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Sensors and Materials, Vol. 35, No. 10 (2023) 4571–4584
MYU Tokyo
S & M 3415

State Synchronous Control for Nerve-cell Systems
via Adaptive Faster Finite-time-stabilized Sliding Mode Method
Ruizhao Yang,1,2 Chi-Hsin Yang,1* Chuanjian Liang,1
Kun-Chieh Wang,1 and Hong-Yi Chen1
1School of Physics and Telecommunication Engineering, Yulin Normal University,

Yulin City, Guang Xi, 53700, China

2School of Physical Science and Technology, Guangxi Normal University,

Guilin City, Guang Xi, 541004, China(Received April 7, 2023; accepted September 22, 2023)

Keywords: adaptive control, state synchronous control, finite-time stabilization, FitzHugh–Nagumo
nerve-cell model, neuronal circuit, sliding mode.

Using a novel integral-type faster finite-time-stabilized sliding mode (IFFSSM), we
developed an adaptive control scheme for solving the state synchronous control problem for two
space-clamped FitzHugh–Nagumo (SFHN) nerve-cell systems under the inf luence ofdisturbances. Because of the characteristics of the SFHN nerve-cell model under the principle ofelectrochemistry, the special stability property of IFFSSM is that two state variables of thesystem are stabilized in sequence on the sliding surface. Two theorems of the stability were also
provided and proven. Numerical experiments were performed. The performance of the proposed
control method was validated for the future practical application to neuronal circuits.

1.

Introduction

Since the chaotic state synchronous control problem was introduced, it has attracted muchattention owing to the broad range of applications in many research fields such as securecommunication,(1) encryption,(2) circuits,(3) biomedical systems,(4) micro-electro-mechanicalsystems,(5,6) nervous systems,(7,8) and signal encoding and transmission.(9–11) Chaotic statesynchronization is the matching of the states between two chaotic systems that start fromdifferent initial conditions. In recent years, state synchronization of the nerve-cell system, whichis conducive to gaining a good comprehension of the behaviors of nerve cells, has been widely
investigated.(7,8,12–22)
In the past, chaotic behaviors were studied using various nerve models such as the Hodglein–
Huxley (HH) model,(23) the Hindmarsh–Rose model,(24) and the FitzHugh–Nagumo (FHN)
models.(25–27) The HH nerve-cell model(23) is a more complete differential equation model for
describing the dynamics of a nerve-cell’s membrane voltage. For practical application, the spaceclamped FHN (SFHN) nerve cell(27) is more convenient for investigating the dynamics of a*Corresponding author: e-mail: yang2020@ylu.edu.cn

https://doi.org/10.18494/SAM4451

ISSN 0914-4935 © MYU K.K.
https://myukk.org/

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Sensors and Materials, Vol. 35, No. 10 (2023)

nerve cell. The typical work of studying chaos and bifurcation of the SFHN nerve-cell model
was addressed in Refs. 12 and 28.
The architecture of circuit-implemented neuron systems was described in Ref. 29. The
frontier viewpoint is that the establishment of neuronal circuits, based on the behavior of thenerve cell system, can give rise to new application themes in artificial intelligence. It highlights
the importance of developing neuronal circuit systems. Moreover, quick adaptability is
demanded of artificial nerve-cell network systems to accommodate unpredictable environments.
Machine learning technologies are applied to overcome the difficult and complex issues of such
systems.(30) FHN neuronal circuits have been constructed of, for example, simple circuits,(31)feedback circuits consisting of operational amplifiers and resistors,(32) field programmable gatearrays,(33) and field-effect transistors (FET).(34) Furthermore, on the framework of the traditional
FHN nerve-cell system, many new types of FHN neuronal circuit system have been proposed,such as the photosensitive neuron,(22,35) the thermosensitive neuron,(36) the piezoelectricneuron,(37) and the neuronal circuit systems with electromagnetic induction.(38–40) Such new
types of neuronal circuit contribute to the development of multidiscipline applications in scienceand technology. The implementation of neuronal circuits in the development of a novel control
system design for nerve cells makes engineering sense. It will be effective in the operation ofnovel and more practical FHN neuronal circuit control systems with voltage-sensing equipment.
A single SFHN nerve-cell model subject to an external electric stimulation (EES), as appliedin the firing of a nerve cell, is defined by the differential equations vs the nondimensionaltime τ,(12)u1 =
−u13 + (a + 1)u12 − au1 − v1 + I 0 + I m cos(ωmτ ),−b(v1 − cu1 ),v1 =

