# The Synchronization Analysis of Cohen–Grossberg Stochastic Neural Networks with Inertial Terms

Authors: Zhi-Ying Li; Wang-Dong Jiang; Yue-Hong Zhang
Journal: Computational Intelligence and Neuroscience
 (2022)

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## Abstract

The exponential synchronization (ES) of Cohen–Grossberg stochastic neural networks with inertial terms (CGSNNIs) is studied in this paper. It is investigated in two ways. The first way is using variable substitution to transform the system to another one and then based on the properties ofit^o integral, differential operator, and the second Lyapunov method to get a sufficient condition of ES. The second way is based on the second-order differential equation, the properties of calculus are used to get a sufficient condition of ES. At last, results of the theoretical derivation are verified by virtue of two numerical simulation examples.


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## Body

## 1. Introduction

The dynamic behavior of neural network (NN) is a popular field in research studies and applications. Synchronization is one of the stability which has been studied a lot. Synchronization is the state in which two or more systems adjust their dynamic characteristics to achieve consistency under external driving or internal interaction.In application, the external interference which can cause great uncertainty is everywhere, and the random interference is always inevitable. So, it is meaningful to consider stochastic term in the systems. The synchronization of stochastic neural networks has caught many scholars’ attention. Li et al. studied the methodology to control the synchronization of stochastic system with memristive [1]. The ES of GSCGNNs is investigated by L Hu by graph-theory and state feedback control technique [2].Synchronization of the systems is studied in [3–16] and so on. However, according to these research studies, the models considered do not contain inertial terms.However, from the point of mathematics and physics, the model without inertial terms can be considered as the model of super damping, but when the damping surpasses the critical point, the dynamic properties of the neuron will change. So, it is meaningful to consider inertial terms in application. Li et al. analyzed the stability and synchronization of INNs delayed by generalized nonlinear measure approach and realized the quasi-synchronization by Halanary inequality and matrix measure (MM) [17]. Zhan et al. and Ke et al. studied the ES of inertial neural networks by using Lyapunov theory [18–20]. And there are other studies on the inertial neural networks [21–26].So far, the neural networks on synchronization have been studied by adding only random terms in the system such as [1–16] or adding only inertial terms such as [17–26]. In application, the NN’s dynamic behavior is not only disturbed by inertia (weak damping) but also influenced by random disturbance.Therefore, it is meaningful to consider both of them in the systems. According to our enquiry, there is no result about synchronization containing both stochastic terms and inertial terms.Motivated by the research studies above, the ES of CGSNNI is studied in this paper. The model is characterized by considering both stochastic factors and inertial factors. Two methods are used to obtain the ES. It will be a new topic and has its value in both theory and application.This paper has an organization as follows: In section1, the CGSNNI model is introduced. In section 2, preliminaries and lemmas are listed. In section 3, two theorems are proved. One is to transform the given second-order differential system into first-order by suitable variable substitution and then using differential operator and the second Lyapunov method to get a sufficient condition. The other one is derived from the second-order differential system, by using the properties of calculus. In section 4, two examples are simulated to verify the theorems. These two sufficient conditions derived are differently in the case of parameters given in the system and can complement each other.We consider a class of CGSNNI as follows:(1)dx˙it=−γix˙itdt−αixithixit−∑j=1naijfjxjt−Iitdt+∑i=1ncijgjxjtdBit,i=1,2,…,n,where t≥0,xit is the state of the ith neuron at time t, αi⋅>0 is the amplification function, hi⋅>0 is the behavior function, γi>0 is the damping coefficient, aij is the connection weights, fj⋅ is the activation function of the jth neuron, Iit is the external input, and Bt=B1t,B2t,…,BntT is the n dimension Brown motion which is defined on complete probability space Ω,F,Ρ, and BT has natural filtering Ftt≥0.Given the initial conditions of system (1) as follows:(2)xis=ψxis,x˙is=χxis,s≤0,where ψxis,χxis are continuous.Consider system (1) as the driven system, then the slave system of system (1) is as follows:(3)dy˙it=−γiy˙itdt−αiyithiyit−∑j=1naijfjyjt−Iitdt+uitdt+∑i=1ncijgjyjtdBit,i=1,2,…,n,where ut=u1t,u2t,…,untT is the control function.Given the initial conditions of system (3) as(4)yis=ψyis,y˙is=χyis,s≤0,where ψyis,χyis are continuous.


