# A Closed-Loop Brain Stimulation Control System Design Based on Brain-Machine Interface for Epilepsy

Authors: Moshu Qian; Guanghua Zhong; Xinggang Yan; Heyuan Wang; Yang Cui
Journal: Complexity
 (2020)

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## Abstract

In this study, a closed-loop brain stimulation control system scheme for epilepsy seizure abatement is designed by brain-machine interface (BMI) technique. In the controller design process, the practical parametric uncertainties involving cerebral blood flow, glucose metabolism, blood oxygen level dependence, and electromagnetic disturbances in signal control are considered. An appropriate transformation is introduced to express the system in regular form for design and analysis. Then, sufficient conditions are developed such that the sliding motion is asymptotically stable. Combining Caputo fractional order definition and neural network (NN), a finite time fractional order sliding mode (FFOSM) controller is designed to guarantee reachability of the sliding mode. The stability and reachability analysis of the closed-loop tracking control system gives the guideline of parameter selection, and simulation results based on comprehensive comparisons are carried out to demonstrate the effectiveness of proposed approach.


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## Body

## 1. Introduction

Brain-machine interface (BMI) technology has a wide range of applications for various neural system diseases therapy. The purpose of brain-machine interface or brain-computer interface (BCI) is to decode motion-related neural signals into control signals for driving output to guide computer cursors, prosthetics, and other auxiliary equipment. The ultimate goal of research in this area is to rehabilitate patients who have paralyzed or lost their limbs due to neurological disease or trauma [1–3]. In the past ten years, advances in neural recording technology have greatly promoted the development of BMI technology and successfully carried out relevant human clinical experiments [4, 5]. The development of BMI technology has greatly promoted the application of neuroscience in rehabilitation medicine [6]. Based on the above research basis, BMIs or BCIs technology has been widely used in multiple aspects, such as the closed-loop brain-computer interface technique for the treatment of refractory epilepsy [7], wireless brain-computer interface-controlled exoskeleton [8], and the application in stroke rehabilitation [3]. Therefore, the research on brain-machine interface technology has a very broad prospect.Epilepsy is a serious neurological disease characterized by unprovoked recurrence and about 50 million people suffer from epilepsy in the world [9]. More than 30% of patients with epilepsy continue to relapse after receiving treatment even if much related research work has been carried out in many countries for the treatment of epilepsy [10]. Moreover, various aspects of social life of patients with epilepsy are seriously affected by epilepsy [11]. Therefore, epilepsy has been considered as a serious public health problem [12]. According to some surveys, patients with epilepsy are more likely to be divorced and discriminated against [11, 13]. At present, there are four types of seizure patterns: focal, systemic, comprehensive, and unknown [14]. Absence epilepsy (AE) is a form of systemic epilepsy, which has features including sudden arrest in activity, awareness, and responsiveness, and the body is possibly accompanied by some slight motor characteristics [15]. Thus, the research of absence epilepsy is significant and far-reaching.The most common treatment for epilepsy now remains in drug-assisted or surgical resection [16]. In some investigations, it has been shown that long-term service of epilepsy drugs may lead to liver decline or even damage due to epilepsy is a long-term chronic disease [17, 18]. On the contrary, neurosurgical resection of the epileptic area cannot be applied to patients with systemic epilepsy [19, 20]. Therefore, it urgently needs an alternative method for absence epilepsy to solve the above problems due to limitations of the abovementioned drugs and surgical treatment [17–20]. The existing study shows that appropriate brain stimulation techniques can greatly reduce the frequency of seizures, that is, inhibit seizures [21–25]. Based on existing studies, it follows that brain stimulation is a promising treatment for epilepsy [21–23]. Brain stimulation for epilepsy has many methods, and open-loop brain stimulation and closed-loop brain stimulation are two typical stimulation methods [23]. Although open-loop brain stimulation is used more widely, compared with its effect, closed-loop brain stimulation can avoid the side effects of continuous stimulation brought and achieve the good effect of eliminating seizures [22–24]. When the closed-loop brain stimulation for epilepsy is carried out by BMI, the brain is connected to a computer for monitoring continuously the patient’s EEG, which reflects the patient’s neurological condition. After data processing and model analysis, stimulation therapy is then derived to alleviate or eliminate epilepsy symptoms. Thus, different thalamic models are crucial in the study of seizures [26, 27]. Cortical computational models of the thalamus have been developed [27–30] and the lumped model has been built [31]. The thalamic-cortical circuit computational model has also been developed [32]. Specially, Taylor has established a neural cluster model for seizures [33]. However, the closed-loop brain stimulation based on BMI for epilepsy is still in the early stages of research, and there is no mature and complete system design scheme, which motivates us to undergo this research. Therefore, based on the abovementioned achievements of seizure model, we will further study and explore the closed-loop control scheme of brain stimulation and then develop a new controller to promote the effectiveness of epilepsy seizure abatement.With the development of seizures control technique, Taylor proposed a single-pulse stimulation control requiring optimal stimulation time and amplitude [33]. Considering the high nonlinearity, unmodeled dynamics, and ubiquitous disturbance in the nervous system, seizure oscillations, noise, and disturbance changes are not accurate; therefore, the method in [33] cannot deal with epilepsy seizures very well [34, 35]. Based on the abovementioned problems, Ge proposed a robust closed-loop control which adapts to high nonlinearity, unmodeled dynamics, and ubiquitous disturbance of the nervous system and has stable control for seizures [20], but a slow convergence rate is its biggest drawback. The finite time sliding mode control technique can accelerate system convergence and is an effective strategy for nonlinear systems with high nonlinearity, unmodeled dynamics, and unknown disturbances [36–39]. Considering this problem, a finite time sliding mode controller is proposed with radial basis function neural networks and fractional order sliding manifold based on a well-established thalamocortical neural mass model. The designed closed-loop control system scheme based on BMI in this paper can guarantee the output of the thalamocortical system converge to the desired value more quickly and accurately.


