# Synchronization Analysis for Stochastic Inertial Memristor-Based Neural Networks with Linear Coupling

Authors: Lixia Ye; Yonghui Xia; Jin-liang Yan; Haidong Liu
Journal: Complexity
 (2020)

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## Abstract

This paper concerns the synchronization problem for a class of stochastic memristive neural networks with inertial term, linear coupling, and time-varying delay. Based on the interval parametric uncertainty theory, the stochastic inertial memristor-based neural networks (IMNNs for short) with linear coupling are transformed to a stochastic interval parametric uncertain system. Furthermore, by applying the Lyapunov stability theorem, the stochastic analysis approach, and the Halanay inequality, some sufficient conditions are obtained to realize synchronization in mean square. The established criteria show that stochastic perturbation is designed to ensure that the coupled IMNNs can be synchronized better by changing the state coefficients of stochastic perturbation. Finally, an illustrative example is presented to demonstrate the efficiency of the theoretical results.


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## Body

## 1. Introduction

The memristor [1] is a kind of a nonlinear resistor with memory and nanoscale, which is widely applied in chaotic circuits, artificial neural networks, and so on. In [2], the relevant mechanisms of neural networks, such as long-term potentiation and spike time-dependent plasticity, are presented by applying basic electric circuits, and more complex mechanisms are constructed to mimic the synaptic connections in a (human) brain. During neuron transmission, synchronous resonance is a very important biological phenomenon. In recent years, a lot of systems have been investigated to realize synchronization such as time-varying switched systems, MNNs, and BAM neural networks [3–19]. In [8], a new switching pinning controller was designed to finite-time synchronization in nonlinear coupled neural networks by regulating a parameter. Therefore, it is necessary for synchronization to design a suitable controller, such as impulsive controller [11, 12, 20–22], nonchattering controller [19], and switching controller [6–8]. Based on parametric uncertainty and state dependency in the connection weight matrices of MNNs, the connection weight matrices jump in certain intervals. Duan and Huang [23] proposed periodicity and dissipativity for memristor-based neural networks with mixed delays involving both time-varying delays and distributed delays via using Mawhin-like coincidence theorem, inclusion theory, and M-matrix properties. The authors established two different types of exponential synchronization criteria for the coupled MNNs based on the master-slave (drive response) concept and discontinuous state feedback controller, and simultaneously, an estimation of the exponential synchronization rate was estimated (see [24]). It is worth pointing out that, the authors in [25, 26] added the linear coupling and interval term into MNNs to achieve two different synchronization via applying the Halanay inequality [27] and the Lyapunov method. However, there were essential differences between the synchronization results established by these two literature studies. In [25], the differential inclusion method was applied to transform the coupled connection weight matrices; moreover, a discontinuous controller was designed to ensure that multiple IMMNs can be synchronized. Li and Zheng [26] demonstrated that the coupled connection weight matrices can be decomposed by interval analysis [28], which by weakening the matrices satisfies the conditions. Besides, the new synchronization criteria for IMMNs with linear coupling were established.As we know, noise plays an important role in synchronization since it can stabilize an unstable system. In recent years, many scholars are very interested in synchronization of stochastic networks with time-varying delays [2, 29–38]. In 2013, the Jensen integral inequality was improved by the so-called Wirtinger-based integral inequality [33]. Furthermore, Gao et al. [29] showed that a state feedback controller and an adaptive updated law used to guarantee stochastic memristor-based neural networks with noise disturbance can be asymptotically synchronized. In [38], the authors investigated the synchronization of a stochastic multilayer dynamic network with time-varying delays and additive couplings by designing two pinning controllers. Therefore, taking stochastic perturbation into complex neural networks is very necessary and important.Note that the stochastic systems were mainly fist-order neural networks in pervious works. In this paper, based on the model of [26], considering xit,xit−τt,fixit,fixit−τt will produce errors, the new model of the stochastic coupled inertial memristor-based neural networks is constructed. Meanwhile, new results on synchronization in mean square are proposed. The main contributions of this paper are high-lighted as follows:(i)
Stochastic perturbation is taken into account in the second-order [39] coupled memristor-based neural networks with inertial term. Synchronization analysis becomes more challenging for the system with higher order and higher dimension.(ii)
The criterion for stochastic inertial memristor-based neural networks with linear coupling is proposed by applying the stochastic analysis techniques and the vector Lyapunov function method to realize synchronization in mean square.(iii)
An illustrative example is given to illustrate that system (1) can be synchronized under the coupled network with five nodes. Besides, system (1) has strong anti-interference.