## 2. Preliminaries

The following assumptions are satisfied fori,j=1,2,…,n:H1:αixit is bounded and derivable. That is, there exit constants α¯i>0,αi¯,Ai>0, which satisfy(5)α¯i≤αixit≤αi¯,αi′xi≤Ai.H2:fj⋅,gj⋅ are bounded in R and satisfy Lipschitz conditions.That is, there exit constants(6)lj>0,mj>0,fj¯>0,which satisfy(7)fju−fjv≤Lju−v,gju−gjv≤mju−v,fj⋅≤fj¯,u,v∈R.H3:kixi=αixihixi,kixi is derivable, and there exist constants ki¯>0,ki¯>0, which satisfy 0≤ki¯≤ki′xi≤ki¯.Definition 1.
If there are constantsλ>0,c>0, which satisfy(8)∑i=1nExit−yit2≤ce−λt−t0,t≥t0,then the drive system (1) and the slave system (3) are ES under the control strategy u(t).Lemma 1.
(see [27]).(9)dxt=ft,xtdt+gt,xtdWt,t≥0,xt0=x0.,where f∈R+×Rn,Rnandg∈R+×Rn,Rn×n are functions which are Borel measurable and W(t) is the standard Brown motion in Rn. We define a differential operator as follows:(10)L=∂∂t+∑j=1nfjt,x∂∂xj+12∑j=1ngt,xgTt,xij∂2∂xi∂xj.
IfVt,x∈C1,2R+×Sh,R+, then(11)LVt,x=∂Vt,x∂t+∂Vt,x∂xft,x+12tracegTt,x∂2Vt,x∂x∂xgt,xij,where Sh=x|x≤h∈Rn,(12)∂V∂x=∂V∂x1,∂V∂x2,…,∂V∂xn,∂2V∂x∂x=∂2V∂xi∂xjn×n.ByIt^o formula, if xt∈Sh, then(13)dVt,xt=LVt,xtdt+∂Vt,x∂xgt,xtdWt.Under the substitutions,(14)zit=x˙it+ηixit,ηi>0,i=1,2,…,n.System (1) and system (2) are transformed into(15)dxit=−ηixit+zitdtdzit=ηiηi−γixitdt−γi−ηizitdt−αixithixit−∑j=1naijfjxjt−Iitdt+∑j=1ncijgjxjtdBt,xis=ψxis,x˙is=χxis,zis=ηiψxis+χxis.Take the substitutions(16)ωit=y˙it+ηiyit,ηi>0,i=1,2,…,n.One sees that system (3) and (4) are transformed into(17)dyit=−ηiyit+ωitdt,i=1,2,…,n,dωit=−ηiηi−γiyitdt−γi−ηiωitdt−αiyithiyit−∑j=1naijfjyjt−Iitdt+uitdt+∑j=1ncijgjyjtdBit,yis=ψyis,y˙is=χyis,ωis=ηiψyis+χyis.Define the synchronization errors:(18)ν1it=yit−xit,ν2it=ωit−zit.And let the control strategy be(19)uit=−πiν1it,πi>0.From (1) and (3), one sees that(20)dν1it=−ηiν1it+ν2itdtdν2it=−ηi2−ηiγi+πiν1itdt−γi−ηiν2itdt−αiyithiyit−αixithixitdt+αiyit∑j=1naijfjyit−fjxitdt+αiyit−αixit∑j=1naijfjxjt+Iitdt+∑j=1ncijgjyjt−gjxjtdBti,i=1,2,...,n.