(1)

where the state variable u1 is the rescaled activation potential and v1 denotes the rescaled
recovery voltage.(27) The positive constants a, b, c dominate the behavior of Eq. (1) where
Imcos(ωmτ) is an EES with amplitude Im and frequency f = ωm/2π , and I0 is an ionic current in
the nerve cell. An individual SFHN nerve-cell model with an EES is expected to exhibit avariety of behaviors. It is reported that the nerve-cell model displays complicated chaotic firingfor certain values of Im.(12,28) Otherwise, the dynamics of the SFHN nerve-cell model aresusceptible to the variation of the ionic current and the EES signal frequency. With the typicalsystem parameters of a = 0.25, b = 0.02, c = 0.25, Im = 0.055, I0 = 0.082, and ωm = 0.1, the SFHN
nerve-cell model shows chaotic dynamics.(12)In previous studies, the problem of chaotic control in a SFHN nerve-cell model has beenaddressed using an adaptive passive control method.(12) A variety of schemes such as adaptive
control, back-stepping control, and LMI-based adaptive control technologies have been
introduced for state synchronizations between two SFHN nerve cells under EES.(13–15) By
considering different EESs and ionic currents between two individual SFHN nerve cells, statesynchronization was achieved,(7) and the adaptive sliding mode control (SMC) with robustness
was proposed. Lu and Chen(8) developed SMC with an input–output linearized approach. For

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coupled FHN nerve-cell networks, synchronization was achieved by Zhang(17) using a typicalrobust SMC scheme with adaptive time-delay feedback control, and Ibrahim et al.(20) achieved
lag synchronization using a feedback control scheme. For the new types of FHN nerve-cellsystems, finite-time synchronization of two dual-memristor-based networks was introduced.(21)State coupling synchronization was developed by Zhang et al.(18) and phase synchronization wasdeveloped by Zhang et al.(19) and Hussain et al.(22) for photosensitive FHN neurons.
This study is motivated by the following observations. In state synchronization between two
SFHN nerve cells, the main idea of SMC reported in past studies is to force the error state
trajectory to arrive at and stay on the sliding surface, which, in Ref. 7, was a line with a negative
slope, and in Ref. 5, a bending curve, in the phase plane. Then, the error states simultaneously
converge to the origin along the defined sliding surface. Finally, a satisfactory performance ofcontrol was obtained by applying the proposed control scheme. In the mathematical model of the
SFHN nerve cell system shown in Eq. (1), it describes that recovery voltage v1 is induced by theactivation potential u1 under the principles of electrochemistry. That is, v̇ 1 + bv1 = bcu1.
Motivated by this, we define a novel sliding mode (SM) to develop an adaptive SMC scheme toachieve state synchronization between two SFHN nerve cells in this study. The novel SM isdefined as having stability such that, on the sliding surface, the error of the activation potential
first converges to and remains at zero in a finite time. Then, according to the second equation of
Eq. (1), the error of the recovery voltage is also exponentially stable.
The novelty and contributions of this study are as follows.
(1) Unlike the past works on the SMC scheme,(5,7) the novel integral-type faster finite-timestabilized SM (IFFSSM), which is defined by the proportional, fractional powered, andintegral terms of two state variables, has the property that two state variables are stabilized
in sequence. That is, a state variable is finite-time stabilized in advance and the other statevariable is subsequently exponentially stabilized on the sliding surface. This leads to a
special shape of the sliding surface defined by the IFFSSM that promotes stability. The
details of the stability are proven by the provided Theorem 1 in Sect. 3.
(2) To control the state synchronization between two SFHN nerve cells, the robust novel
adaptive SMC scheme based on the IFFSSM is performed in accordance with Theorem 2 inSect. 3. The main idea of the developed control scheme is divided in two parts. First, theerror state trajectory is controlled to arrive at and remain on the sliding surface defined by
the IFFSSM. Second, on the basis of the stable property of the IFFSSM, the error of theactivation potential tends to and remains at zero at a finite time in advance. Then, the errorof the recovery voltage is exponentially stabilized in sequence.
(3) Three state-feedback gains are included in the proposed adaptive IFFSSM control scheme tocope with nonlinear dynamics without eliminating any nonlinear terms. Meanwhile, thethree gains are updated online adaptively using an established algorithm. The stable statesynchronization is presented on the basis of the Lyapunov stability theory.
(4) The validation and feasibility of the developed control scheme are verified by numericalsimulations. They can mainly support the future implementation of neuronal circuits. In
Sect. 4, the numerical experiments we performed are described. The performance of the
proposed scheme is further checked. Although, in accordance with the stable properties of4574