## 2. Model Formulation and Preliminaries

In this paper, we consider the model of stochastic coupled inertial memristor-based neural networks (IMMNs for short) withN coupled identical nodes described by the following equation:(1)ddxitdt=−Ddxitdt−Cxit+Axitfxit+Bxitfxit−τt+c∑j=1NGijΓdxitdt+xjtdt+σ1xit+σ2xit−τt+σ3fxit+σ4fxit−τtdwt,i∈N,where xit=xi1t,xi2t,…,xint⊤∈Rn,i∈N, D=diagβ1,β2,…,βn, C=diagc1,c2,…,cn, and σi=diagσi1,σi2,…,σin,i=1,2,3,4, are the constant positive definite matrices and 0<τt<τ; the connection memristive weight matrix Axit=akjxijtn×n and the delayed connection of the current voltage characteristics Bxit=bkjxijtn×n satisfy the following conditions:(2)akjxijt=akj∗,xij<Tj,bkj∗∗,xij>Tj,bkjxijt=akj∗,xij<Tj,bkj∗∗,xij<Tj,where T>0 is the switching jump and akj∗,akj∗∗,,bkj∗,andbkj∗∗ are all constants, k,j∈n. The network coupling strength c>0 is a constant, and Γ=diagγ1,γ2,…,γn∈Rn×n is the inner coupling matrix. G=GijN×N is the constant coupling configuration matrix representing the topological structure of the system. Gij>0 is defined as a link from node i to node j, otherwise, Gij=0. Besides, G satisfies(3)Gii=−∑j=1,j≠iNGij,i∈N.The initial condition associated with system (1) is given as xis=φis∈c1−τ,0,Rn,i∈N. And, fxit=f1xi1t,f2xi2t,…,fnxint⊤ denotes the output of the neuron unit, which satisfies the following assumption:H1: for any two different u,v∈R, there exists a positive scalar li>0i∈n such that fiu−fiv≤liu−v.For stochastic systems, the Ito^ formula plays an important role in the synchronization. Consider a general stochastic system dxt=fxt,tdt+gxt,tdwt on t>t0 with an initial value xt0=x0∈Rn, where f:∈Rn×R+⟶Rn×m and g:∈Rn×R+⟶Rn×m. Denote a general nonnegative function Vx,t on Rn×R+ to be continuously twice differentiable in x and once differentiable in t, an stochastic differential operator tdVx,t=ℒVx,tdt+Vxx,tgx,tdwt,whereℒVx,t=Vtx,t+Vxx,tfx,t+1/2tracg⊤x,tVxxx,tgx,t, Vtx,t=∂Vx,t/∂t,Vxx,t=∂Vx,t/∂x1,∂Vx,t/∂x1,…,∂Vx,t/∂xn, Vxxx,t=∂2Vx,t/∂xi∂xjn×m, and EdVx,t=EℒVx,tdt.Consideringakjxij and bkjxij are bounded, therefore, Axit∈A¯,A¯,Bxit∈B¯,B¯, where A¯=a¯kjn×n,A¯=a¯kjn×n,B¯=b¯kjn×n,andB¯=b¯kjn×n with a¯kj=minakj∗,akj∗∗,a¯kj=maxakj∗,akj∗∗,b¯kj=minbkj∗,bkj∗∗, and b¯kj=maxbkj∗,bkj∗∗.By