Fixed-time Stabilized Adaptive Sliding Mode Control for Synchronizations of Neuro-cell Circuit Systems

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Introduction
The architecture of circuits implemented for neuronal systems has recently been explored. (1) The frontier viewpoint is that the different types of neuronal circuit are the elementary components of an artificial neuro-cell network system, highlighting the importance of developing neuronal circuits based on the behavior of a neuro-cell. The establishment of neuronal circuits can accelerate the application of themes in artificial intelligence. Basically, an artificial neuro-cell network system requires the ability to rapidly adapt to unpredictable environments. This requirement has motivated the use of machine learning technologies to solve the difficult and complex problems of such systems. (2) To study the behavior of a neuro-cell, various biomathematical models such as the Hodgkin-Huxley (HH), (3) Hindmarsh-Rose, (4) and FitzHugh-Nagumo (FHN) (5)(6)(7) neuron models have been proposed. The HH neuron model is a relatively complete mathematical model and consists of a set of fourth-order nonlinear differential equations that describe the dynamics of the voltage of the neuron membrane base using the principles of electrochemistry. However, it contains four state variables and many system parameters, increasing the difficulty of its application. To simplify the analysis of the neuro-cell system, the traditional space-clamped FitzHugh-Nagumo (SFHN) neuron model (7) is more convenient for investigating the dynamic behavior of a neurocell system. Previous studies on the chaos and bifurcation of the SFHN neuron model are reviewed in Refs. 8 and 9. To extend the practical application of the traditional SFHN neuron model, circuits are implemented using various technologies, such as analogical electronic circuits, (10)(11)(12) feedback circuits consisting of operational amplifiers and resistors, (13) very large scale integration (VLSI), (14) field-programmable gate arrays (FPGAs), (15) and field-effect transistors (FETs). (16) Moreover, in the framework of the traditional SFHN neuro-cell system, many new types of neuronal circuit system have been proposed, such as thermosensitive neuron, (17) piezoelectric neuron, (18) photosensitive neuron, (19,20) and neuronal circuit systems with electromagnetic induction. (21)(22)(23) Such new types of neuronal circuit are contributing to the development of multidiscipline applications in science and technology. The various types of neuronal circuit are helpful for realizing novel and more practical neuronal circuit control systems. It is therefore of practical significance to study new control schemes of neural cell circuit systems.
Because the HH neuron model (3) is difficult to apply, the alternative SFHN neuro-cell model, which has two state variables, was developed. The dynamical system adopts the nonlinear circuit reported in Ref. 20 for analogical simulation, as shown in Fig. 1. The through current iL(t) of the induction coil L and the cross voltage v(t) of the nonlinear resistor NR are the state variables of the system. C, L, R, and RS are the system parameters and VS(t) is the input excitation.
For brevity of this paper, the derivation of the nondimensional dynamical system is omitted (see Ref. 20 for details). The state variables of an SFHN neuron are analogized by the corresponding voltages in the circuits. Therefore, the time responses of an electrical SFHN neuronal circuit for feedback control can be sensed by applying voltage-sensing equipment. Furthermore, the control schemes developed for synchronization are realized by applying a hardware circuit with sensing equipment. (24)  The SFHN neuro-cell model can be represented by nonlinear non-autonomous differential equations with respect to the normalized time t: (8) (1) The variable x 1 is a fast state, which models the membrane potential of the neuron cell. The variable y 1 is a slow state, which represents the recovery of the membrane potential. The system parameters α, β, and γ govern the dynamics of the neuro-cell model. The system uncertainty Δ(x 1 , y 1 ) and the external disturbance δ(t) are assumed to be bounded and to satisfy I ion_m in Eq. (1) represents the ionic current inside the master neuro-cell and I m cos(ω 1 t) represents external electrical stimulation (EES) with amplitude I m and frequency f m = ω m /2π. As shown in Fig. 2, the SFHN neuron cell exhibits chaotic dynamics for the parameter value set (8)  In previous studies on control problems for SFHN neuron models, chaotic control in an SFHN neuron model by an adaptive passive control method was addressed. (8) State synchronization between two SFHN neurons subjected to EES by applying different schemes such as back-stepping control, (25) adaptive control, (26) and linear matrix inequality (LMI)-based adaptive control technologies was introduced. (27) By considering different EESs and ionic currents between two SFHN neurons, state synchronization was achieved in Ref. 28, in which robust adaptive sliding mode control (SMC) was proposed, and in Ref. 29, in which SMC with an input-output linearized approach was developed. For coupled FHN neuron networks, synchronizations were achieved by employing a typical robust SMC scheme, (30) by adaptive time-delay feedback control, (31) or by lag synchronization using a feedback control scheme. (32) For the new types of FHN neuron system, the finite-time synchronization of a dual-memristorbased network has been introduced. (33) State-and phase-coupling synchronizations were developed for photosensitive FHN neurons in Refs. 34 and 35.
The motivation of this study emerged from the following observations. To achieve state synchronization between two SFHN neuro-cell systems, the main idea of the SMCs reported in past studies is to follow the error state trajectory to arrive at and remain on the sliding surface in the phase plane, which in Ref. 28 was a line with a negative slope and in Ref. 36 was a curve. Then, the error states were simultaneously converged to the origin along the defined sliding surface. The mathematical model of the SFHN neuro-cell system given by Eq. (1) describes how the recovery voltage y 1 is induced by the activation potential x 1 . That is, 1 . Motivated by this fact, in this study, we define a novel sliding mode (SM) to develop an adaptive SMC scheme that achieves the goals of state synchronization and anti-synchronization between two SFHN neuro-cell systems. The novel SM is defined such that the error of the activation potential converges to zero at a fixed time and remains on the sliding surface. Then, according to the second equation of Eq. (1), the error of the recovery voltage is also exponentially stabilized.
The novelty, features, and contributions of this study are summarized in the following. (1) In contrast to previous works on SMC schemes, (28,36) a novel integral-type fixed-time stabilized sliding mode (IFSSM) is introduced. This IFSSM consists of the fractionalpowered, proportional, and integral terms of two state variables. Furthermore, the two state variables are stabilized one by one. That is, on the sliding surface, one of the state variables becomes stable at a fixed time, then the other state variable is exponentially stabilized in sequence. The stability is proven by Theorem 1 in Sect. 3. (2) To control the state synchronization and anti-synchronization between two SFHN neuro-cell circuits, the robust adaptive SMC scheme based on the novel IFSSM is performed on the basis of Theorem 2 in Sect. 3. The developed control scheme is divided into two parts. First, the error state trajectory is controlled to arrive at and remain on the sliding surface, which is defined by the IFSSM. Second, on the basis of the stable property of the IFSSM, the error of the activation potential first tends to zero at a fixed time, then remains on the sliding surface. The error of the recovery voltage is then exponentially stabilized. (3) In previous studies, the problems of state synchronization and anti-synchronization were solved using different types of control method to complete control missions. (37)(38)(39)(40)(41)(42) In contrast, in this work, the robust adaptive IFSSM control approach is introduced to simultaneously solve the control problems of state synchronization and anti-synchronization by suitably adjusting the design parameters, which are included in the control scheme. (4) The developed adaptive IFSSM control approach, which contains four feedback gains, can dispose of the nonlinear dynamics without directly eliminating the nonlinear terms commonly adopted in the previous control approaches. (37)(38)(39)(40)(41)(42) Meanwhile, the four gains are updated online using the proposed algorithm. We analyze the stability for both types of synchronization using Lyapunov stability theory.
(5) The validity and feasibility of the developed control scheme are verified by numerical simulation, demonstrating that the scheme can support most future implementations of neuronal circuits. In Sect. 4, numerical experiments are carried out to evaluate the performance of the proposed control approach. The remainder of this paper contains the following. The control problems of state synchronization and anti-synchronization between two SFHN neuro-cell circuit systems are formulated in Sect. 2. In Sect. 3, the design procedures of the adaptive IFSSM control scheme are developed to achieve the two types of synchronization. In addition, proofs of the stability of the control systems are provided. Numerical simulations performed to verify the validity of the proposed control scheme are reported in Sect. 4. Finally, some concluding remarks are made in Sect. 5.