Sensors and Materials, Vol. 35, No. 10 (2023)

the IFFSSM, the error states are stabilized in sequence in the simulation results, by tuning
the parameter involved in the proposed control scheme, the two error states are almost
simultaneously converged to the origin in the phase plane. However, the almost simultaneousconvergence leads to a good control performance of state synchronization. The details are
given in Sect. 4.
The content of the study is summarized. In Sect. 2, the state synchronous control problem for
two SFHN nerve cells is formulated. In Sect. 3, the definition of the novel IFFSSM and its stableproperty are provided. In addition, the concept of the design and the proof of stability of theadaptive IFFSSM control scheme are described. In Sect. 4, the numerical experiments are
discussed, and the conclusions are presented.

2.

Definition of State Synchronous Control Problem

The state synchronous control problem between two SFHN nerve cells is formulated byregarding the different EESs and I0. The master SFHN nerve cell is defined by Eq. (1) by adding
an external disturbance, d(τ), and a system uncertainty, Δ(u1, v1). The slave SFHN nerve cell isalso formulated as Eq. (1) with the addition of the input ϕ(τ), and is modeled asu2 =
−u23 + (a + 1)u22 − au2 − v2 + I 0_s + I s cos(ωsτ ) + φ (τ ),−b(v2 − cu2 ),v2 =

(2)

where u2, v2 are the state variables. Iscos(ωsτ) is the EES with amplitude Is and frequency
fs = ωs/2π, and I0_s is the ionic current. In the control problem of this study, the two EESs andtwo different I0 in Eqs. (1) and (2) are considered. We set I0_s ≠ I0, Is ≠ Im, ωs ≠ ωm. Furthermore,
we assume that the master and slave nerve cells of Eqs. (1) and (2), respectively, have uniquesolutions in the time interval τ ∈ [0, ∞) for any given initial condition. Both SFHN nerve cells
still show the behavior of bounded state trajectories under the influences of external disturbanced(τ), system uncertainty Δ(u1, v1), and designed control ϕ(τ).
The synchronous error states are defined asθu (t ) =−u2 (t ) u1 (t ), θ v (t ) =−v2 (t ) v1 (t ) .

(3)

The control scheme ϕ(t) in Eq. (2) is proposed for the state synchronous control problem so thatfor given different initial conditions of nerve cells of Eqs. (1) and (2), the responses of the slave
nerve cell approach those of the master nerve cell.
By taking the time derivative of Eq. (3), the dynamical system of the synchronous error states
is obtained as

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θu =[− F1 (u1 , u2 ) + F2 (u1 , u2 )]θu − aθu − θ v + ( I 0_s − I 0 )
+ I s cos(ωsτ ) − I m cos(ωmτ ) − ∆(u1 , v1 ) − d (τ ) + φ (τ ),−b(θv − cθu ),θv =(4)

where F1(u1, u2) = u12 + u1u2 +u22 and F2(u1, u2) = (a + 1)(u1 + u2) are bounded nonlinear
functions owing to both the master and slave nerve cells being chaotic with bounded phasetrajectories of state variables. In the design of the control scheme, F1(u1, u2) and F2(u1, u2) aretreated as variable gains of θu(τ) and are assumed to have upper bounds and to satisfy
0 < F1 (u1 , u2 ) ≤ M1 , 0 < F2 (u1 , u2 ) ≤ M 2 .