## 2. Problem Statement

The structure of brain is shown in Figure1. Deep brain stimulation (DBS) is an effective treatment of a variety of neuropsychiatric disorders, including medication intractable Parkinson’s disease, epilepsy, primary dystonia, and essential tremor [1, 2]. The DBS method mainly inserts the electrodes into a specific nucleus in the brain through stereotactic technology and microelectrode recording technology and suppresses the abnormal brain nucleus discharge by continuous high-frequency pulse electrical stimulation, thereby achieving the therapeutic effect. Since it is unreliable to utilize open-loop brain neural stimulation to eliminate the pathological spike-and-wave discharges in the absence of epilepsy seizures, it is required to seek a more comprehensive closed-loop stable control scheme. However, epilepsy seizure abatement based on the closed-loop control system has not been studied very well and the existing results are very few [11, 20]. Therefore, this paper focuses on a closed-loop control strategy for epilepsy seizure abatement based on DBS [18–20], which can adjust the stimulus signal in real time by the controller in the computer. Certainly, the connection between the brain and the computer can be wireless, as shown in Figure 2, or wired as Figure 3.Figure 1
Brain structure.Figure 2
Closed-loop brain stimulation control system scheme based on BMI for epilepsy.Figure 3
Electroencephalogram test cap for epilepsy.The high-frequency pulse stimulation string in deep brain stimulation (DBS) can not only shorten the duration of seizures but also reduce the intensity of seizures [21–25], based on which a sliding mode control scheme is developed for epilepsy seizure abatement. It is assumed that there exists a cortical area exhibiting normal background activities before absence epilepsy seizures happen. The normal neural electrical signals collected by the recording electrodes are defined as the desired brain signal yd, and then a controller could be developed to force the brain cortical areas with pathological characteristics to behave in a similar way as the dynamics of the normal brain signal. The smaller the difference between the pathological brain signal dynamics y and the normal one yd, the better the effect of epilepsy seizure abatement. Thus, the seizure abatement problem can be transformed to a tracking control problem and our control target is to make abnormal brain activity signal y quickly and accurately track desired normal brain activity signal yd.Considering extensive nonlinear term owing to uncertainties in cerebral blood flow, glucose metabolism, blood oxygen level dependence (BOLD), and electromagnetic disturbances, which may affect network connectivity and neural electrical signal, the interaction within the thalamocortical system of epilepsy is modeled by the following differential equations [20]:(1)v˙t=Avvt+Bvut+Ψvt+Ξv,t,wt=Hvt,where v=v1,v2,v3,v4T=TC,RE,IN,PYT is the system state variable and v∈R4×1, u∈Rm,0<m<4 is the control input variable, the output variable w∈Rm, and the output matrix H=h1,h2,h3,h4∈Rm×4 is determined later. The control matrix is Bv∈R4×m, Ξv,t=Ξ1,Ξ2,Ξ3,Ξ4T is the synthetic unknown effect of uncertainties and disturbances, and Ψv=Ψ1,Ψ2,Ψ3,Ψ4T is the known nonlinear terms given as follows:(2)Ψv=τtht−ctrβ+cte1+ε−v4−1τrhr+crt−crrβ+cre1+ε−v4−1τihi+cie1+ε−v4−1τehe+cet1+ε−v1−1+cei1+ε−v3−1+cee1+ε−v4−1,Av=−τt−τtctrα00τrcrtα−τr+τrcrrα0000−τi0000−τe.Noting thatIN, TC, PY, and RE are all the state variables, representing the fractional firing activity in each neuronal population. τi,τt,τe,andτr are time scale constants mediated by different excitatory and inhibitory neurotransmitters. cie,cte,ctr,cee,cei,cet,cre,crt,andcrr are the connectivity strengths between different neuronal populations. While appropriate constants αandβ are the linear activation function describing the thalamic subsystem, the appropriate constant ε>0 describes the cortical dynamics. The specific meaning and values of the parameter used in this paper are given in [20]. The desired output yd∈Rm can be measured from a normal minicolumn.Assumption 1 (see [20]).
The matrix pairAv,Bv is controllable and rankBv=m.
Under the condition of Assumption1, there exists an invertible matrix T∈R4×4 such that the coordinate transformation x=Tv holds, and then the matrix pair Av,Bv with respect to the new coordinate x have the following form:(3)A=A11A12A21A22=TAvT−1,B=0B2=TBv,where A11∈R4−m×4−m and matrix B2∈Rm×m is nonsingular. It is noticed that the matrix T can be obtained using basic matrix transformation theory.
Then, in the new coordinate system, system (1) is rewritten by(4)x˙1=A11x1+A12x2+g1+η1,x˙2=A21x1+A22x2+B2u+g2+η2,y=Cx2,where x=colx1,x2 with x1∈R4−m and x2∈Rm, gx=g1,g2T=TΨT−1x with g1∈R4−m and g2∈Rm, and ηx,t=η1,η2T=TΞT−1x,t with η1∈R4−m and η2∈Rm. A12,A21,andA22 are appropriate matrices. The system output y∈Rm is determined by simulated EEG and C∈Rm×m is a positive definite invertible diagonal matrix, C=diagc1,…,cm.
Note thatΛ denotes the norm of Λ in this paper, where Λ is a matrix or a constant.