introducing the variable transformation,(4)rit=dxitdt+xit.System (1) can be transformed into(5)dxit=−xit+ritdt,drit=−Θxit−Λrit+A¯,A¯fxit+B¯,B¯fxit−τt+c∑j=1NGijΓritdt+σ1xit+σ2xit−τt+σ3fxit+σ4fxit−τtdwt,i∈N,where Θ=I+C−D,Λ=D−I.Based on interval uncertainty theory, the intervalsA¯,A¯andB¯,B¯ can be decomposed into A¯,A¯=A0+−1,1HAandB¯,B¯=B0+−1,1HB, where A0=A¯+A¯/2,HA=1/2A¯−A¯andB0=B¯+B¯/2,HB=1/2B¯−B¯.Then, system (5) can be equivalently expressed as(6)dxit=−xit+ritdt,drit=−Θxit−Λrit+A0fxit+B0fxit−τt+EΔt+c∑j=1NGijΓritdt+σ1xit+σ2xit−τt+σ3fxit+σ4fxit−τtdwt,i∈N,where(7)EΔt=12−1,1A¯−A¯fxit+12−1,1B¯−B¯fxit−τt=−1,1HAfxit+HBfxit−τt,and EΔt is satisfied:(8)EΔt⊤EΔt≤HAfxit⊤HAfxit+HBfxit−τt⊤HBfxit−τt.Letxt=x1t⊤,x2t⊤,…,xNt⊤⊤,rt=r1t⊤,r2t⊤,…,rNt⊤⊤,fxt=fx1t⊤,fx2t⊤,…,fxNt⊤⊤,Θ=IN⊗Θ,Λ=IN⊗Λ,G=G⊗Γ,A0=IN⊗A0,B0=IN⊗B0,EΔt=IN⊗EΔt,HA=IN⊗HA,HB=IN⊗HB, and σi=IN⊗σi,i=1,2,3,4. For simplicity, we use x,r,andxτ instead of xt,rt,andxt−τt in the following sections. Then, system (6) can be written as(9)dx=−x+rdt,dr=−Θx−Λr+A0fx+B0fxτ+EΔt+cGrdt+σ1x+σ2xτ+σ3fx+σ4fxτdwt,i∈N.Definition 1.
The stochastic coupled IMMNs (5) are said to be globally synchronized in the mean square sense if Exit−xjt2⟶0 as t⟶+∞ for any given initial conditions ψi0, where i,j=1,2,…,N.Lemma 1 (see [1]).
LetG be an N×N matrix in the set TR;k. Then, the N−1×N−1 matrix H defined by H=MGJ satisfies MG=HM, where G and J are given, respectively, by(10)M=1−10⋯0001−1⋯00⋮⋮⋮⋱⋮⋮000⋯1−1N−1×N,J=111⋯1011⋯1⋮⋮⋱⋮⋮00⋯1100⋯0100⋯00N×N−1,and TR^,K is the set of matrices with entries in R such that the sum of the entries in each row is equal to R.Lemma 2.
For any positive definite symmetric constant matrixM∈Rn×n, if there exist the scalars r1<r2 and vector function w:r1,r2⟶Rn such that the concerned integrations are well defined, then the following inequality holds:(11)∫r1r2wsds⊤M∫r1r2wsds≤r12∫r1r2w⊤sMwsds,where r12=r2−r1.Lemma 3.
Given any real matricsXandY and Q>0 with appropriate dimensions, then the following matrix inequality holds:(12)X⊤Y+Y⊤X≤X⊤QX+Y⊤Q−1Y.Lemma 4 (see [40]).
The LMIS11xS12xS12⊤xS22x>0, where S11x=S11⊤x,S22x=F⊤x, and S12x depend on x, is equivalent to each of the following conditions:(i)
S11x>0,S22x−S12⊤xS11−1xS12x>0(ii)
S22x>0,S11x−S12⊤xS22−1xS12x>0