(5)Definition 1:
The accomplishment of state synchronization between the two SFHN nerve cells of Eqs. (1)and (2) is equivalent to the synchronous error states in Eq. (4) approaching zero. This means thatlim θu (τ ) → 0 and lim θ v (τ ) → 0.

τ →∞

3.

τ →∞

Design of Adaptive Integral-type Sliding Mode Control

It has been shown that the state synchronous control problem is altered to stabilize thedynamical system by Eq. (4) by applying the appropriate control scheme ϕ(τ). On the basis of thedefinition of IFFSSM, an adaptive control scheme for solving the state synchronous control
problem between two SFHN nerve cells is developed in this section. To develop the main
theorem, the lemma of finite-time stability is provided in advance.
Lemma:(41)For the nonlinear differential equationnms (τ ) =x (τ ) + ρ x(τ ) + λ [ x(τ )]

=0,

(6)

where λ > 0, ρ > 0, the positive n, m are odd integers that satisfy 0 < n/m < 1. If ∀x(τ =0) =x0 ≠ 0,x(τ) can reach the origin x = 0 at the fixed time of=Ts ρ ( x )( m − n ) m

m
0
ln 
+ 1 > 0 ,ρ ( m − n) λand remain there for ∀τ ≥ Ts > 0. Then, this state x(τ) is globally finite-time stable.

(7)

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Sensors and Materials, Vol. 35, No. 10 (2023)

Equation (6) is the definition of the faster terminal sliding surface in Ref. 41. The origin x = 0
is a terminal attractor. The proof of the lemma is omitted here but can be found in Ref. 41.
The concept for developing the adaptive IFFSSM control scheme is described in sequence.
Firstly, an IFFSSM σ(τ) is introduced and the finite-time stability of σ(τ) = 0 is proven. Secondly,on the basis of the IFFSSM, the adaptive control scheme is developed such that, in the phaseplane, the synchronous error states θu(t) and θ v(t) are forced to maintain σ(τ) = 0 and σ (τ ) = 0
under the influence of internal and external disturbances.
Definition 2:
The definition of the novel IFFSSM σ(τ) is given by
τ
τ
β 1

σ (τ ) =[θu (τ )] p / q + α
θv (t )dt  ,[θu (t )] p / q dt +  θ v (τ ) +
t 0=
t
=
0
c b


∫

∫

(8)

where α, β > 0 and positive p and q are odd integers with 1 < p/q < 2.
The introduced SM is defined by the proportional, fractional powered, and integral terms of
θu(t) or θ v(t), and the finite-time stability of the IFFSSM on the sliding surface σ(τ) = 0 is
obtained in accordance with the following Theorem 1.
Theorem 1.
For the second equation of Eq. (4) with the IFFSSM σ(τ) in Eq. (8), the synchronous error
state θu(τ) is globally finite-time stable with σ(τ) = 0 and σ (τ ) = 0. The finite time τs taken forθu(τ) to approach the origin θu = 0 and remain there, ∀τ ≥ τ s > 0, is given by
=
τs

α

p
( p − q )/ q
+ 1 + τ 0 ,ln  [θu (τ 0 ) ]α ( p − q)  β(9)where τ = τ0 > 0 is the time at which the synchronous error states θu(t), θ v(t) arrive at the sliding
surface σ(τ) = 0. Then, the synchronous error state θv(τ) is exponentially stabilized.
Proof.
For the given θu(0), θ v(0), when the synchronous error states θu(t) and θ v(t) reach the sliding
surface σ(τ) = 0 with σ (τ ) = 0, the dynamical system in Eq. (8) with the second equation in Eq.
(4) is equivalently altered asp( p / q ) −1 
( p / q ) −1 β  1 +  θ=θu (τ )]θu (τ ) + α [θu (τ )][v (τ ) + θ v (τ )  0qc bdθu (τ ) α q
βq
0,
⇒
+
θu (τ ) +[θu (τ )]2−( p / q ) =
dτ
p
p

=
σ (τ )

(10)