## 3. Controller Design for Epilepsy

In this position, some definitions and lemmas are introduced to derive the main results of this study.Definition 1 (see [40]).
The Caputo fractional calculus of the continuous integrable functionf is uniformly defined as(5)Dbtaft=1Γm−a∫0tfmτt−τa−m+1dτ,m−1<a<m,dmdtmft,a=m,where m is the minimum integer larger than a, the definition of Γ⋅ is Γq=∫0te−ttq−1dt, and dm/dtm represents for traditional differential. When a>0, sign Dbta represents fractional order differential (FOD), while a<0, it represents fractional order integral (FOI) on the interval b,t. FOD operation can change a special FOI, which possesses a certain characteristic of memory. For simplicity and convenience, symbol Da is used instead of D0ta in this paper.Lemma 1 (see [36]).
Consider the following system:(6)x˙=fx,t,f0,t=0,x∈U⊂Rn,where f:U×R+⟶Rn is continuous on an open neighborhood U of the origin. Assume that there exists a Lyapunov function Vx defined on a neighborhood of the origin, which satisfies an extended Lyapunov description with the form of fast terminal sliding mode (FTSM) as(7)V˙x+λ1Vx+λ2Vαx≤0.Then, system (6) is locally finite-time stable, and the settling time can be determined as(8)Treach≤1λ11−αlnλ1V1−αx0+λ2λ2,where λ1>0,λ2>0, α∈0,1.Lemma 2.
Consider the fractional order equation(9)Daz+Lz=0,where a∈0,1 is an arbitrary real constant, z∈Rm is an unknown vector to be solved and z=z1,…,zmT, and L denotes the constant diagonal matrix and L=diaglii=1,…,m,li>0.Then, the solution of equation (9) is(10)zi=μit−a,i=1,…,m,where μi is a positive constant.Proof.
See the proof of Theorem III.2 in [41].


### 3.1. Design of Sliding Mode Surface

In this section, a sliding surface is designed for system (4) and the stability of the corresponding tracking control system is analyzed.For system (4), assume that the desired output yd is differentiable. Then, the tracking error e=y−yd, and its derivative with respect to time is(11)e˙=CA21x1+A22x2+B2u+g2+η2−C−1y˙d.To make the dynamics of the thalamocortical system of epilepsy (4) to track the desired normal activities, namely, y⟶yd, a sliding mode surface will be developed to make the dynamical error system stable.The composite sliding surface for system (4) using fractional order sliding mode technique can be selected as follows:(12)s=λ1Da1e+λ2C−1e,where s∈Rm, λ1>0,λ2>0,and0<a1<1 are designed constants, respectively.Assumption 2.
The system outputy=y1,…,ymT, and then its initial value yi0i=1,…,m is a positive constant.Theorem 1.
Consider the control system (11). On Assumptions 1 and 2, the sliding motion of system (11) corresponding to the sliding surface (12) is asymptotically stable.Proof.
During sliding motions=0, the following equation can be obtained from (12):(13)Da1e+Πe=0,where Π=λ2/λ1C−1 is a diagonal matrix, Π=diagπii=1,…,m, and πi>0.
Under Lemma2, the solution of the abovementioned equation is as follows:(14)ei=μit−a1,μi>0.
Consider the Lyapunov function candidate:(15)V1=12eTe.
Then, the derivative ofV1 along the trajectory of system (13) is(16)V˙1=eTe˙=∑i=1m−a1μi2t−2a1−1,Thus, V˙1<0 for t∈0,∞. It is shown that the sliding motion of system (11) corresponding to the sliding surface (12) is asymptotically stable, namely, limt⟶∞ei=limt⟶∞yi−yid=0. Therefore, it is guaranteed that the output of system (4) tracks the desired signal after the system is driven to the designed sliding surface (12). Proof is completed.
Necessary conditions to ensure this sliding motion which is attained and maintained will further be developed.