Control Problems of Synchronization
We consider the control problems of state synchronization and anti-synchronization between two SFHN neuro-cell systems with different ionic currents and EESs. The master SFHN neurocell system with a system uncertainty and an external disturbance is given the form in Eq. (1). The slave SFHN neuro-cell system has a similar form to Eq. (1) and is described as where x 2 , y 2 are the state variables of the slave neuro-cell, I ion_r is the ionic current in the slave neuro-cell, and I r cos(ω r t) represents the ESS with amplitude Ir and frequency f r = ω r /2π. In the control problems of the study, we take into account the different ionic currents and EESs of the SFHN neuro-cells in Eqs. (1) and (3), that is, I ion_r ≠ I ion_m , I r ≠ I m , ω r ≠ ω m , and u(t) is the control scheme to be designed. Furthermore, we assume that the master and slave neuro-cells in Eqs. (1) and (3), respectively, have unique solutions in the time interval [t 0 , ∞), t 0 > 0 for any given initial conditions. In addition, under the influences of the system uncertainty Δ(x 1 ,y 1 ), external disturbance δ(t), and control input u(t), the two systems can still process bounded state trajectories. The control problems for state synchronization and anti-synchronization are formulated in the following. The synchronous error states between the systems in Eqs. (1) and (3) are defined as is the scaling factor defining the relation between the synchronous systems. The control goal of the current problems is to design an appropriate control scheme u(t) such that for any initial conditions of the two neuro-cells in Eqs. (1) and (3), the behavior of the response neuro-cell system converges to that of the master one, that is, x t x t y t y t λ λ →∞ →∞ → → , where λ = 1 for state synchronization and λ = −1 for anti-synchronization.
For the case of λ = 1, by taking the derivative of Eq. (4) with respect to the normalized time t, the dynamics of the synchronous error states in the state synchronization are expressed as where the functions are bounded because of the bounded state trajectories of x 1 (t), x 2 (t) for the master and slave neuron cell systems. f 1 (x 1 ,y 1 ) and f 2 (x 2 ,y 2 ) are, respectively, the state-dependent and time-varying coefficients of the synchronous error state e x (t).
For the case of anti-synchronization with λ = −1, by taking the derivative of Eq. (4) with respect to t, the dynamics of the synchronous error states are obtained as 3 where the functions are bounded owing to the bounded state trajectories of x 1 (t), x 2 (t) for the master and slave neuro-cell systems.
In this case, f 4 (x 1 ,x 2 ) in Eq. (6) is different from f 2 (x 1 ,x 2 ) in Eq. (5). f 4 (x 1 , x 2 ) is regarded as a bounded external disturbance in the design of the control scheme.