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4577

where 1 < p/q < 2 and 0 < 2 − (p/q) < 1. The corresponding initial condition is θu(τ0) ≠ 0, where
τ0 > 0 is the time of θu(t) and θ v(t) in Eq. (4) to reach σ(τ) = 0 from the given θu(0), θ v(0).
In accordance with the aforementioned lemma, by replacing the corresponding parameterswith λ = βq/p, ρ = αq/p, n/m = 2 – (p/q), we obtain the finite-time stability of θu(τ). The finite timeτs for approaching the origin θu = 0 and holding constant ∀τ ≥ τ s > 0 is calculated using Eq. (9).
Then, it is quite clear that θv(τ) is exponentially stabilized for θ v(τs) ≠ 0 under θu = 0, ∀τ ≥ τ s > 0.
That is,1 θv (τ ) + θv (τ ) = 0 ⇒ θv (τ ) = θv (τ s ) exp(−bτ ), ∀τ ≥ τ s > 0, b > 0.
b

(11)This proves Theorem 1.
Remark 1.
From Theorem 1, the sliding surface σ(τ) = 0 defined by the IFFSSM has a special shape. For
various sets of α, β, the shapes of sliding surfaces are shown in Fig. 1.
Remark 2.
The value of the finite time τs in Eq. (9) is mainly caused by tuning the parameter α > 0. In
addition, the performance of the controlled system depends on the chosen value of α > 0, as
discussed in Sect. 4.
On the basis of the IFFSSM in Eq. (8), the adaptive control with the corresponding adaption
algorithms is designed in Theorem 2 by taking into account the existence of system uncertainty
and external disturbance.
Theorem 2.
On the basis of the defined IFFSSM σ(τ) in Eq. (8), the control input in Eq. (4) is designed in
the form of ϕ(τ) = ϕeq(τ) + ϕsw(τ), where

Fig. 1.

(Color online) Shapes of sliding surfaces σ(τ) for various sets of α, β.

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Sensors and Materials, Vol. 35, No. 10 (2023)

φeq (τ ) = −βq2−( p / q ),θu (τ )][
p(12)

−  g 0 (τ ) + g1 (τ ) θu (τ ) + g 2 (τ ) θ v (τ )  ⋅ sign(σ (τ )),φsw (τ ) =and sign(‧) is the sign function. The restrictions of system parameters are β > 0 and positive p, q
are odd integers with 1 < p/q < 2. The adaptive gains gi(τ), i = 0, 1, 2 are respectively updated by
the following algorithms.
g=
0 (t ) γ 0 σ (τ ) θu (τ )( p / q ) −1g1=(t ) γ 1 σ (τ ) θu (τ )

p/q

g=
2 (t ) γ 2 σ (τ ) θu (τ )( p / q ) −1

, g o=(0) 0, γ 0 > 0, g1=(0) 0, γ 1 > 0(13)(0) 0, γ 2 > 0
θv (τ ) , g 2=

Then, the synchronous error states θu(τ), θ v(τ) in Eq. (4) can approach σ(τ) = 0 and remain
constant, that is, σ (τ ) = 0. Then θu(τ) is forced to tend to zero at a finite time in advance and θv(τ)
is stabilized exponentially as below. Finally, the state synchronous control problem of the two
SFHN nerve cells of Eqs. (1) and (2) is solved.
Proof.
There are two steps to prove Theorem 2 on the basis of the Lapunov stability theory. First, thepositive definite function, V(τ) ≥ 0, which consists of σ(τ) and gi(τ), i = 0, 1, 2, is selected. Then, it
is proven that the selected positive definite function is a decreasing function, that is, V̇ (τ) < 0.
Finally, it is ensured that σ(τ) = 0 and gi(τ), i = 0, 1, 2 are converging to fixed values.
The positive definite function is selected as

V (τ=
)

2
1 2
p
σ (τ ) + ∑( gi (τ ) − Gi )2 ≥ 0 ,2
i =0 2qγ i(14)where Gi > 0, i = 0, 1, 2 and satisfies

G0 > I 0 + I 0 _ s + I m + I s + ∆(u1 , u2 ) + d (τ ) ,G1 > M1 + M 2 + α q p + a ,(15)