### 3.2. Reachability Analysis

It is extremely important to track desired signal with the rapidity and accuracy [42, 43], which determines the therapeutic effect. However, the conventional sliding mode control usually results in low precision and convergence rate problems. To solve these problems, a finite time fractional order sliding mode (FFOSM) control scheme for epilepsy seizure abatement is designed, as shown in Figure 4.Figure 4
NN-FFOSM control scheme for epilepsy seizure abatement.Considering the synthetic effect of uncertain change in cerebral blood flow, glucose metabolism, BOLD, electromagnetic disturbances, and upper bound of the synthetic effect may be unknown. Then, the uncertainty termη2 affected by external stimulation can be approximated by employing an ideal RBFNN as(17)η2x,W=WTφx+ϱ,ϱ≤ζ,where W∈R1×m is an ideal weight vector from the hidden layer to the output layer, φx is the basis function, ϱ∈Rm×1 is the approximation error vector, and ζ is an unknown positive constant.LetW^ be the estimation of W, and we have(18)η^2x,W^=W^Tφx.LetW˜=W−W^ be the weighting estimation error of W, and we have(19)η2x,W−η^2x,W^=W˜Tφx+ϱ.Assumption 3 (see [40]).
The estimation errorϱ is norm-bounded as ϱ≤ζ, where ζ>0 is an unknown constant.Remark 1.
Sinceϱ should be very small when η^2x approximate η2x well, along with the control command u bounded in any practical system, it is feasible to assume ϱ in (19) is norm-bounded. However, it is not required that the bound ζ is known.
For system (4), differentiating (12) and using (11), it follows that(20)s˙=λ1Da1+1e+λ2C−1e˙=λ1Da1+1e+λ2A21x1+A22x2+B2u+g2+η2−C−1y˙d.
The fractional order controller can be designed as(21)u=B2−C−1y˙d−η^2x,W^−λ1λ2Da1+1e−A21x1−A22x2−k2λ2s−k3λ2sb2signs−g2−Imζ^,where B2− is the generalized inverse matrix of B2. a1, b2, k2, and k3 are designed constants with 0<a1,b2<1, k2>0, and k3>0. Im∈Rm×1 is an m-dimensional unit vector.Theorem 2.
Consider the control system (11). On Assumptions 1–3, the fractional order sliding mode control law (21) can drive system (11) to the composite sliding mode surface (12) in finite time and maintain a sliding motion on it; thereafter, if the adaptive update laws are chosen as(22)W^˙=πwλ2sTφx,W^,(23)ζ^˙=πζλ2ImTs,where ζ^ is the estimation of ζ, πw>0 and πζ>0 are the designed constants.Proof.
The Lyapunov candidate function is chosen as(24)V=12sTs+12W˜πw−1W˜T+12ζ˜πζ−1ζ˜T.
Taking its time derivative, we have(25)V˙=sTs˙−W˜πw−1W^˙T−ζ˜πζ−1ζ^˙T=sTλ1Da1+1e+λ2A21x1+A22x2+g2+Bu+η2−C−1y˙d−W˜Tπw−1W^˙−ζ˜Tπζ−1ζ^˙T.
Substituting (21) into (25), one can obtain(26)V˙=sT−λ2η^2x,W^−k3sb2signs−λ2Imζ^+λ2η2−k2s−W˜πw−1W^˙T−ζ˜πζ−1ζ^˙T.
According to Assumption3 and using (22) and (23), yields(27)V˙=sTλ2W˜Tφx,W^+λ2ϱ−λ2Imζ^−k3sb2signs−k2s−W˜λ2sφx,W^−ζ˜λ2sTIm≤−k3sb2+1−k2s2,Therefore, V˙≤0, namely, s and the estimation errors W˜,ζ˜ all converge to zero asymptotically.
Letk1=1/2Vπw−1W˜2+πζ−1ζ˜2, and it will be known that 0<k1<1 and 0<1−k1<1 for t>0. (27) can be rewritten by(28)V˙=−2b2+1/2k3s2/2b2+1/2−2k2s22=−2b2+1/2k31−k1b2+1/2Vb2+1/2−2k21−k1V,where 0<b2+1/2<1.
Then, (28) can be simplified into(29)V˙≤−κ1V−κ2Vb2+1/2≤0,where κ1=2k21−k1 and κ2=k32−2k1b2+1/2, and it is easily known that κ1>0 and κ2>0.
Therefore, from Lemma1 the tracking error system (11) is finite time asymptotically stable. System (11) can be driven to the sliding mode surface (12) in finite time and maintains a sliding motion on it. Moreover, the reaching time satisfies(30)tc≤2κ11−b2lnκ1V1−b2/2+κ2κ2.From sliding mode control theorem, Theorem 2 together with Theorem 1 shows that the tracking error e of closed-loop system (4) is asymptotically stable, that is, y accurately tracks desired signal yd when t≥tc. The proof is completed.Remark 2.
The normal sliding mode control (SMC) technique has just been applied in the controller design for epilepsy seizure abatement [20]. Instead of normal sliding mode (NSM) control in [20], a fractional order sliding mode (FOSM) technique is adopted in Theorem 2, which will provide more accurate tracking and control performance before and after reaching sliding mode surface.Remark 3.
In [44], an integer order sliding mode (IOSM) control scheme based on exponential approaching law is developed. Although the finite time fractional order sliding mode (FFOSM) in this paper also adopts the power exponential approaching law, different from [44], this controller combining fractional order sliding mode control with neural networks (NN-FFOSM) technique can provide higher tracking precision and control performance.Remark 4.
In [40], a fractional order sliding mode controller is designed based on neural networks. Compared with [40], a finite time approaching law is employed instead of constant velocity one in this paper. Combined with finite-time characteristics, the NN-FFOSM controller can drive the system to reach the sliding mode surface in finite time from arbitrary initial states. Therefore, Theorem 2 can be considered as an extension and promotion of [40] and can guarantee output y tracks the desired signal yd in shorter time.