## 3. Main Results

Theorem 1.
Under the assumptionH1, 0<τt<τ, and τ˙t≤μμ>0, the stochastic coupled IMMNs (1) are globally synchronized in mean square sense if there exist positive definite symmetric matrices P,Q,Vi∈Rn×n,i=1,2,3,4,5,6, and positive diagonal matrices R,S,T,R˜,S1,S2,S3,S4,S5∈Rn×n, such that the following matrix inequalities hold: Φ<0 and Π−ΨΦ−1Ψ>0, where(13)Φ=Φ11Φ12Φ13⋆Φ22Φ23⋆⋆Φ33,Ψ=P−QΘ1−τ2R−V1+1τV1S−1V4V1+1τV1S−1V5V1+1τV1S−1V6−V2+1τV2S−1V4V2+1τV2S−1V5V2+1τV2S−1V6−V3+1τV3S−1V4V3+1τV3S−1V5V3+1τV3S−1V6,Π=Π44+γ44V4+1τV4S−1V5V4+1τV4S−1V6⋆2V5+1τV5S−1V5V5+1τV5S−1V6⋆⋆2V6−R˜+1τV6S−1V6,with(14)Φ11=−2P+τ2−1R+H1⊤Q+τ2SH1+1τV1S−1V1+τ2Θ1⊤R˜Θ1+γ11,Φ12=H1⊤Q+τ2SH2+1τV1S−1V2,Φ13=R+1τV1S−1V3,Φ22=H2⊤Q+τ2SH2+1τV2S−1V2+γ22,Φ23=1τV2S−1V3,Π44=τ2cH−Λ1⊤R˜cH−Λ1+τ2R+2QcH−Λ1−2V4+1τV4S−1V4,γ44=Q+τ2cH⊤−Λ1⊤R˜A0N−1+HAN−1S1A0N−1⊤+HAN−1⊤Q+τ2R˜cH−Λ1+Q+τ2cH⊤−Λ1⊤R˜B0N−1⊤+HBN−1⊤S2B0N−1+HBN−1Q+τ2R˜cH−Λ1,γ11=2H1⊤Q+τ2SH3+τ2Θ1⊤R˜A0N−1+HAN−1L+L⊤S1−1+S4−1+TL+L⊤H3⊤Q+τ2SH3+τ2A0N−1⊤+HAN−1⊤R˜A0N−1+HAN−1L+L⊤H3⊤Q+τ2SH4+τ2A0N−1⊤+HAN−1⊤R˜B0N−1+HBN−1×S5H4⊤Q+τ2SH3+τ2B0N−1⊤+HBN−1⊤R˜A0N−1+HAN−1L+H1⊤Q+τ2SH4+τ2Θ1⊤R˜B0N−1+HBN−1S3×H4⊤Q+τ2SH1+τ2B0N−1⊤+HBN−1⊤R˜Θ1,γ22=L⊤S2−1+S3−1+S5−1+H4⊤Q+τ2SH4−1−μTL+2H2⊤Q+τ2SH4L+τ2L⊤B0N−1⊤+HBN−1⊤R˜B0N−1+HBN−1L+H2⊤Q+τ2SH3S4H3⊤Q+τ2SH2.Proof.
For convenience, we set(15)gt=−Θx−Λr+A0fx+B0fxτ+EΔt+cGrt,yt=σ1x+σ2xτ+σ3fx+σ4fxτ.
Consider the following Lyapunov–Krasovskii functional:(16)Vt,x=∑i=16Vit,x,where(17)V1t,x=xt⊤M⊤PMxt,V2t,x=rt⊤M⊤QMrt,V3t,x=∫t−τttfxs⊤M⊤TMfxsds,V4t,x=τ∫−τ0∫t+θtx˙s⊤M⊤RMx˙sdsdθ,V5t,x=τ∫−τ0∫t+θtys⊤M⊤SMysdsdθ,V6t,x=τ∫−τ0∫t+θtgs⊤M⊤R˜Mgsdsdθ,for simplicity, we use x˙t instead of dxt/dt in the paper.