## 3. Main Results

In this part, by using the properties ofit^o integral, differential operator, and stability theory of Lyapunov and the properties of calculus, two sufficient conditions for the ES of CGSNNI are derived.Theorem 1.
In system (1), ifH1−H3 are satisfied, Iit is bounded, which means there exits Ii>0 and πi>0, which satisfies Iit≤Ii, and let the control strategy be(21)uit=−πiyit−xit.
If(22)pi=2ηi−γiηi−ri−∑j=1nα¯jajili−∑k=1n∑j=1nckj2mi2>0,qi=2γi−ηi−γiηi−ri−∑j=1nα¯iaijlj>0,where ri=1+ηi2−πi+k¯i+Ai∑i=1naijf¯j+AiIi,i=1,2,…,n then the drive system (1) and the slave system (3) are ES under the control strategy u(t).Proof of Theorem 1.
Let(23)νt=ν11t,ν12t,…,ν1nt,ν21t,ν22t,…,ν2ntT,for any ε>0, define a Lyapunov function as follows:(24)Vt,νt=∑i=1neεtν1i2t+ν2i2t.
One can see that(25)Vtt,νt=∑i=1neεtεν1i2t+ν2i2t,Vνtt,νt=2eεtνt,Vνtνtt,νt=2eεtE2n×2n,where E2n×2n is the 2n×2n order identity matrix, and Vvtt,vt,Vvtvtt,vt is the first and second derivatives with respect to vt.
From Lemma1 and (20),(26)LVt,νt=∑i=1nν1iteεtεν1i2t+ν2i2t+2−ηiν1it+ν2it−2ν2it−ηi2−γiηi+πiν1it+γi−ηiν2it−2ν2itαiyithiyit−αixithixit+2ν2itαiyit∑j=1naijfjyjt−fjxjt+2ν2itαiyit−αixit∑j=1naijfjxjt+Iit+∑i=1ncijgjyjt−gjxjt2.
AsH1−H3 are satisfied, one can see that(27)αiyit−αixit=αi′ξityit−xit,kiyit−kixit=αiyithiyit−αixithixit=ki′ξi∗tyit−xit,where ξit and ξi∗t are between yit and xit.
Derive from (26),(28)LVt,νt≤∑i=1neεtεν1i2t+ν2i2t+2−ηiν1i2t+ν1itν2it−2−ηi2−γiηi+πiν1itν2it+γi−ηiν2i2t+2k¯iν1itν2it+2α¯i∑j=1naijljν1itν2it+2Aiν1itν2it∑j=1naijf¯j+Iit+∑j=1ncijmjν1jt2≤∑i=1neεtε−2ηi+∑k=1n∑j=1nckj2mi2ν1i2t−ε+2γi−2ηiν2i2t+21+ηi2+γiηi−πiν1itν2it+2k¯i+Ai∑j=1naijf¯j+AiIiν1itν2it+2α¯i∑j=1naijljν1jtν2it≤∑i=1neεtε−2ηi+γiηi+1+ηi2−πi+k¯i+Ai∑j=1naijf¯j+AiIi+∑j=1nα¯jajili+∑k=1n∑j=1nckj2mi2ν1i2t+ε−2γi+2ηi+γiηi+1+ηi2−πi+k¯i+Ai∑j=1naijf¯j+AiIi+∑j=1nα¯jaijljν2i2t.
According to the conditions in Theorem1, if there exits ε>0, which satisfy(29)ε−2ηi+γiηi+1+ηi2−πi+k¯i+Ai∑j=1naijf¯j+AiIi+∑j=1nα¯jajili+∑k=1n∑j=1nckj2mi2≤0,ε−2γi+2ηi+γiηi+1+ηi2−πi+k¯i+Ai∑j=1naijf¯j+AiIi+∑j=1nα¯jaijlj≤0.
From that one has(30)LVt,νt≤0.
In addition,(31)dVt,νt=LVt,νtdt+∂Vt,νt∂νtgyt−gxtdBt,Vt,νt=V0,ν0+∫0tLVs,νsds+2∫0t∑i=1n∑j=1ncijν2isgjyjs−gjxjsdBis.
