Utilization of fixed-time integral super twisting sliding mode controller for suppression of epileptic activity via stimulus current with DBS method

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Highlights

  • A controller is proposed to make the epileptic state reach a healthy state in a fixed and finite time.

  • The controller is applied to the dynamic model of a neuron in the CA3 region via implanted electrodes in the DBS method.

  • The time required to achieve zero tracking error occurs within a limited time.

  • The problems of classical sliding mode controller such as chattering and singularity are solved.

  • The proposed controller controls the value of DBS current over the time of stimulation, along with epilepsy control.

Abstract

Epileptic seizures should be controlled in a fixed time. Deep brain stimulation (DBS) has many advantages in treating neurological disorders such as epilepsy. In the present study, a fixed-time integral supertwisting sliding mode controller was proposed to apply to the Pinsky-Rinzel (PR) dynamic model via the DBS method to avoid epileptic seizures. First, the current which was generated by the electrodes implanted into the brain in the DBS method, was applied to the state variable of soma membrane potential in the PR model. Then, the proposed controller was used to the combined system including the DBS current and the PR model. Based on the results, the fixed-time sliding mode controller caused the system to approach a zero tracking error in a bounded and fixed time. In addition, the super-twisting sliding mode controller prevented chattering by producing a continuous control signal. Further, the integral sliding mode controller eliminated the singularity problem caused by derivation and made the system asymptotic stable. Finally, the tracking error of the healthy state could reach zero within 2.5 milliseconds. Thus, it is suggested that the epileptic seizures be controlled exceeding this time.

Introduction

Epilepsy is considered as a set of chronic medical or long-term neurological disorders characterized by epileptic seizures. The epileptic attacks may be very mild and almost unrecognizable or, on the contrary, be prolonged with severe vibrations [1,2]. Drug treatment can be useful for controlling seizures in about 70% of the cases [3]. However, alternative methods have been proposed to control epileptic seizures given the prevalence of drug-resistant epilepsy during recent years. Brain surgery was a relatively common method to treat neurologic and neuropsychiatric disorders during the 1950 and 1960s [4]. Epilepsy surgery is a type of neurosurgery which removes an area of the brain involved in seizures. In this surgery, however, the patient may be affected by the infection, speech and movement disorders, visual disturbances, and even stroke or more seizures [5]. Therefore, physicians have attempted to treat uncontrolled seizures with DBS due to the success of deep brain stimulation (DBS) such as programmability, reversibility, and low risk of complications for treating neurological and movement diseases [6]. In the mid-1980s, Benabid et al. used DBS to treat Parkinson’s disease [7]. Later, this method was approved in 1997 by the US Food and Drug Administration (USFDA) as a treatment for neurological disorders [8].

The studies on epilepsy control by DBS have been conducted on clinical patients in some areas of the brain where epilepsy may occur such as the anterior nucleus of the thalamus (ANT) [[9], [10], [11]], subthalamic nucleus (STN) [12], hippocampus [13,14], cerebellum [15], the centromedian nucleus of the thalamus (CMT) [16], and caudate nucleus (CN) [17]. For example, Krishna et al. used DBS of ANT in 16 patients, among whom nine patients indicated a decrease in the frequency of epilepsy immediately after electrode implantation [18]. Similar studies were performed on mice [[19], [20], [21]]. Feng et al. focused on transgenic mice and suggested that the dorsal part of the STN can play an important role in regulating central MC4R signaling and motility so that the stimulation of the dorsal part of the subthalamic nucleus can be considered as an appropriate treatment in epilepsy [22].

According to Lytton, dynamic modeling brain regions is practical for controlling epilepsy [23]. Thus, during recent years, several dynamic models have been introduced for the brain and various control algorithms have been implemented on these models. The Hodgkin-Huxley (HH) model’s Hopf bifurcation, which is directly related to epilepsy, is controlled by permanent or interval Washout filters (WF), which, in turn, act as a dynamic feedback controller. These filters can transform the subcritical bifurcations into supercritical bifurcations by an interval current in order to stabilize the HH equations [24]. In [25], a human cortical electrical activity model, which produced the activity characteristics of a seizure by appropriate parameters, was considered. Then, linear feedback, differential, and filter controllers were incorporated into the model dynamics for preventing such seizures. The results indicated that these controllers can operate to some extent because they are not robust against variations of pathological parameters, although these three controllers can be used to eliminate seizing activity in the model system. Jansen’s Neural Mass Model (NMM) was taken as a testbed to develop a PI-type closed-loop controller for suppressing epileptic activity. A graphical stability analysis method was employed to determine the stabilizing region of the PI controller in the control parameter space, which provided a theoretical guideline to choose the PI control parameters. However, the main limitation was that the linearized approximation might provide a conservative estimate of the PI controller stabilizing region [26]. The input-output linearization method was utilized for controlling the seizures in Pinsky-Rinzel (PR) model, by which the epileptic waves were controlled during 2.3 s [27]. Then, feedback decoupling was introduced as a robust seizure control strategy and applied to Jansen’s neuronal mass model to suppress seizures. Nevertheless, only equal bidirectional coupling was considered in this study due to the limitations of the model, although there might be unequal (directional) interactions in the brain [28]. In addition, the control algorithms presented so far have not been well expanded and not robust against external perturbation or alteration of pathological parameters. Further, tracking error was not completely zero at a fixed time. In [29], the robust sliding mode controller was utilized to prevent epileptic seizures in the human cortex model. The advantages of the sliding mode controller included rapid impact, non-sensitivity to perturbation and parameter changes, easy implementation, and no need for online system detection [30,31]. Despite providing a robust control algorithm in [29], the usual problems of the sliding mode controller such as singularity and chattering persisted and the tracking error was not absolutely zero. In addition, the control process in [29] was not performed through DBS.