By the Ito^ formula, we can calculate ℒVt,x along system (9), and then we have(18)ℒVt,x=∑i=16ℒVit,x,and ℒVit,x,i=1,2,3,4,5,6 are calculated along system (9) as follows:(19)ℒV1t=2x⊤M⊤P−x+r≜ξt⊤Ω1ξt,where(20)ξt⊤=xt⊤M⊤,xt−τt⊤M⊤,xt−τ⊤M⊤,rt⊤M⊤,rt−τ⊤M⊤,∫t−τtMgsds⊤,Ω1=−2P00P00000000000000P00000000000000000,(21)ℒV2t=2r⊤M⊤QMgt+yt⊤M⊤QMyt≤ξt⊤Ω2ξt+2x⊤M⊤H1⊤QH3Mfx+2x⊤M⊤H1⊤QH4Mfxτ+2r⊤M⊤QA0N−1+HAN−1Mfx+2r⊤M⊤QB0N−1+HBN−1Mfxτ+2xτ⊤M⊤H2⊤QH3Mfx+2xτ⊤M⊤H2⊤QH4Mfxτ+fx⊤M⊤H3⊤QH3Mfx+2fx⊤M⊤H3⊤QH4Mfxτ+fxτ⊤M⊤H4⊤QH4Mfxτ,where(22)Mσi=HiM,i=1,2,3,4,MΘ=Θ1M,MΛ=Λ1M,MA0=A0N−1M,MB0=B0N−1M,MHA=HAN−1M,MHB=HBN−1M,Ω2=H1⊤QH1H1⊤QH20−QΘ100H1⊤QH2H2⊤QH20000000000−QΘ1002QcH−Λ100000000000000,(23)ℒV3t=fx⊤TMfx−1−τ˙tfxτ⊤M⊤TMfxτ≤fx⊤M⊤TMfx−1−μfxτ⊤M⊤TMfxτ,(24)ℒV4t=τ2xt.⊤M⊤RMxt.−τ∫t−τtx˙s⊤M⊤RMx˙sds,where τ˙t is the derivative of τt.
By applying Lemma2, one has(25)−τ∫t−τtx˙s⊤M⊤RMx˙sds≤−∫t−τtMx˙sds⊤R∫t−τtMx˙sds,then ℒV4t≤ξt⊤Ω3ξt,where(26)Ω3=τ2−1R0R−τ2R00000000R0−R000−τ2R00τ2R00000000000000,ℒV5t=τ2yt⊤M⊤SMyt−τ∫t−τtys⊤M⊤SMysds.
Recalling (9) and (15), it is easy to see that the following equalities hold:(27)0=2ξt⊤Vrt−rt−τ−∫t−τtgsds−∫t−τtysdws,where V=V1t⊤V2t⊤V3t⊤V4t⊤V5t⊤V6t⊤⊤.
By Lemma3, we have(28)2ξt⊤V∫t−τtysdws≤1τξt⊤VS−1V⊤ξt+τ∫t−τtysdws⊤S∫t−τtysdws.
Then,(29)ℒV5t≤ξt⊤Ω4ξt+τ∫t−τtMysdws⊤∫t−τtMysdws−τ∫t−τtys⊤M⊤SMysds+2τ2x⊤M⊤H1⊤SH3Mfx+2τ2x⊤M⊤H1⊤SH4Mfxτ+2τ2xτ⊤M⊤H2⊤SH3Mfx+2τ2xτ⊤M⊤H2⊤SH4Mfxτ+τ2fx⊤M⊤H3⊤SH3Mfx+2τ2fx⊤M⊤H3⊤SH4Mfxτ+τ2fxτ⊤M⊤H4⊤SH4Mfxτ,where(30)Ω4=τ2H1⊤SH1τ2H1⊤SH20−V1V1V1⋆τ2H2⊤SH20−V2V2V2⋆⋆0−V3V3V3⋆⋆⋆−2V4V4V4⋆⋆⋆⋆2V5V5⋆⋆⋆⋆⋆2V6+1τVS−1V⊤.
Using the property of the Ito^ isometry, we have(31)E∫t−τtMysdws⊤S∫t−τtMysdws=E∫t−τtys⊤M⊤SMysds.