Definition 1
The achievement of state synchronization or anti-synchronization between the two SFHN neuro-cells given by Eqs. (1) and (3) is equivalent to the state variables in Eqs. (5) and (6)  The state synchronization or anti-synchronization for the two neuro-cells given by Eqs. (1) and (3) is clearly equivalent to the stabilization of the synchronous error systems in Eqs. (5)  For the design of the control scheme for the state synchronization or anti-synchronization of two SFHN neuron cell systems, the effects of the nonlinear functions f i (x 1 , x 2 ), i = 1, 2, 3, 4 are key points to be considered. Many approaches to compensating for nonlinear effects in chaotic control problems have been proposed, such as active control (37,38) and active SMC methods. (39)(40)(41)(42) In this study, an IFSSM is proposed for the control, and a robust adaptive SMC with timevarying feedback gains is introduced to realize such control.

Design of Adaptive Control Scheme
In this section, an adaptive IFSSM control is proposed to achieve state synchronization or anti-synchronization between the neuro-cells given by Eqs. (1) and (3). The design approach of the adaptive IFSSM control scheme includes two basic steps. First, the IFSSM with the prescribed sliding motion is selected. The defined SM is designed such that on the sliding surface the synchronous error state e x (t) is first stabilized at a fixed time. Then, the synchronous error state e y (t) is exponentially stabilized to complete the synchronization. Second, adaptive control u(t) is performed such that the state trajectory of e x (t), e y (t) in the phase plane arrives at and remains on the sliding surface despite the system uncertainties Δ(x 1 ,y 1 ) and the external disturbances δ(t).