G2 > 1.
Taking the derivative of Eq. (14) vs τ combined with the dynamical system in Eq. (4), the defined
IFFSSM in Eq. (8), and the adaptive control scheme in Eqs. (12) and (13), we obtain

Sensors and Materials, Vol. 35, No. 10 (2023)

4579

2

p
V (τ ) =σ (τ )σ (τ ) + ∑( gi (τ ) − Gi )g i (τ )i = 0 qγ ip
  p 2 g (τ ) − Giβ  θ
p q −1 
p q
θu + α [θu ] +  v + θv   + ∑ i
g i (τ )= σ  [θu ]
γi
c b
 q
  q i = 0pαqp q −1 =⇒ V (τ ) σ [θu ]⋅  − F1 ( x1 , x2 ) + F2 ( x1 , x2 ) +− a θu − θ vqp+ ( I 0_s − I 0 ) + I s cos(ωsτ ) − I m cos(ωmτ ) − ∆(u1 , u2 ) − d (τ )− ( g 0 (τ ) + g1 (τ ) θu + g 2 (τ ) θ v ) ⋅ sign(σ ) +

⇒ V (τ ) ≤(16)

p 2 gi (τ ) − Gig i (τ )∑ γq i =0ipp q −1θuσ ⋅  − G0 − I 0 _ s − I 0 − I s − I m − ∆ − d (τ )

q

(

)

− ( G1 − M1 − M 2 − α q p − a ) θu − ( G2 − 1) θ v  < 0.

(17)

Up to this stage, it is proven that the positive definite function, Eq. (14), is a decreasing function,
which confirms that σ = 0 and gi(τ) = Gi > 0, i = 0, 1, 2 are guaranteed. By applying the proposed
scheme in Eqs. (12) with (13), it is shown that θu(τ), θ v(τ) of the dynamical system in Eq. (4) are
asymptotically converging to σ(τ) = 0 and maintaining σ (τ ) = 0 from the given θu(0), θ v(0).
Besides, with σ(τ) = 0, θu(τ) is stabilized in advance at a finite time τs using Eq. (9) withsuitably selected values of α, β > 0 and the positive odd integers p and q with 1 < p/q < 2. Then,for θ v(τ), the exponential stabilization is achieved according to Theorem 1. Finally, the statesynchronous control problem between the two SFHN nerve cells of Eqs. (1) and (2) is solved.
This completes the proof.
Remark 3.
The control scheme in Eq. (12) includes the discontinuous term, i.e., the sign function sign(‧).
For the implementation of control input and to diminish the chattering phenomenon, sign(‧) in
Eq. (12) is replaced by tanh(σ/δ) with a small constant of δ = 10−6.

4.

Numerical Experiments

At present, the existing hardware implementation of neuronal circuits is a mature technology.
Before the future practical realization of our proposed control scheme, its validation andfeasibility are required. The preceding numerical experiments can provide the support for the
implementation of neuronal circuits with voltage-sensing equipment.
In the following section, the numerical experiments for the two SFHN nerve cells of Eqs. (1)and (2) are performed for the developed adaptive IFFSSM control. The Runge–Kutta numericalsolver for the differential equations with a time-step size of 10−4 is applied in the computation.