Based on a basal ganglia-thalamocortical model, a type of the DBS voltage was employed on the STN to study the control mechanism of absence epilepsy seizures [32]. In the DBS method, a constant current is applied by implanted electrodes without decreasing or increasing over time. After some time since the beginning of stimulation, it may not be necessary to apply the stimulation current with the initial value. Therefore, a solution is needed to control the generated current from the electrodes to make the waveform of epileptic state track the healthy state in a fixed and finite time.

In this paper, a fixed time integral super-twisting sliding mode controller is proposed to suppress the simulated seizures in the PR model via the DBS method in order to solve the above-mentioned problems. The PR model describes the CA3 region of the hippocampus as an important area of the outbreak and epileptic seizures propagation in the brain [28]. By applying the mentioned controller to the system, it is anticipated that the system convergence occurs at a finite and fixed time, the singularity and chattering problems be eliminated, and the time required to control the system be shortened.

The remainder of this paper is organized as follows. Section 2 explains PR model. Section 3 represents the analysis of the DBS current and its role in the PR model. To control epileptic seizures through the DBS method, a robust fixed-time integral super-twisting sliding mode controller, along with the analysis of fixed-time stability with the proof of the resistance against uncertainties, is proposed in Section 4. The numerical simulation is discussed and a comparison is made between the proposed controller and classical sliding mode schemes in Section 5. Finally, Section 6 presents the concluding remarks.

Section snippets

Pinsky-Rinzel (PR) model

CA3 region has a special strong functional synaptic connectivity. In addition, the excitatory synapses are strong enough for a burst in a single neuron to elicit a burst in another connected neuron [33]. In other words, a synchronous neural activity at the single-cell level indicating epileptic activity or an interictal discharge, can spread to other cells through local recurrent paths. This synchronized discharge transition occurs in several steps until activating the entire population

DBS current

Epileptic seizures can be suppressed by changing the soma membrane potential [27] to control neuronal firing patterns via the current applied to the epileptic region in the DBS method. DBS generates a current to regulate the abnormal brain signal such as the epileptic seizures wave by involving implanting electrodes within specific regions of the brain. The current generated by implanted electrodes is described as follows [45]:IDBS=iDHsin2πt/ρD[1-Hsin2πt+δD/ρD]where iD indicates the

Controller design: A fixed-time integral super-twisting sliding mode approach

In the DBS method, the DBS current is applied to the epileptic area of the brain with a fixed and certain value without any change over the time of stimulation. However, after some time of DBS, it may no longer be necessary to apply this current with the same value. On the other hand, various controllers for epilepsy control have been introduced during the recent years, most of which are simple in structure and are not resistant against parameter uncertainties or perturbation. Therefore, in the

Simulation results

This section explains the results of applying the fixed-time integral super-twisting sliding mode controller to the PR model for controlling the DBS current applied by electrodes in DBS method. Then, a classical sliding mode controller is designed and applied to this system for more accurate comparison.

Fig. 2, Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7 demonstrate the results of the fixed-time integral super-twisting sliding mode controller utilization in the PR model stimulated with DBS current.

Conclusions

Due to the prevalence of epilepsy in recent years, patients are resistant to drug treatment in many cases. On the other hand, epileptic seizures can be diagnosed near the attack times. The control algorithms presented for controlling epilepsy in dynamic models of the brain, have a simple structure and are not resistant to the changes in model parameters and perturbations.

The present study aimed to provide a controller which can control stimulated epileptic seizures in a dynamic brain model via

CRediT authorship contribution statement

Samira Rezvani-Ardakani: Software, Formal analysis, Writing. Sajad Mohammad-Ali-Nezhad: Conceptualization, Supervision, Project administration, Review & editing. Reza Ghasemi: Supervision.

Declaration of Competing Interest

The authors report no declarations of interest.

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