Definition 2
The proposed IFSSM s(t) is defined by where ρ > 0, p > 0, q > 0 are odd integers with 1 < p/q < 2 to escape the singularity of the equivalent control. The novel IFSSM consists of the fractional-powered, proportional, and integral terms of two state variables. Theorem 1 ensures the fixed-time stability of s(t) = 0.
Then, e x (t) = 0 is maintained 0 s t T ∀ ≥ > , where t = t 0 > 0 is the time required for the state trajectory in the phase plane to arrive at s(t) = 0 from initial values e x (0), e y (0). Then, e y (t) is exponentially stabilized.

Proof
Assume that the system is controlled such that the state trajectory of e x (t), e y (t) arrives at s(t) = 0, then remains there, so that ( ) 0 s t =  is satisfied. By substituting the second equation in Eq. (5) or (6) Integrating with respect to the normalized time , where t 0 > 0 is the time required for the state trajectory of e x (t), e y (t) in the phase plane to arrive at s(t) = 0 from the initial values e x (0), e y (0), yields The fixed time T s taken to move from thus proving Theorem 1.
In the following, the robust and adaptive IFSSM control u(t) in Eq. (5) or (6) for state synchronization or anti-synchronization is provided in Theorem 2. In the literature, (37,39,40) the developed control methods only solved a defined nonlinear chaotic control problem. In contrast, the proposed control scheme can be applied to solve both types of synchronization problem by simply suitably tuning the designed parameters. Theorem 2. The adaptive IFSSM control scheme u(t) = u eq (t) + u sw (t) in Eq. (5) or (6) where the designed parameters ρ > 0, 0 < n < 1, p > 0, q > 0 are integers with 1 < p/q < 2, s(t) is the SM defined in Eq. (7), and sign( )  expresses the sign function. The adaptive feedback gains K i (t), i = 0,1,2,3 are updated online in accordance with the adaption algorithms . This causes e x (t) to first converge to zero at a fixed time T s approximated by Eq. (8). Then, e y (t) is exponentially stabilized in sequence. Thus, the state synchronization or antisynchronization of the SFHN neuro-cells given by Eqs. (1) and (3) is accomplished.

Proof
(1) For the case of state synchronization, λ = 1, the candidate positive Lyapunov function for the dynamical system given by Eq. (5) is chosen as where k i > 0, i = 0, 1, 2, 3 are positive constants satisfying Taking the derivative of Eq. (14) with respect to t with the solutions of the synchronous error system in Eq. (5), the selected SM in Eq. (7), and the adaptive IFSSM control scheme in Eqs. (12) and (13) From Eq. (15), it is proved that V 1 (t) is a positive definite function, and from Eq. (17), it is proved that it is a decreasing function. Thus, the zero equilibriums (s = 0, K i (t) = k i , i = 0, 1, 2, 3) are globally asymptotically stable. Therefore, e x (t), e y (t) in Eq. (5) will asymptotically approach the sliding surface s(t) = 0 with ( ) 0 s t =  and remain there. On this surface, e x (t) is stabilized at a fixed time T s by suitably choosing odd integers ρ > 0, p > 0, q > 0 with 1 < p/q < 2 in accordance with Eq. (8). Then, the exponential stabilization of e y (t) is guaranteed by Eq. (11), thus realizing the state synchronization between the systems given by Eqs. (1) and (3) and completing the proof.
(2) For the case of anti-synchronization, λ = −1, the positive Lyapunov function for the system in Eq. (6) is selected to have the form in Eq. (14), where g i > 0, i = 0, 1, 2, 3 are positive constants satisfying The remainder of the proof is similar to that for state synchronization and is omitted.