As(32)Vt,νt=∑i=1neεtν1i2t+ν2i2t,and LVt,νt≤0, one sees that(33)∑i=1nν1i2t+ν2i2t≤e−εt∑i=1nν1i20+ν2i20+2e−εt∫0t∑i=1n∑j=1ncijν2isgjyjs−gjxjsdBis.
By taking expectations,(34)∑i=1nEν1i2t+ν2i2t≤e−εt∑i=1nEν1i20+ν2i20.
Therefore,(35)∑i=1nEν1i2t+ν2i2t≤ce−εt,ε>0,c>0,t≥0,where(36)c=∑i=1nEν1i20+ν2i20.
It comes to(37)∑i=1nExit−yit2≤ce−εt,ε>0,c>0,t≥0.
According to Definition1, system (1) and system (3) are ES under the control strategy u(t).Theorem 2.
IfH1−H3 are satisfied, Iit is bounded; that is, there exit Ii>0 and πi>0, which satisfy Iit≤Ii; let the control strategy be uit=−πiyit−xit.If(38)2πi−2−γi−πi−3k¯i−α¯i∑j=1naijlj−3Ai∑j=1naijf¯j+Ii>0,2γi−2−2−γi−πi−k¯i−∑j=1nα¯jajili−∑j=1nα¯iaijlj−Ai∑j=1naijf¯j+Ii>0,then the drive system (1) and the slave system (3) are ES under control strategy u(t).Proof of Theorem 2.
Let(39)νit=yit−xit.
From (1) and (3),(40)dν˙it=−γiν˙itdt−πiνitdt−αiyithiyit−αixithixitdt+αiyit∑j=1naijfjyjt−fjxjtdt+αiyit−αixit∑j=1naijfjxjt+Iitdt+∑j=1ncijgjyjt−gjxjtdBit,i=1,2,…,n.
For anyε>0,(41)Vt=∑i=1nνi2t+νit+ν˙it2eεt.
From the two formulas above,(42)dVt=∑i=1nενi2t+νit+ν˙it2eεtdt+2νitν˙it+νit+ν˙itν˙it+ν¨iteεtdt=eεt∑i=1nενi2t+νit+ν˙it2dt+2νitν˙it+2νit+ν˙itν˙it+2νit+ν˙it−γiν˙itdt−πiνitdt−αiyithiyit−αixithixitdt+αiyit∑j=1naijfjyjt−fjxjtdt+αiyit−αixit∑j=1naijfjxjt+Iitdt+∑j=1ncijgjyjt−gjxjtdBit.
Integral both sides byt,(43)Vt≤V0+∑i=1n∫0teεs2ε−2πiνi2s+ε+2−2γiν˙i2s+2ε+4−2γi−2πiνisν˙is+2k¯iνis+ν˙isνis+2νis+ν˙isα¯i∑j=1naijljν˙js+2Aiνis+ν˙isνis∑j=1naijf¯j+Iids+2∑i=1n∫0teεs∑j=1ncijνis+ν˙isgjyjs−gjxjsdBis=V0+∑i=1n∫0teεs2ε−2πi+ε+2−γi−πi+3k¯i+α¯i∑j=1naijlj+3Ai∑j=1naijf¯j+Iiνi2s+ε+2−2γi+ε+2−γi−πi+k¯i+∑j=1nα¯jajili+∑j=1nα¯iaijlj+Ai∑j=1naijf¯j+Iiνi2sds+2∑i=1n∫0teεs∑j=1ncijvis+v˙isgjyjs−gjxjsdBis.
According to conditions in the theorem, there exitsε>0 which satisfy(44)2ε−2πi+ε+2−γi−πi+3k¯i+α¯i∑j=1naijlj+3Ai∑j=1naijf¯j+Ii≤0,ε+2−2γi+2−γi−πi+k¯i+∑j=1nα¯jajili+∑j=1nα¯iaijlj+Ai∑j=1naijf¯j+Ii≤0.
Derive from (14) that(45)Vt≤∑i=1nνi20+νit+ν˙i02+2∑i=1n∫0teεs∑j=1ncijvis+v˙isgjyjs−gjxjsdBis.
Then,(46)∑i=1nνi2t+νit+ν˙it2≤C0e−εt+2∑i=1n∫0teεs−t∑j=1ncijνis+ν˙isgjyjs−gjxjsdBis.
Taking expectation of it,(47)∑i=1nEνi2t+νit+ν˙it2≤e−εtEC0.
Then,(48)∑i=1nEyit−xit2≤e−εtEC0.where ε>0,(49)C0=∑i=1nψyi0−ψxi02+ψyi0−ψxi0+χyi0−χxi02.
According to Definition1, system (1) and system (3) are ES under control strategy u(t).