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The initial conditions (u1, v1) = (−0.25, 0.8), (u2, v2) = (0.25, 0.5) are set in sequence. The positive
constants of the nerve cell system are a = 0.25, b = 0.02, c = 0.25. Different amplitudes andfrequencies of the EESs for the master and slave SFHN nerve cell systems are taken as
Im = 0.055, ωm = 0.1, Is = 0.06, ωs = 0.15. The ionic currents applied to the two SFHN nerve cellsystems are set as I 0 _ s = 0.082 and I0 = 0.1, respectively. The external disturbance and thesystem uncertainty of the master nerve cell are modeled by d(τ) = 0.25‧sin(0.1πτ) andΔ(u1, v1) = 0.5‧sin(u1)sin(v1), respectively.
The first numerical experiment with the positive design parameters of p = 13, q = 11, α = 0.6,
β = 1.5, γ0 = 0.0015, γ1 = 0.001, γ2 = 0.01 for the adaptive IFFSSM control system in Eq. (4)associated with Eqs. (12) and (13) is proposed. The control input at τ = 0 triggers the statesynchronization between two nerve cells. The responses of IFFSSM σ(τ) with respect to thesynchronous errors θu(τ) and θ v(τ) are shown in Fig. 2. It is depicted that, when σ(τ) = 0 is
maintained, θu(τ) tends to zero in a finite time in advance. Next, θ v(τ) approaches zero whenσ(τ) = 0 and θu = 0. The stable characteristics of the defined IFFSSM in Eq. (8) with Theorem 1
are verified. In Fig. 3, the responses of θu(τ) and θ v(τ) vs τ are depicted. It is shown that θu(τ)
tends to zero faster than θ v(τ) converges to zero. Nevertheless, the performance of control in thisexperiment is unacceptable owing to the convergent process in θ v(τ) → 0 being sustained for aconsiderably long time.
Indeed, the performance of control is modified by suitably adjusting the designed parameter
α. For the same initial conditions, system parameters, and other designed parameters, except forα, as in the first experiment, the state trajectories corresponding to three values of α in the phaseplane are depicted in Fig. 4. It is demonstrated that, for the larger value of α, θu(τ) more quickly
approaches zero in advance when the state trajectories start to stay on σ(τ) = 0. Then, θ v(τ) issequentially convergent. By selecting a smaller value of α, almost identical convergences of θu(τ)
and θv(τ) are achieved. It is shown that the performance of control is improved by choosing asuitable value of α.
The second numerical experiment is conducted under the same simulation conditions as in
the first experiment except for choosing α = 0.45. In Figs. 5 and 6, the responses of θu(τ), θ v(τ)

Fig. 2.

Responses of σ(τ) versus θu(τ) and θ v(τ) for α = 0.6.

Sensors and Materials, Vol. 35, No. 10 (2023)

Fig. 3.

Fig. 4.

Time responses of θu(τ) and θ v(τ) for α = 0.6.

(Color online) State trajectories for various values of α.

Fig. 5.

Time responses of θu(τ) and θ v(τ) for α = 0.45.

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Sensors and Materials, Vol. 35, No. 10 (2023)

Fig. 8.

Fig. 6.

Time responses of σ(τ) and ϕ(τ) for α = 0.45.

Fig. 7.

Time responses of gi(τ), i = 0, 1, 2 for α = 0.45.

Time responses of state variables for two SFHN nerve cells.

Sensors and Materials, Vol. 35, No. 10 (2023)

4583

and the IFFSSM σ(τ) and the control input signal ϕ(τ) vs τ are shown, respectively. It is shownthat θu(τ) and θ v(τ) have nearly identical convergence to the origin while σ(τ) = 0 is held, andgood performance of control is achieved. In addition, the control signal ϕ(τ) of the proposed
adaptive IFFSSM control is continuous without chatter.
The responses of three adaptive feedback gains vs τ are depicted in Fig. 7, and the gains areseen to finally become constant. It is indicated that θu(τ), θ v(τ), and the IFFSSM σ(τ) all tend tozero in accordance with the adaption algorithms in Eq. (13). In Fig. 8, the responses of statevariables vs τ for two SFHN nerve cells are shown. It is apparent that the state synchronous
control problem in the presence of external disturbances and system uncertainties is solved.

5.

Conclusions

The novel adaptive IFFSSM control scheme with robustness has been developed to solve thestate synchronous control problem between two SFHN nerve cell systems in this study. The
IFFSSM is defined by the proportional, the fractional powered, and the integral terms of statevariables. Additionally, the special stability property, in which two state variables are stabilized
in sequence on the sliding surface, is described. Two theorems that support the stability of thecontrolled system are provided and proven in the framework of the Lyapunov stability theory.
For the future practical realization of the present control scheme in neuronal circuits, theproposed scheme is validated through numerical experiments.

Acknowledgments
We acknowledge the support of the School of Physics and Telecommunication Engineering,
Yulin Normal University. This research was funded in part by the Scientific Research Fund
Project (Grants: G2020ZK15, G2020ZK16, and G2021ZK19) of Yulin Normal University and the
Major Cooperative Project (Grant: YLSXZD2019015) of Yulin Municipal Government andYulin Normal University.

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