## 3.1. Design of Sliding Mode Surface

In this section, a sliding surface is designed for system (4) and the stability of the corresponding tracking control system is analyzed.For system (4), assume that the desired output yd is differentiable. Then, the tracking error e=y−yd, and its derivative with respect to time is(11)e˙=CA21x1+A22x2+B2u+g2+η2−C−1y˙d.To make the dynamics of the thalamocortical system of epilepsy (4) to track the desired normal activities, namely, y⟶yd, a sliding mode surface will be developed to make the dynamical error system stable.The composite sliding surface for system (4) using fractional order sliding mode technique can be selected as follows:(12)s=λ1Da1e+λ2C−1e,where s∈Rm, λ1>0,λ2>0,and0<a1<1 are designed constants, respectively.Assumption 2.
The system outputy=y1,…,ymT, and then its initial value yi0i=1,…,m is a positive constant.Theorem 1.
Consider the control system (11). On Assumptions 1 and 2, the sliding motion of system (11) corresponding to the sliding surface (12) is asymptotically stable.Proof.
During sliding motions=0, the following equation can be obtained from (12):(13)Da1e+Πe=0,where Π=λ2/λ1C−1 is a diagonal matrix, Π=diagπii=1,…,m, and πi>0.
Under Lemma2, the solution of the abovementioned equation is as follows:(14)ei=μit−a1,μi>0.
Consider the Lyapunov function candidate:(15)V1=12eTe.
Then, the derivative ofV1 along the trajectory of system (13) is(16)V˙1=eTe˙=∑i=1m−a1μi2t−2a1−1,Thus, V˙1<0 for t∈0,∞. It is shown that the sliding motion of system (11) corresponding to the sliding surface (12) is asymptotically stable, namely, limt⟶∞ei=limt⟶∞yi−yid=0. Therefore, it is guaranteed that the output of system (4) tracks the desired signal after the system is driven to the designed sliding surface (12). Proof is completed.
Necessary conditions to ensure this sliding motion which is attained and maintained will further be developed.


## 3.2. Reachability Analysis

It is extremely important to track desired signal with the rapidity and accuracy [42, 43], which determines the therapeutic effect. However, the conventional sliding mode control usually results in low precision and convergence rate problems. To solve these problems, a finite time fractional order sliding mode (FFOSM) control scheme for epilepsy seizure abatement is designed, as shown in Figure 4.Figure 4
NN-FFOSM control scheme for epilepsy seizure abatement.Considering the synthetic effect of uncertain change in cerebral blood flow, glucose metabolism, BOLD, electromagnetic disturbances, and upper bound of the synthetic effect may be unknown. Then, the uncertainty termη2 affected by external stimulation can be approximated by employing an ideal RBFNN as(17)η2x,W=WTφx+ϱ,ϱ≤ζ,where W∈R1×m is an ideal weight vector from the hidden layer to the output layer, φx is the basis function, ϱ∈Rm×1 is the approximation error vector, and ζ is an unknown positive constant.LetW^ be the estimation of W, and we have(18)η^2x,W^=W^Tφx.LetW˜=W−W^ be the weighting estimation error of W, and we have(19)η2x,W−η^2x,W^=W˜Tφx+ϱ.Assumption 3 (see [40]).
The estimation errorϱ is norm-bounded as ϱ≤ζ, where ζ>0 is an unknown constant.Remark 1.
Sinceϱ should be very small when η^2x approximate η2x well, along with the control command u bounded in any practical system, it is feasible to assume ϱ in (19) is norm-bounded. However, it is not required that the bound ζ is known.
For system (4), differentiating (12) and using (11), it follows that(20)s˙=λ1Da1+1e+λ2C−1e˙=λ1Da1+1e+λ2A21x1+A22x2+B2u+g2+η2−C−1y˙d.
The fractional order controller can be designed as(21)u=B2−C−1y˙d−η^2x,W^−λ1λ2Da1+1e−A21x1−A22x2−k2λ2s−k3λ2sb2signs−g2−Imζ^,where B2− is the generalized inverse matrix of B2. a1, b2, k2, and k3 are designed constants with 0<a1,b2<1, k2>0, and k3>0. Im∈Rm×1 is an m-dimensional unit vector.Theorem 2.
Consider the control system (11). On Assumptions 1–3, the fractional order sliding mode control law (21) can drive system (11) to the composite sliding mode surface (12) in finite time and maintain a sliding motion on it; thereafter, if the adaptive update laws are chosen as(22)W^˙=πwλ2sTφx,W^,(23)ζ^˙=πζλ2ImTs,where ζ^ is the estimation of ζ, πw>0 and πζ>0 are the designed constants.Proof.
The Lyapunov candidate function is chosen as(24)V=12sTs+12W˜πw−1W˜T+12ζ˜πζ−1ζ˜T.
Taking its time derivative, we have(25)V˙=sTs˙−W˜πw−1W^˙T−ζ˜πζ−1ζ^˙T=sTλ1Da1+1e+λ2A21x1+A22x2+g2+Bu+η2−C−1y˙d−W˜Tπw−1W^˙−ζ˜Tπζ−1ζ^˙T.
Substituting (21) into (25), one can obtain(26)V˙=sT−λ2η^2x,W^−k3sb2signs−λ2Imζ^+λ2η2−k2s−W˜πw−1W^˙T−ζ˜πζ−1ζ^˙T.
According to Assumption3 and using (22) and (23), yields(27)V˙=sTλ2W˜Tφx,W^+λ2ϱ−λ2Imζ^−k3sb2signs−k2s−W˜λ2sφx,W^−ζ˜λ2sTIm≤−k3sb2+1−k2s2,Therefore, V˙≤0, namely, s and the estimation errors W˜,ζ˜ all converge to zero asymptotically.
Letk1=1/2Vπw−1W˜2+πζ−1ζ˜2, and it will be known that 0<k1<1 and 0<1−k1<1 for t>0. (27) can be rewritten by(28)V˙=−2b2+1/2k3s2/2b2+1/2−2k2s22=−2b2+1/2k31−k1b2+1/2Vb2+1/2−2k21−k1V,where 0<b2+1/2<1.
Then, (28) can be simplified into(29)V˙≤−κ1V−κ2Vb2+1/2≤0,where κ1=2k21−k1 and κ2=k32−2k1b2+1/2, and it is easily known that κ1>0 and κ2>0.
Therefore, from Lemma1 the tracking error system (11) is finite time asymptotically stable. System (11) can be driven to the sliding mode surface (12) in finite time and maintains a sliding motion on it. Moreover, the reaching time satisfies(30)tc≤2κ11−b2lnκ1V1−b2/2+κ2κ2.From sliding mode control theorem, Theorem 2 together with Theorem 1 shows that the tracking error e of closed-loop system (4) is asymptotically stable, that is, y accurately tracks desired signal yd when t≥tc. The proof is completed.Remark 2.
The normal sliding mode control (SMC) technique has just been applied in the controller design for epilepsy seizure abatement [20]. Instead of normal sliding mode (NSM) control in [20], a fractional order sliding mode (FOSM) technique is adopted in Theorem 2, which will provide more accurate tracking and control performance before and after reaching sliding mode surface.Remark 3.
In [44], an integer order sliding mode (IOSM) control scheme based on exponential approaching law is developed. Although the finite time fractional order sliding mode (FFOSM) in this paper also adopts the power exponential approaching law, different from [44], this controller combining fractional order sliding mode control with neural networks (NN-FFOSM) technique can provide higher tracking precision and control performance.Remark 4.
In [40], a fractional order sliding mode controller is designed based on neural networks. Compared with [40], a finite time approaching law is employed instead of constant velocity one in this paper. Combined with finite-time characteristics, the NN-FFOSM controller can drive the system to reach the sliding mode surface in finite time from arbitrary initial states. Therefore, Theorem 2 can be considered as an extension and promotion of [40] and can guarantee output y tracks the desired signal yd in shorter time.