Based on Lemma2 and (15), we have(32)ℒV6t=τ2gt⊤M⊤R˜Mgt−τ∫t−τtgs⊤M⊤R˜Mgsds≤τ2gt⊤M⊤R˜Mgt−∫t−τtMgsds⊤R˜∫t−τtMgsds≤ξt⊤Ω5ξt+2τ2x⊤M⊤Θ1⊤R˜A0N−1+HAN−1Mfx+2τ2x⊤M⊤Θ1⊤R˜B0N−1+HBN−1Mfxτ+2τ2r⊤M⊤cH⊤−Λ1⊤R˜A0N−1+HAN−1Mfx+2τ2r⊤M⊤cH⊤−Λ1⊤R˜B0N−1+HBN−1Mfxτ+τ2fx⊤M⊤A0N−1⊤+HAN−1⊤R˜A0N−1+HAN−1Mfx+2τ2fx⊤M⊤A0N−1⊤+HBN−1⊤R˜B0N−1+HBN−1Mfxτ+τ2fxτ⊤M⊤B0N−1⊤+HBN−1⊤R˜B0N−1+HBN−1Mfxτ,where(33)MG=HM,Ω5=τ2Θ1⊤R˜Θ100−τ2Θ1⊤R˜cH−Λ100000000000000−τ2Θ1⊤R˜cH−Λ100τ2cH⊤−Λ1⊤R˜cH−Λ10000000000000−R˜.
Under assumptionH1 and Lemma 3, we obtain(34)2r⊤M⊤Q+τ2cH⊤−Λ1⊤R˜A0N−1+HAN−1Mfx=2∑i=1N−1ri−ri+1⊤Q+τ2cH⊤−Λ1⊤R˜A0N−1+HAN−1⊤fxi−fxi+1≤2∑i=1N−1ri−ri−1⊤Q+τ2cH⊤−Λ1⊤R˜A0N−1+HAN−1Lxi−xi+1≤∑i=1N−1ri−ri+1⊤Q+τ2cH⊤−Λ1⊤R˜A0N−1+HAN−1S1A0N−1⊤+HAN−1⊤×Q+τ2R˜cH−Λ1ri−ri+1+∑i=1N−1xi−xi+1⊤L⊤S1−1Lxi−xi+1=r⊤M⊤Q+τ2cH⊤−Λ1⊤R˜A0N−1+HAN−1S1A0N−1⊤+HAN−1⊤×Q+τ2R˜cH−Λ1Mr+x⊤M⊤L⊤S1−1LMx.
Similarly, we have(35)2r⊤M⊤Q+τ2cH⊤−Λ1⊤R˜B0N−1+HBN−1Mfxτ≤∑i=1N−1ri−ri+1⊤Q+τ2cH⊤−Λ1⊤R˜B0N−1+HBN−1S2×B0N−1⊤+HBN−1⊤Q+τ2R˜cH−Λ1ri−ri+1+∑i=1N−1xit−τt−xi+1t−τt⊤L⊤S2−1Lxit−τt−xi+1t−τt=r⊤M⊤Q+τ2cH⊤−Λ1⊤R˜B0N−1+HBN−1S2B0N−1⊤+H0N−1⊤×Q+τ2R˜cH−Λ1Mr+xτ⊤M⊤L⊤S2−1LMxτ.
Next, it is easy to verify that(36)2x⊤M⊤H1Q+τ2SH3+τ2Θ1⊤R˜A0N−1+HAN−1Mfx≤x⊤M⊤2H1⊤Q+τ2SH3+2τ2Θ1⊤R˜A0N−1+HAN−1LMx,(37)2x⊤M⊤H1Q+τ2SH4+τ2Θ1⊤R˜B0N−1+HBN−1Mfxτ≤x⊤M⊤H1⊤Q+τ2SH4+τ2Θ1⊤R˜B0N−1+HBN−1S3×H4⊤Q+τ2SH1+τ2B0N−1⊤+HBN−1⊤R˜Θ1Mx+xτ⊤M⊤L⊤S3−1LMxτ,(38)2xτ⊤M⊤H2⊤Q+τ2SH3Mfx≤x⊤M⊤L⊤S4−1LMx+xτ⊤M⊤H2⊤Q+τ2SH3S4H3⊤Q+τ2SH2Mxτ,(39)2xτ⊤M⊤H2⊤Q+τ2SH4Mfxτ≤2xτ⊤M⊤H2⊤Q+τ2SH4LMxτ,(40)fx⊤M⊤τ2A0N−1⊤+HAN−1⊤R˜A0N−1⊤+HAN−1⊤+H3⊤Q+τ2SH3+TMfx≤x⊤M⊤L⊤T+H3⊤Q+τ2SH3+τ2=A0N−1⊤+HAN−1⊤R˜A0N−1⊤+HAN−1⊤LMx,(41)2fx⊤M⊤τ2A0N−1⊤+HAN−1⊤R˜B0N−1+HBN−1+H