is described by the following equations: $\{\begin{matrix} \dot{x} (t) = A x (t) + f (x) + B u (t) \\ y (t) = C x (t) \end{matrix}$

In (1), $x (t) \in R^{6}$ represents the state vector, $u (t) \in R^{2}$ is the input

Unknown input observers for the proposed model

Reference [13] studied observability and controllability of approximated model (2) without the unknown input $v (t)$ . However, in order for a system to be observable in presence of unknown inputs, some additional criteria (stated in Theorem 1, 2) are required [18]. Two major types of PI [18,22,23] and disturbance decoupling [[18], [19], [20], [21]] state observers are commonly used for unknown input observation. PI observers use an additional variable (or vector) to estimate the unknown input(s) [

Controller formulation

Since (1) is unstable [13] a control strategy is required to stabilize the system. Firstly, note that the input matrix $B$ given in (3) has a large fifth entry which can adversely amplify the effect of control input on the system output. Therefore, in order to confine the system response, we firstly use an inner control loop of the following form [13]: $u_{2} (t) = a g {\hat{x}}_{2}$

The inner control loop (19) represents the magnitude of GPi stimulation signal. This double loop control structure is deemed to enhance

Conclusion

In this paper, a proportional integral unknown input observer is designed to enhance the estimation and tracking performance of the feedback deep brain stimulation for tremor control. The proposed observer compensates for the effect of nonlinearity in the BG model via introducing an additional state variable which estimates the approximation error of the nonlinear function. The simulation results shown significant enhancement to the most recent control scheme which used a proportional observer.

CRediT authorship contribution statement

Mohammad Mahdi Share Pasand: Software, Supervision, Writing - review & editing. Ali Tavakoli Golpaygani: Software, Supervision, Writing - review & editing.

Declaration of Competing Interest

None.

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View full text

Technical noteFeedback deep brain stimulation for rehabilitation in Parkinson’s disease via unknown input observer

Abstract

Graphical abstract

Introduction

Section snippets

Modelling the tremor in PD

Unknown input observers for the proposed model

Controller formulation

Conclusion

CRediT authorship contribution statement

Declaration of Competing Interest

Parkinsonism Relat. Disord.

Clin. Neurophysiol.

J. Theor. Biol.

Projected number of people with Parkinson disease in the most populous nations, 2005 through 2030

Neurology

Computational stimulation of the basal ganglia neurons with cost effective delayed Gaussian waveforms

Front. Comput. Neurosci.

Towards frequency adaptation for delayed feedback deep brain stimulations

Neural Regen. Res.

Adaptive deep brain stimulation in advanced Parkinson disease

Ann. Neurol.

Designing a robust backstepping controller for rehabilitation in Parkinson’s disease: a simulation study

IET Syst. Biol.

Modeling and automatic feedback control of tremor: adaptive estimation of deep brain stimulation

PLoS One

Technical note
Feedback deep brain stimulation for rehabilitation in Parkinson’s disease via unknown input observer