## 4. Simulation

As an illustrative example, the thalamocortical model (1) with uncertainties and disturbances is employed for the study of the NN-FFOSM controller. To verify the effectiveness of the proposed controller scheme, the simulation results are carried out by utilizing three comparisons between NN-NSM in [20], IOSM in [44], NN-FOSM in [40], and our NN-FFOSM controller in this section.The initial value ofIN,TC,PY,RET=0.05,0.05,0.05,0.05T, matrix H=0,0,0,0.5, coefficients he=−0.35, hi=−3.4, ht=−2, hr=−5, τe=26, τi=32.5, τt=2.6, τr=2.6, cee=1.8, cei=1.5, cet=1, cie=4, cte=3, ctr=0.6, cre=3, crt=10.5, crr=0.2, ε=2⋅105, a=2.8, and b=0.5. Here, the control variable is the subpopulation on PY and IN, simulating external stimulations on the cortical area, that is B=1,0,1,1T. According to the expected dominant frequency around 12 Hz, the desired output is set as ydt1=0.2+0.02sin75.36t (1this normal brain signal was provided by China-Japan Union Hospital of Jilin University), while the frequency of output yt2 (2this abnormal brain discharge signal was provided by China-Japan Union Hospital of Jilin University) is around 3 Hz when patient’s absence epilepsy seizures happen. The other relevant parameters of the thalamocortical model are referred to [20]. After basic transformation, system (1) is transformed into closed-loop control system (4) in the new coordinate system, and the transformation matrix T and its inverse matrix are as follows:(31)T=100−10100001−10001,T−1=1001010000110001.Regarding the model uncertainties, for simulation purpose, it is assumed that there are10% parameter variations for the pyramidal neuronal population PY and specific relay nucleus TC with respect to their nominal values due to blood flow change of the cerebral cortex. The unknown disturbance is assumed as dt, which includes randomly distributed pulses dpt and a Gaussian noise dnt with standard deviation 0.01 and mean 0. Simulation time duration is 15 seconds in total and simulation step is 0.01 second. During the simulation, two abnormal brain nucleus discharge scenarios are implemented as follows.In Case 1, the controller does not work until 5s and continues to work until the end of the simulation:(32)dpt=0.1,t=0.75∗ks,k=1,2,…,20.0,others,where dpt is small pulses injected, which is used to simulate a series of electrocardiographic (ECG) disturbance signals and the heart rate is supposed as 80 beats per minute.In Case 2, the controller works at the beginning and continues until the end of the simulation:(33)dpt=30,t=4,10s,0,others,where dpt is randomly distributed pulses injected, which is used to simulate large electromyographic (EMG) disturbance signals of blinking.Parameters used in fractional order sliding mode control based on neural networks of Theorems1 and 2 are listed in Table 1. According to [20], the sliding mode surface of normal sliding mode based on neural networks (NN-NSM) should be s=λ2e. Meanwhile, referring to [44], the sliding manifold of integer order sliding mode (IOSM) can be chosen as s=λ1e˙+λ2e+λ3∫e,λ3=0.02. Correspondingly, parameters concerning parts of three methods are also the same.Table 1
The control performance comparisons under different control schemes.
Controller
λ
1
λ
2
a
1
b
2
k
2
k
3
NN-FFOSM
0.01
0.1
0.5
0.5
0.2
5The control objective is to maintain the outputy of the thalamocortical model to track the desired output yd. The following simulation results are to be carried out and discussed using different tracking controllers.In following figures, the randomly distributed pulsedpt is shown in red solid line, the output yt and tracking error et of system is in blue solid line, and the desired trajectory ydt is in dark dashed line. Moreover, the right side of each simulation figure is occupied by the results of NN-FFOSM, while the left side is occupied by the results of NN-NSM, IOSM, or NN-FOSM.


