FitzHugh-Nagumo神经元同步的新型输入输出线性化最小滑模误差反馈控制资源-CSDN文库

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FitzHugh-Nagumo神经元模型是一种简化了的神经元模型，用以模拟神经元在受刺激后产生的兴奋和抑制行为。由于其在研究神经脉冲发放和信号传输中的重要性，FitzHugh-Nagumo模型成为了神经科学和非线性动力学领域研究的一个基本模型。论文中提到的“输入输出线性化最小滑模误差反馈控制”（I/OMSMEFC）是一种控制策略，用于实现两个未耦合的FitzHugh-Nagumo神经元在不同离子电流和外部电刺激作用下的同步。本文将详细探讨该控制策略的核心概念和实现过程。让我们从FitzHugh-Nagumo模型本身讲起。FitzHugh-Nagumo模型是一个由两个非线性微分方程构成的系统，这两个方程模拟了神经元膜电位的变化及其动力学。在数学上，模型可以表示为： dx/dt = ax - bx^3 - y + I dy/dt = cx - dy 在这里，x代表膜电位，y代表恢复变量，I代表外部电流刺激，而a、b、c和d为模型参数。该模型能够表现出振荡和脉冲发放等复杂的动态行为。接着，我们来看输入输出线性化。这是一种控制理论方法，旨在将一个非线性系统通过适当的状态反馈和输入变换转化为线性系统。在FitzHugh-Nagumo模型中应用输入输出线性化，可以将非线性模型通过特定的变换转化为等价的线性形式，简化了控制过程。最小滑模误差反馈控制（Minimum Sliding Mode Error Feedback Control, SMC）是一种鲁棒控制方法，用于处理系统的不确定性和外部干扰。滑模控制的核心在于引入一个滑动面，该滑动面上系统状态的动态行为会按照预定的方式进行。滑模控制能够保证系统到达滑动面后，在有限时间内达成最终的稳态，并且对系统的参数变化和外部扰动具有较强的鲁棒性。在本文中，研究者提出了一个新颖的控制策略，即I/OMSMEFC，这一策略结合了输入输出线性化和最小滑模误差反馈控制的特点，其核心是等效控制误差的概念。等效控制误差指的是在滑模控制中，考虑到系统不确定性和外部扰动时的一个误差估计，这种误差估计被反馈到控制输入中去，以实现系统状态的精确控制。为了实现等效控制误差的估计和反馈，研究者基于最小滑模协方差约束原理构建了一个成本函数。通过对成本函数进行最小化处理，可以得到控制输入的最优值，即等效控制误差。当应用了I/OMSMEFC后，系统将能够补偿各种类型的系统不确定性和外部干扰，并且相比于传统的滑模控制，I/OMSMEFC能更大程度地减少稳态误差。此外，I/OMSMEFC使得滑模运动趋向于理想的滑模运动状态，从而提高控制性能和质量。文章最后给出了基于Lyapunov稳定性定理的充分条件，并通过数值仿真验证了所提出的控制策略对于混沌同步的准确有效性。在研究论文的引言部分，作者简要介绍了混沌现象的广泛存在及其在非线性科学领域中的重要性。混沌系统对初始条件的高度敏感性和状态空间内某些区域的遍历性质，表明了其动态行为的复杂性。本文提出的I/OMSMEFC控制策略在神经元同步控制领域中具有创新意义，它通过结合线性化技术与滑模控制的鲁棒性，增强了控制效果，并且在面对复杂系统不确定性和外部干扰时，能提供更为精确的同步控制。这对于神经网络动力学、生物信息学以及相关领域的研究与应用具有重要的参考价值。

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Lu Cao

Assistant Research Fellow

The State Key Laboratory of

Astronautic Dynamics (ADL),

China Xi’an Satellite Control Center,

Xi’an 710043, China

e-mails: caolu_space2015@163.com;

lu.cao2@mail.mcgill.ca

Xiaoqian Chen

Professor

College of Aerospace Science and Engineering,

National University of Defense Technology,

Changsha 410073, China

e-mail: chenxiaoqian@nudt.edu.cn

A Novel Input–Output

Linearization Minimum Sliding

Mode Error Feedback Control

for Synchronization of

FitzHugh–Nagumo Neurons

A novel input–output linearization minimum sliding mode error feedback control (I/OMS-

MEFC) is proposed for the synchronization between two uncoupled FitzHugh–Nagumo

(FHN) neurons with different ionic currents and external electrical stimulations. To esti-

mate and offset the system uncertainties and external disturbances, the concept of equiva-

lent control error is introduced, which is the key to utilization of I/OMSMEFC. A cost

function is formulated on the basis of the principle of minimum sliding mode covariance

constraint; then the equivalent control error is estimated and fed back. It is shown that

the proposed I/OMSMEFC can compensate various kinds of system uncertainties and

external disturbances. Meanwhile, it can reduce the steady-state error more than the con-

ventional sliding mode control (SMC). In addition, the sliding mode after the I/OMS-