## 4. Numerical Examples

In this section, two examples are given to illustrate the theorems.The CGSNNI is considered as follows:(50)dx˙it=−γix˙itdt−αixithixit−∑j=12aijfjxjt−Iitdt+∑i=12cijgjxjtdBit,i=1,2.The corresponding slave system is as follows:(51)dy˙it=−γiy˙itdt−αiyithiyit−∑j=12aijfjyjt−Iitdt+uitdt+∑i=12cijgjyjtdBit,i=1,2.The control strategy is given as follows:uit=−πiyit−xit,πi>0,i=1,2.Example 1.
Let the parameters and the functions in system Example1 be(52)γ1=0.8,γ2=1.1,a11=0.3,a12=0.5,a21=−0.4,a22=0.15.(53)fjxjt=sinxjt,gjxjt=cosxjt,Iit=e−t,i,j=1,2,η1=0.5,η2=0.7,τ11=0.01,τ12=0.2,τ21=0.1,τ22=0.02,π1=1.25,π2=1.49,h1x1=2.6x1,h2x2=6x2.
α1x1=1/1002+1/1+x12andα2x2=1/1002−1/1+x22. After calculating, one has(54)α1¯=0.02,α¯1=0.03,α2¯=0.01,α¯2=0.02,A1=A2=0.01,f¯j=lj=I¯i=1,i,j=1,2,k1¯=0.048,k¯1=0.078,k2¯=0.11,k¯2=0.1125.
One can see that assumptionsH1−H3 are satisfied and(55)p1=2η1−γ1η1−1+η12−π1−k¯1−A1∑j=12a1jf¯j−A1I1−∑j=12α¯jaj1l1−∑k=12∑j=12ckj2m12=0.0555>0,p2=2η2−γ2η2−1+η22−π2−k¯2−A2∑j=12a2jf¯j−A2I2−∑j=12α¯jaj2l2−∑k=12∑j=12ckj2m22=1.496>0,q1=2γ1−η1−γ1η1−1+η12−π1−k¯1−A1∑j=12a1jf¯j−A1I1−∑j=12α¯ja1jlj=0.1905>0,q2=2γ2−η2−γ2η2−1+η22−π2−k¯2−A2∑j=12a2jf¯j−A2I2−∑j=12α¯ja2jlj=0.606>0,which satisfy Theorem 1. Therefore, system (50) and system (51) are ES.
On the other hand, let the initial conditions be(56)x10,x˙10,y10,y˙10=1,0.3,−0.2,0.5;x20,x˙20,y20,y˙20=0.9,0.6,0.3,0.7.
According to the simulation, one can see the instant response and the synchronization of the state variable in the drive system and the slave system in Example1 (Figures 1–3).
Obviously, the simulation and Theorem1 are consistent.Figure 1
The state of the drive variable 1 and the response variable 1 in example 1.Figure 2
The state of the drive variable 2 and the response variable 2 in example 1.Figure 3
The state of the error variable 1 and the error variable 2 in example 1.Example 2.
Let the parameters and the functions in system Example1 be γ1=2.1andγ2=2.2.
Others parameters and functions are the same as Example1. One sees that(57)2π1−2−γ1−π1−3k¯1−α¯1∑j=12a1jlj−3A1∑j=12a1jf¯j+I1=0.598>0,2π2−2−γ2−π2−3k¯2−α¯2∑j=12a2jlj−3A2∑j=12a2jf¯j+I2=0.918>0,2γ1−2−2−γ1−π1−k¯1−∑j=12α¯jaj1l1−∑j=12α¯1a1jlj−A1∑j=12a1jf¯j+I1=0.435>0,2γ2−2−2−γ2−π2−k¯2−∑j=12α¯jaj2l2−∑j=12α¯2a2jlj−A2∑j=12a2jf¯j+I2=0.5725>0,which satisfy Theorem 2. Therefore, system (50) and system (51) are ES.
On the other hand, let the initial conditions be(58)x10,x˙10,y10,y˙10=2,0.3,0.7,0.5;x20,x˙20,y20,y˙20=1,0.4,0.6,0.7.
According to the simulation, one can see the track of the instant response and the synchronization error of Example2 (Figuers 4–6).
Obviously, the simulation is consistent with Theorem 2.Figure 4
The state of the drive variable 1 and the response variable 1 in example 2.Figure 5
The state of the drive variable 2 and the response variable 2 in example 2.Figure 6
The state of the error variable 1 and the error variable 2 in example 2.


## 5. Conclusions

The ES of CGSNNI is studied in this paper. According to the definition of synchronization, there is an error system by the drive system and the slave one. Proper substitution of variable is used to transform the second-order system into a first one. In Theorem1, properties of it^o integral, differential operator, and the second Lyapunov method are used to get a sufficient condition for the ES. In Theorem 2, the properties of calculus are used on the second-order differential equation to get a sufficient condition of exponential synchronization. At last, two examples are given to illustrate the theorems. The conditions in two theorems are different and can complement each other. They are different ways to decide if there is synchronization between the drive system and the slave system. In the examples simulated, Theorem 1 is suitable for Example 1 but not suitable for Example 2. Theorem 2 is suitable for Example 2 but not suitable for Example 1. The effectiveness of the theorems is verified. They provide two different ways. In application, we can choose one of them according to the parameters given in the system. Also, the method we used in the proof of two theorems can be adopted in other models with inertial terms and stochastic terms.



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