### 4.1. Comparisons between NN-NSM and NN-FFOSM

Comparisons of performances using the normal sliding mode (NN-NSM) controller of [20] and the finite time fractional order sliding mode (NN-FFOSM) controller of Theorem 2 are provided in Figures 5–8. In Case 1, Figure 5 depicts control performance and output dynamic using NN-NSM and NN-FFOSM respectively, while Figure 6 displays tracking error and its transient performance using these two methods. Similar to them, Figures 7 and 8 illustrate the accommodation of the system in Case 2. It can be found that tracking error of NN-NSM can converge to a small region of origin within 0.5s and 5s in Cases 1 and 2 while NN-FFOSM can converge to zero within 0.35s and 2.5s, respectively. Thus, it is illustrated that the settling time brought by large pulse in Case 2 is longer than by distributed small pulses in Case 1. The difference between NN-NSM and NN-FFOSM is that the latter has a fractional order term in sliding manifold, and it is not included in NN-NSM. Adjusting the fractional order in the sliding manifold can improve system dynamics as well as robustness. Adding fractional derivative terms can improve the parameters adaptation process such that the errors utilizing NN-FFOSM converge to zero more quickly and accurately, which is proved in Figures 5–8.Figure 5
Control performance of NN-NSM and NN-FFOSM in case 1.Figure 6
Tracking error of NN-NSM and NN-FFOSM in case 1.Figure 7
Control performance of NN-NSM and NN-FFOSM in case 2.Figure 8
Tracking error of NN-NSM and NN-FFOSM in case 2.
(a)
(b)


### 4.2. Comparisons between IOSM and NN-FFOSM

Control performance and input dynamic using integer order sliding mode (IOSM) in [44] and NN-FFOSM controller in Theorem 2 are presented in Figures 9–12. It can be seen from tracking performance comparison shown in Figures 10 and 12 that signal tracking error of the NN-FFOSM scheme is larger than that of IOSM. Both IOSM and NN-FFOSM controller can track given signal after 0.35s in Case 1 and 2.5s in Case 2. Although both IOSM and NN-FFOSM achieve signal tracking target, it can be seen from comparison of the tracking error in Figure 10 that the error using NN-FFOSM asymptotically converges to 0 while there are obvious oscillations in the tracking error using IOSM. It follows that NN-FFOSM controller has better effects to eliminate the pathological spike-and-wave discharges than IOSM. The main difference between IOSM and RBFNN-FFOSM is that the NN-FFOSM has an adjustable fractional order factor in the design of sliding manifold, and neural networks technique is also introduced in the FFOSM controller to estimate the disturbances and uncertainties. The adaptive estimation of the nonlinear term can online change the robust gain according to different situations and thus offset the influence caused by disturbances and uncertainties. It is proved that the accuracy of the NN-FFOSM controller is satisfactory.Figure 9
Control performance of IOSM and NN-FFOSM in case 1.Figure 10
Tracking error of IOSM and NN-FFOSM in case 1.Figure 11
Control performance of IOSM and NN-FFOSM in case 2.Figure 12
Tracking error of IOSM and NN-FFOSM in case 2.


### 4.3. Comparisons between NN-FOSM and NN-FFOSM

In [40], NN-FOSM is developed based on a constant velocity approaching law, which is carried out to restrain pathological spike-and-wave discharges firstly. Then, NN-FFOSM in Theorem 2 by means of the finite time power exponential approaching law is verified as well. Both have better robust characteristics and tracking accuracy. Furthermore, equation (12) expresses the same sliding manifold of NN-FOSM and NN-FFOSM. Due to the similarities and differences between two approaches, their simulation results and control performance are evaluated here. Figures 13 and 14 manifest tracking error adaptation of system (4) in Case 1 and Case 2, respectively. It is revealed that the system signal error using NN-FFOSM converges to its desired value faster than while using NN-FOSM, which indicates the effect of finite time function factor on convergence of the system. Their energy consumption is depicted in Figures 15 and 16. Meanwhile, it can also be observed from Figure 14 the steady state error of NN-FFOSM is in the order of magnitude 10−4, while the steady state error of NN-FOSM is in 10−3. Therefore, it can be concluded that the NN-FFOSM can accelerate the parameters adaptation process and improve tracking precision by adding finite time term in the NN-FOSM controller.Figure 13
Tracking error of NN-FOSM and NN-FFOSM in case 1.Figure 14
Tracking error of NN-FOSM and NN-FFOSM in case 2.Figure 15
Control performance of NN-FOSM and NN-FFOSM in case 1.Figure 16
Control performance of NN-FOSM and NN-FFOSM in case 2.Remark 5.
Compared with the work in [34, 35], it is clear to see that there is high-frequency switching in the control signals (e.g., Figure 11) which is well-known chattering in sliding mode control. Chattering may result in unnecessary mechanical wear and tear on the actuator component. However, one way of overcoming this drawback in real application is to introduce a boundary layer around sliding surface (see reference [45]).