3⊤Q+τ2SH4Mfxτ≤x⊤M⊤L⊤H3⊤Q+τ2SH4+τ2A0N−1⊤+HAN−1⊤R˜B0N−1+HBN−1×S5H4⊤Q+τ2SH3+τ2B0N−1⊤+HBN−1⊤R˜A0N−1+HAN−1LMx+xτ⊤M⊤L⊤S5−1LMxτ,(42)fxτ⊤M⊤τ2B0N−1⊤+HBN−1⊤R˜B0N−1+HBN−1Mfxτ+fxτ⊤M⊤H4⊤Q+τ2SH4−1−μTMfxτ≤xτ⊤M⊤L⊤H4⊤Q+τ2SH4−1−μTLMxτ+xτ⊤M⊤L⊤τ2B0N−1⊤+HBN−1⊤R˜B0N−1+HBN−1LMxτ.
Substituting (19)–(42) into (18), we arrive at(43)EℒVt≤Eξt⊤Ξξt≤λmaxΞEξt2,where(44)Ξ=Ξ11Ξ12Ξ13Ξ14Ξ15Ξ16⋆Ξ22Ξ23Ξ24Ξ25Ξ26⋆⋆Ξ33Ξ34Ξ35Ξ36⋆⋆⋆Ξ44Ξ45Ξ46⋆⋆⋆⋆Ξ55Ξ56⋆⋆⋆⋆⋆Ξ66,with(45)Ξ11=−2P+τ2−1R+H1⊤Q+τ2SH1+1τV1S−1V1+τ2Θ1⊤R˜Θ1+γ11,Ξ12=H1⊤Q+τ2SH2+1τV1S−1V2,Ξ13=R+1τV1S−1V3,Ξ14=P−QΘ1−τ2R+1τV1S−1V4−V1,Ξ15=V1+1τV1S−1V5,Ξ16=V1+1τV1S−1V6,Ξ22=H2⊤Q+τ2SH2+1τV2S−1V2+γ22,Ξ23=1τV2S−1V3,Ξ24=−V2+1τV2S−1V4,Ξ25=V2+1τV2S−1V5,Ξ26=V2+1τV2S−1V6,Ξ33=−R+1τV3S−1V3,Ξ34=−V3+1τV3S−1V4,Ξ35=V3+1τV3S−1V5,Ξ36=V3+1τV3S−1V6,Ξ44=τ2cH−Λ1⊤R˜cH−Λ1+τ2R+2QcH−Λ1−2V4+1τV4S−1V4+γ44,Ξ45=V4+1τV4S−1V5,Ξ46=V4+1τV4S−1V6,Ξ55=2V5+1τV5S−1V5,Ξ56=V5−R˜+1τV5S−1V6,Ξ66=2V6−R˜+1τV6S−1V6,γ11=2H1⊤Q+τ2SH3+τ2Θ1⊤R˜A0N−1+HAN−1L+L⊤S1−1+S4−1+TL+L⊤H3⊤Q+τ2SH3+τ2A0N−1⊤+HAN−1⊤R˜A0N−1+HAN−1L+L⊤H3⊤Q+τ2SH4+τ2A0N−1⊤+HAN−1⊤R˜B0N−1+HBN−1×S5H4⊤Q+τ2SH3+τ2B0N−1⊤+HAN−1⊤R˜A0N−1+HAN−1L+H1⊤Q+τ2SH4+τ2Θ1⊤R˜B0N−1+HBN−1S3×H4⊤Q+τ2SH1+τ2B0N−1⊤+HBN−1⊤R˜Θ1,γ22=L⊤S2−1+S3−1+S5−1+H4⊤Q+τ2SH4−1−μTL+2H2⊤Q+τ2SH4L+τ2L⊤B0N−1⊤+HBN−1⊤R˜B0N−1+HBN−1L+H2⊤Q+τ2SH3S4H3⊤Q+τ2SH2,γ44=Q+τ2cH⊤−Λ1⊤R˜A0N−1+HAN−1S1A0N−1⊤+HAN−1⊤Q+τ2R˜cH−Λ1+Q+τ2cH⊤−Λ1⊤R˜B0N−1⊤+HBN−1⊤S2B0N−1+HBN−1Q+τ2R˜cH−Λ1.
Based on Lemma4 and the conditions of Theorem 1, we have the matrix Ξ<0; then, λmaxΞ<0.
Obviously, we obtainEℒVt<0. Hence, it follows the stochastic stability theory that the stochastic coupled IMMNs (1) are globally synchronized in mean square sense.