MEFC will tend to be the ideal SMC, resulting in improved control performance and

quantity. Sufﬁcient conditions are given based on the Lyapunov stability theorem and nu-

merical simulations are performed to verify the effectiveness of presented I/OMSMEFC

for the chaotic synchronization accurately. [DOI: 10.1115/1.4032074]

Keywords: synchronization, minimum sliding mode error feed back control, input–output

linearization, FitzHugh–Nagumo (FHN) neuron, different external electrical stimulations

1 Introduction

As we know, the chaotic phenomenon exists widely in nonlin-

ear science ﬁelds, which is a very complex dynamical nonlinear

system and exhibits some characteristic, such as excessive sensi-

tivity to initial conditions, ergodic property in some areas of state

space. That is, small variances in the initial condition can induce

large deviations in system states. Hence, chaotic system has the

widespread application values in many ﬁelds [1–4]. Speciﬁcally,

the chaos synchronization control exhibits tremendous utilization

potential and development, which have received great attentions.

Neural network is one of the complex dynamical networks with

strong backgrounds and various potential real-world applications

[5]. Therefore, it has been a fascinating and important subject of

many research ﬁelds [6]. During the last few years, there emerged

many results about the coupled FHN neuron system, which is usu-

ally utilized to study neural ﬁrings to investigate processing of

information in brain due to the simplicity. The FHN neuron model

was ﬁrst derived as simpliﬁed model of the Hodglein–Huxley neu-

ron model by FitzHugh [7] and Nagumo et al. [8]. There are some

literatures [9,10] that have done many qualitative study of the

FHN model for its nonlinear phenomena. Especially, the synchro-

nization of chaotic neurons under external electric stimulations

(EES) is attracted many interests during the last decade. The pur-

pose of synchronization is to synchronize the states of slave sys-

tem identical or opposite to the states of master system [11]. With

the development of control theory, various control schemes have

been successfully applied to control and synchronization of cha-

otic neurons [12–16]. In Ref. [12], controlling chaos in FHN neu-

ron by the adaptive passive method was introduced.

Synchronization of two uncoupled FHN neurons in EES was

found in Ref. [13] where a nonlinear control technology was

addressed. A novel internal model control method was proposed

for the robust output synchronization [14]. Ambrosio and Aziz

investigated the synchronization and control of coupled

reaction–diffusion FHN systems in Ref. [15]. In addition, Yu

et al. discussed the synchronization of two coupled neurons in the

external electrical stimulation based on adaptive backstepping

SMC method in Ref. [16].

In the past, the chaotic synchronization was discussed under the

assumption of two FHN neurons had the same ionic current and

EES [13,16–18]. In this study, the chaotic synchronized problem

with the existence of system uncertainties and external disturban-

ces between two uncoupled FHN neurons is considered with

different ionic currents and external electrical stimulations. Due

to the existence of system uncertainties and external disturbances,

the SMC is a suitable control methodology. It is well-known that

the SMC is a robust nonlinear feedback control methodology, in

which the structure between the switching surfaces is changed to

achieve the desired performance. The excellent theoretical foun-

dation endows SMC with several advantages, such as rapid

response, low sensitivity to parameter variations and external dis-

turbances, and low computational cost [19]. In Refs. [20,21], a

simple type sliding mode was deﬁned to design the controllers for

synchronization between two identical uniﬁed chaotic systems

where the driven system was assumed to have a single input. Syn-

chronization of two different uncertain chaotic systems with

unknown parameters was realized in Refs. [22,23]. However, for

the practical usage in engineering, the system uncertainties and

external disturbances are always time-varying, which hinders the

control precision of conventional SMC, even to inspire the chat-

tering phenomenon.

With a view to tackle the above limitation, the main goal of this

study is to introduce the I/OMSMEFC to accomplish chaotic syn-

chronization between two uncouple FHN neurons with high

Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the

OURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received February

28, 2014; ﬁnal manuscript received November 1, 2015; published online December

11, 2015. Assoc. Editor: Gabor Stepan.

Journal of Computational and Nonlinear Dynamics JULY 2016, Vol. 11 / 041011-1

2016 by ASME

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control accuracy and robustness. For the proposed I/OMSMEFC,

the concept of equivalent control error is introduced to estimate

and offset the system uncertainties and external disturbances,

which is the key to utilization of I/OMSMEFC. To obtain the opti-

mal equivalent control error, a cost function is formulated on the

basis of the principle of minimum sliding mode covariance con-

straint; then the equivalent control error is estimated and fed back

to the conventional SMC. It is shown that the I/OMSMEFC can

compensate various kinds of system uncertainties and external

disturbances. Meanwhile, it can also reduce the steady-state error

compared with conventional SMC, which is ﬁrst discussed in this

paper.