Numerical Simulation Studies
The hardware implementation of neuronal circuits is a mature technology. Before the future practical realization of the developed control scheme, its validity and feasibility must be verified. An advantage of the following numerical experiments is that the performance of the overall SFHN neuro-cell control system can be evaluated by simulations, thus supporting the further implementation of neuronal circuits.
In this section, numerical simulations are carried out to verify the validity of the proposed adaptive IFSSM control scheme. The fourth-order Runge-Kutta method with a time step of 0.0001 and initial conditions (x 1 , y 1 ) = (−0.5, 0.75), (x 2 , y 2 ) = (1.0, 0.6) is applied. The system parameters of the neuron cell are α = 0.25, β = 0.02, and γ = 0.25. The EESs of the master and slave SFHN neuro-cells are assumed to have different current amplitudes and frequencies, and the corresponding values are taken as I m = 0.055, ω m = 0.1, I r = 0.06, ω r = 0.15. The ionic currents of the two SFHN neuro-cells are selected to be I ion_m = 0.1 and I ion_r = 0.082. The system uncertainty and the external disturbance of the master neuro-cell system are assumed to be Δ(x 1 ,y 1 )= 0.15sin(x 1 )cos(y 1 ) and d(t) = 0.15sin(0.05πt), respectively. The numerical simulations are first performed with the master and slave neuro-cells running without control. Then, at t = 320, the control input of the slave neuro-cell system is implemented for state synchronization or anti-synchronization between the two neuro-cell systems.
For state synchronization, λ = 1, the positive design parameters of the proposed adaptive IFSSM control scheme in Eq. (5) described by Eqs. (12) and (13) where K x (t) and K y (t) are, respectively, the adaptive feedback gains updated using the following adaptive rules:

Remark 1
The controllers in Eqs. (12) and (21) provide discontinuous control. To reduce the phenomenon of chattering, the sign function in the control is modified as tanh(s/ε), where ε is a sufficiently small design constant. This modification is valid in all the numerical simulations.

Remark 2
Different from adaptive control scheme #A in Eq. (21), the synchronous error state e y (t) in Eq. (5) is compensated adaptively without directed elimination by utilizing the equivalent control part u eq (t) in Eq. (12). Figure 3 shows that e x (t), e y (t), and s(t) behave irregularly when the control input is turned off, and when the control is implemented at t = 320, e x (t), e y (t) converge to zero and the state synchronization is completed. Figure 4 shows a comparison of the proposed adaptive IFSSM control scheme with adaptive control scheme #A in Ref. 28, which indicates that e x (t) is first stabilized at a fixed time. Then, e y (t) is exponentially stabilized for the adaptive IFSSM control scheme. Figure 5 shows that the control input signals of both the proposed adaptive IFSSM control scheme and adaptive control scheme #A are continuous and chatter free. The peak value for the proposed control scheme is less than that for adaptive control scheme #A. The time responses of the adaptive feedback gains are shown in Fig. 6; the gains finally become constants. The figure indicates that e x (t), e y (t), and s(t) all tend to zero in accordance with the adaption algorithm in Eq. (13). The case of state synchronization is achieved by the proposed control scheme in the aforementioned illustrated example. In the following, it is shown that the case of antisynchronization is also achieved by the same control scheme with only suitable tuning of the designed parameters.
For the anti-synchronization, λ = −1, the positive design parameters of the proposed adaptive IFSSM control scheme of the system in Eq. (6) described by Eqs. (12) and (13) are taken as p = 31, q = 19, ρ = 0.01, n = 1/8, γ 0 = 0.5, γ 1 = 0.005, γ 2 = 0.01, and γ 3 = 0.01. Figures 7 and 8 show that anti-synchronization is achieved with stable characteristics similar to those in the case of state synchronization. That is, ex(t) is first stabilized at a fixed time, then e y (t) is exponentially stabilized. Figure 9 illustrates the time responses of the state variables for the two SFHN neuro- cells given by Eqs. (1) and (3). As expected, the state trajectories of the two neuro-cells separate from each other when different initial conditions are chosen. After the implementation of control at t = 320, both state variables become anti-synchronized, with the master neuro-cell system exhibiting system uncertainties and external disturbances.

Conclusions
We have proposed an adaptive IFSSM control scheme to solve the control problems of state synchronization and anti-synchronization between two SFHN neuro-cell systems in the presence of system uncertainty and external disturbance. For the control design, the novel defined IFSSM consists of the fractional-powered, proportional, and integral terms of two state variables. The characteristics of the IFSSM on the sliding surface are also proven and experimentally verified. That is, the error state e x (t) is first stabilized at a fixed time, then the error state e y (t) is exponentially stabilized in sequence. The stability of both types of synchronization by applying the proposed adaptive IFSSM control scheme is proven in detail. Toward the future practical realization of neuronal circuits, numerical simulations are performed to verify the effectiveness of the present control scheme. It is also shown that the two control problems of state synchronization and anti-synchronization are solved by the proposed control scheme by simply suitably tuning the designed parameters.