## 4. Conclusion

Based on interval uncertainty theory, the stochastic analysis techniques and the vector Lyapunov function method are applied to realize the global synchronization in mean square sense. Nevertheless, the criterion given in Theorem1 is different from the results in the existing literature. Moreover, the time delay is dependent, and the upper bound of the delayed derivative is 0 or less than 1. Hence, Theorem 1 would be feasible and less conservative. It is worth noting that the upper bound of the random factor can be calculated when the state parameter of the system is selected based on Theorem 1. Similarly, the stochastic coupled IMMNs (1) have strong anti-interference. In addition, the coupling of various neural nodes can be described. The example in Section 5 fully illustrates these two points.


## 5. Numerical Simulations

Now, we perform some numerical simulations to illustrate our analysis.Example 1.
Consider the following stochastic inertial memristor-based neural networks with five coupled identical nodes:(46)ddxitdt=−Ddxitdt−Cxit+Axitfxit+Bxitfxit−τt+c∑j=1NGijΓdxitdt+xjtdt+σ1xit+σ2xit−τt+σ3fxit+σ4fxit−τtdwt,i=1,2,3,4,5,where the activation function fxit=0.6tanhxi, the time delay τt=et/et+1, and the coupling strength c=0.5. The system parameters are taken as(47)D=6006,C=2002,Γ=1001,G=−2.21000.21−3.21.20002−3.50.50002−4.41.40.2003−4.2,σ1=2002,σ2=1.8001.8,σ3=1.6001.6,σ4=1.4001.4,Axit=a11xi1ta12xi2ta21xi1ta22xi2t,Bxit=b11xi1tb12xi2tb21xi1tb22xi2t,with the memristor connection weights:(48)a11x=0.2,x≤0.1,−0.2,x>0.1,a12x=0.6,x≤0.1,−0.6,x>0.1,a21x=0.4,x≤0.1,−0.4,x>0.1,a22x=0.4,x≤0.1,−0.4,x>0.1,b11x=0.2,x≤0.1,−0.2,x>0.1,a12x=0.4,x≤0.1,−0.4,x>0.1,b21x=0.3,x≤0.1,−0.3,x>0.1,b22x=0.3,x≤0.1,−0.3,x>0.1.
Obviously, by calculation, we can get the Lipschitz constantsL=0.6∗I2, and the upper bound of the delay τ=1.
In order to show the effectiveness of Theorem1, we display the synchronization of each node xijti=1,2,3,4,5;j=1,2 in Figure 1. Moreover, Figures 2 and 3 depict the synchronization error trajectories of xi1t−x11t and xi2t−x12t,i=1,2,3,4.Figure 1
The synchronization of each nodexijt in Example 1.Figure 2
The synchronization error trajectories ofxi1t−x11t in Example 1.Figure 3
The synchronization error trajectories ofxi2t−x12t in Example 1.



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*Source: 5430410-2020-07-23.xml*