The rest of this paper is organized as follows. In Sec. 2, the

description of uncouple FHN neurons is introduced. Under the

existence of system uncertainties and external disturbances, the

I/OMSMEFC is developed to accomplish the synchronization of

uncouple FHN neurons in Sec. 3. In addition, the further theoreti-

cal analyses are also presented in the corresponding section. In

Sec. 4, numerical simulations are performed to demonstrate the

effectiveness of the proposed I/OMSMEFC. Some concluding

remarks are made in Sec. 5.

2 Descriptions of Dynamics for FitzHugh–Nagumo

Neuron

The FHN neuron subject to ESS is described by the following

second-order nonautonomous differential equations:

¼x

þða þ 1Þx

 ax

 y

þ I

cos ðx

tÞ

¼ cx

 by

(1)

where the ﬁrst state variable, x

is the action potential, that is, the

potential difference across the membrane. The second state vari-

able, y

, is the recovery variable which represents the effects of

changes in those ion-channel conductance which tend to return

the membrane potential to its resting level [24]. System parame-

ters a; b and c are positive constants; I

stands for the ionic current

inside the cell, and I

cos ðx

tÞ represents the ESS with the ampli-

tude I

and period T

¼ 2p=x

. The dynamics of the FHN neuron

with EES was studied in Refs. [24,25]. It is concluded that with

the certain value of I

the neuron experiences complicated chaotic

ﬁring. Meanwhile, the dynamic behavior of FHN neuron

is very sensitive to the variation in the stimulation signal

frequency. The FHN neuron depicting chaotic dynamics for the

parameter values sets, a ¼ 0:25, b ¼ 0:02, c ¼ 0:005, I

¼ 0:082,

¼ 0:055, x

¼ 0:1[25], with the initial conditions ðx

ð0Þ; y

ð0ÞÞ

¼ð0:2; 0:12Þ,isasshowninFig.1.

3 I/OMSMEFC Design for the Synchronization

of FitzHugh–Nagumo Neuron

In what follows, the synchronized problem between two

uncoupled FHN neuron with the case of different ionic currents

and external electrical stimulations is considered. The master neu-

ron is taken to be the form of Eq. (1). The slave neuron is chosen

in the similar structure of Eq. (1) with system uncertainty, external

disturbance, and can be described in the following:

¼x

þða þ 1Þx

 ax

 y

þ I

cos ðx

tÞ

þ nðx

; y

; tÞþuðtÞ

¼ cx

 by

(2)

where x

; y

are state variables of the slave system. I

is the ionic

current inside the cell and I

cos ðx

tÞ represents the ESS with

amplitude I

and period T

¼ 2p=x

. In this case, it is assumed

that I

6¼ I

, I

6¼ I

, x

6¼ x

. Considering the existence of sys-

tem uncertainties and external disturbances in the practical appli-

cation, the slave system in Eq. (2) is assumed to be subject to the

system uncertainty n

ðx

; y

Þ2R and the external disturbance

ðtÞ2R. The total effect in Eq. (2) can be expressed as

nðx

; y

; tÞ¼n

ðx

; y

Þþn

ðtÞ and uðtÞ is the controller to be

designed. Without loss of generality, the system uncertainty and

the external disturbance are bounded and satisﬁed

ðx

; y

Þj  n

2 R

; jn

ðtÞj  n

2 R

jnðx

; y

; tÞj  jn

ðx

; y

Þþn

ðtÞj  n

þ n

 n

2 R

(3)

It is also reasonable to assume that nðx

; y

; tÞ satisﬁes the con-

ditions required to ensure system (2) has a unique solution in the

time interval ½t

; 1Þ; t

> 0 for any given initial condition. By

taking account of the different ionic currents and the external elec-

trical stimulations between systems (1) and (2), the synchronized

problem is formulated in the following.

To proceed, the synchronized error states between systems (1)

and (2) are deﬁned as follows:

ðtÞ¼x

ðtÞx

ðtÞ; e

ðtÞ¼y

ðtÞy

ðtÞ (4)

As discussed earlier, the goal of the control problem is to derive

the appropriate controller, which can drive the states of slave

system (2) converge to the state of master system (1). In other

words, the controller needs to satisfy the condition of

Fig. 1 State trajectory of the FHN neuron

041011-2 / Vol. 11, JULY 2016 Transactions of the ASME

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