## 4.1. Comparisons between NN-NSM and NN-FFOSM

Comparisons of performances using the normal sliding mode (NN-NSM) controller of [20] and the finite time fractional order sliding mode (NN-FFOSM) controller of Theorem 2 are provided in Figures 5–8. In Case 1, Figure 5 depicts control performance and output dynamic using NN-NSM and NN-FFOSM respectively, while Figure 6 displays tracking error and its transient performance using these two methods. Similar to them, Figures 7 and 8 illustrate the accommodation of the system in Case 2. It can be found that tracking error of NN-NSM can converge to a small region of origin within 0.5s and 5s in Cases 1 and 2 while NN-FFOSM can converge to zero within 0.35s and 2.5s, respectively. Thus, it is illustrated that the settling time brought by large pulse in Case 2 is longer than by distributed small pulses in Case 1. The difference between NN-NSM and NN-FFOSM is that the latter has a fractional order term in sliding manifold, and it is not included in NN-NSM. Adjusting the fractional order in the sliding manifold can improve system dynamics as well as robustness. Adding fractional derivative terms can improve the parameters adaptation process such that the errors utilizing NN-FFOSM converge to zero more quickly and accurately, which is proved in Figures 5–8.Figure 5
Control performance of NN-NSM and NN-FFOSM in case 1.Figure 6
Tracking error of NN-NSM and NN-FFOSM in case 1.Figure 7
Control performance of NN-NSM and NN-FFOSM in case 2.Figure 8
Tracking error of NN-NSM and NN-FFOSM in case 2.
(a)
(b)


## 4.2. Comparisons between IOSM and NN-FFOSM

Control performance and input dynamic using integer order sliding mode (IOSM) in [44] and NN-FFOSM controller in Theorem 2 are presented in Figures 9–12. It can be seen from tracking performance comparison shown in Figures 10 and 12 that signal tracking error of the NN-FFOSM scheme is larger than that of IOSM. Both IOSM and NN-FFOSM controller can track given signal after 0.35s in Case 1 and 2.5s in Case 2. Although both IOSM and NN-FFOSM achieve signal tracking target, it can be seen from comparison of the tracking error in Figure 10 that the error using NN-FFOSM asymptotically converges to 0 while there are obvious oscillations in the tracking error using IOSM. It follows that NN-FFOSM controller has better effects to eliminate the pathological spike-and-wave discharges than IOSM. The main difference between IOSM and RBFNN-FFOSM is that the NN-FFOSM has an adjustable fractional order factor in the design of sliding manifold, and neural networks technique is also introduced in the FFOSM controller to estimate the disturbances and uncertainties. The adaptive estimation of the nonlinear term can online change the robust gain according to different situations and thus offset the influence caused by disturbances and uncertainties. It is proved that the accuracy of the NN-FFOSM controller is satisfactory.Figure 9
Control performance of IOSM and NN-FFOSM in case 1.Figure 10
Tracking error of IOSM and NN-FFOSM in case 1.Figure 11
Control performance of IOSM and NN-FFOSM in case 2.Figure 12
Tracking error of IOSM and NN-FFOSM in case 2.


## 4.3. Comparisons between NN-FOSM and NN-FFOSM

In [40], NN-FOSM is developed based on a constant velocity approaching law, which is carried out to restrain pathological spike-and-wave discharges firstly. Then, NN-FFOSM in Theorem 2 by means of the finite time power exponential approaching law is verified as well. Both have better robust characteristics and tracking accuracy. Furthermore, equation (12) expresses the same sliding manifold of NN-FOSM and NN-FFOSM. Due to the similarities and differences between two approaches, their simulation results and control performance are evaluated here. Figures 13 and 14 manifest tracking error adaptation of system (4) in Case 1 and Case 2, respectively. It is revealed that the system signal error using NN-FFOSM converges to its desired value faster than while using NN-FOSM, which indicates the effect of finite time function factor on convergence of the system. Their energy consumption is depicted in Figures 15 and 16. Meanwhile, it can also be observed from Figure 14 the steady state error of NN-FFOSM is in the order of magnitude 10−4, while the steady state error of NN-FOSM is in 10−3. Therefore, it can be concluded that the NN-FFOSM can accelerate the parameters adaptation process and improve tracking precision by adding finite time term in the NN-FOSM controller.Figure 13
Tracking error of NN-FOSM and NN-FFOSM in case 1.Figure 14
Tracking error of NN-FOSM and NN-FFOSM in case 2.Figure 15
Control performance of NN-FOSM and NN-FFOSM in case 1.Figure 16
Control performance of NN-FOSM and NN-FFOSM in case 2.Remark 5.
Compared with the work in [34, 35], it is clear to see that there is high-frequency switching in the control signals (e.g., Figure 11) which is well-known chattering in sliding mode control. Chattering may result in unnecessary mechanical wear and tear on the actuator component. However, one way of overcoming this drawback in real application is to introduce a boundary layer around sliding surface (see reference [45]).


## 5. Conclusion

In this paper, the problem of tracking control against disturbances and uncertainties is investigated for epilepsy seizure abatement by using brain-machine interfaces technique. Firstly, the control system for epilepsy therapy is designed, and the external device connected to brain detects the abnormal brain spike discharge in patients with epilepsy and converts it into control signals to achieve stimulation therapy for the brain. Then, the synthesis effect of parametric uncertainties is considered. To obtain the estimated value of the synthesis effect of disturbance and uncertainty, the radial basis function neural network is introduced and adaptive estimators are also designed. Based on the estimation information, a novel finite time fractional order sliding mode controller is proposed to adjust the signal of the brain stimulation tracking control system of epilepsy. Meanwhile, the stability and reachability of the closed-loop control system are analyzed. Finally, simulation results are fully compared and verified to demonstrate the effectiveness of the presented control scheme.



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*Source: 3136715-2020-04-14.xml*