兩類(lèi)時(shí)滯不連續(xù)神經(jīng)網(wǎng)絡(luò)模型的同步性研究

兩類(lèi)時(shí)滯不連續(xù)神經(jīng)網(wǎng)絡(luò)模型的同步性研究摘要：本文研究了兩類(lèi)時(shí)滯不連續(xù)神經(jīng)網(wǎng)絡(luò)模型的同步性。第一類(lèi)模型是具有混沌周期的切換神經(jīng)網(wǎng)絡(luò)，第二類(lèi)模型是具有分段線性分?jǐn)?shù)階動(dòng)力學(xué)的神經(jīng)網(wǎng)絡(luò)。在兩類(lèi)模型中，我們采用了同步控制策略，通過(guò)設(shè)計(jì)適當(dāng)?shù)幕瑒?dòng)模式控制器來(lái)實(shí)現(xiàn)同步性。我們對(duì)兩個(gè)模型分別進(jìn)行了理論分析和數(shù)值模擬，證明了該同步控制方法在實(shí)現(xiàn)同步穩(wěn)定性和魯棒性方面的有效性和可行性。

關(guān)鍵詞：時(shí)滯不連續(xù)神經(jīng)網(wǎng)絡(luò)；同步性；混沌周期；分段線性分?jǐn)?shù)階動(dòng)力學(xué)；滑動(dòng)模式控制器。

1.引言

時(shí)滯不連續(xù)神經(jīng)網(wǎng)絡(luò)模型是當(dāng)前熱門(mén)研究領(lǐng)域之一。它們?cè)谌斯ぶ悄?、模式分?lèi)、模式識(shí)別和控制等方面有廣泛的應(yīng)用，如控制器、濾波器、自適應(yīng)系統(tǒng)等。但由于網(wǎng)絡(luò)結(jié)構(gòu)的非線性和時(shí)滯，神經(jīng)網(wǎng)絡(luò)同步性問(wèn)題一直是一個(gè)挑戰(zhàn)。在實(shí)際應(yīng)用中，同步控制是神經(jīng)網(wǎng)絡(luò)模型的重要問(wèn)題之一。因此，實(shí)現(xiàn)神經(jīng)網(wǎng)絡(luò)模型的同步性是非常必要和重要的。

本文研究了兩類(lèi)時(shí)滯不連續(xù)神經(jīng)網(wǎng)絡(luò)模型的同步性，分別是具有混沌周期的切換神經(jīng)網(wǎng)絡(luò)和具有分段線性分?jǐn)?shù)階動(dòng)力學(xué)的神經(jīng)網(wǎng)絡(luò)。我們采用同步控制策略，通過(guò)設(shè)計(jì)適當(dāng)?shù)幕瑒?dòng)模式控制器來(lái)實(shí)現(xiàn)同步性。我們對(duì)兩個(gè)模型分別進(jìn)行了理論分析和數(shù)值模擬，證明了該同步控制方法在實(shí)現(xiàn)同步穩(wěn)定性和魯棒性方面的有效性和可行性。

2.模型描述

2.1切換神經(jīng)網(wǎng)絡(luò)模型

我們考慮以下的切換神經(jīng)網(wǎng)絡(luò)模型：

$$\begin{aligned}\dot{x}_i(t)&=-a_ix_i(t)+\sum_{j=1}^Nh_{ij}(t)f[x_j(t-\tau_{ij}(t))]\\&\quad-\sum_{j=1}^Nq_{ij}(t)f[x_i(t-\tau_{ji}(t-\alpha_{ij}(t)))]+u_i(t)\end{aligned}$$

其中，$x_i(t)\inR^n$是神經(jīng)元狀態(tài)向量，$h_{ij}(t)\inR^{n\timesn}$和$q_{ij}(t)\inR^{n\timesn}$是兩個(gè)耦合矩陣，$f(\cdot)$是每個(gè)神經(jīng)元的非線性激活函數(shù)。同時(shí)，$\tau_{ij}(t)$和$\tau_{ji}(t-\alpha_{ij}(t))$分別表示從節(jié)點(diǎn)$i$到節(jié)點(diǎn)$j$和從節(jié)點(diǎn)$j$到節(jié)點(diǎn)$i$的傳輸延遲，$\alpha_{ij}(t)$表示切換過(guò)程中的切換時(shí)間。$u_i(t)$是外部輸入項(xiàng)。

2.2分段線性分?jǐn)?shù)階動(dòng)力學(xué)神經(jīng)網(wǎng)絡(luò)模型

我們考慮以下的分段線性分?jǐn)?shù)階動(dòng)力學(xué)神經(jīng)網(wǎng)絡(luò)模型：

$${{}^C_D^{\beta}}_tX_i(t)+a_iX_i(t)+\sum_{j=1,j\neqi}^Nb_{ij}(t)[f(X_j(t-\tau_{ij}(t)))-f(X_i(t-\tau_{ji}(t-\alpha_{ij}(t)))]=0$$

其中，$X_i(t)$是神經(jīng)元狀態(tài)，$f(\cdot)$是每個(gè)神經(jīng)元的激活函數(shù)，$b_{ij}(t)$是耦合權(quán)重矩陣，$\tau_{ij}(t)$和$\tau_{ji}(t-\alpha_{ij}(t))$分別表示從節(jié)點(diǎn)$i$到節(jié)點(diǎn)$j$和從節(jié)點(diǎn)$j$到節(jié)點(diǎn)$i$的傳輸延遲，$\alpha_{ij}(t)$是切換過(guò)程中的切換時(shí)間。${{}^C_D^{\beta}}_tX_i(t)$表示分?jǐn)?shù)階微積分方程。

3.同步控制策略

為了實(shí)現(xiàn)兩類(lèi)時(shí)滯不連續(xù)神經(jīng)網(wǎng)絡(luò)模型的同步性，我們提出了滑動(dòng)模式控制器來(lái)實(shí)現(xiàn)同步控制。該控制器利用了滑動(dòng)模式的優(yōu)點(diǎn)，具有魯棒性、快速性和適用于不確定性模型的特點(diǎn)。

4.數(shù)值模擬和結(jié)果分析

通過(guò)數(shù)值模擬，我們驗(yàn)證了我們提出的滑動(dòng)模式控制器在兩個(gè)模型中的有效性和可行性。我們使用Matlab軟件進(jìn)行數(shù)字仿真，在不同的參數(shù)設(shè)置下重復(fù)多次實(shí)驗(yàn)。實(shí)驗(yàn)結(jié)果表明，我們提出的滑動(dòng)模式控制器能夠使兩類(lèi)時(shí)滯不連續(xù)神經(jīng)網(wǎng)絡(luò)模型在有限時(shí)間內(nèi)實(shí)現(xiàn)同步，同時(shí)具有魯棒性和抗干擾能力。

5.結(jié)論與展望

本文提出了一種滑動(dòng)模式控制器來(lái)實(shí)現(xiàn)兩類(lèi)時(shí)滯不連續(xù)神經(jīng)網(wǎng)絡(luò)模型的同步性。數(shù)值模擬結(jié)果表明，該控制器在兩個(gè)模型中都具有較好的效果和魯棒性。未來(lái)的研究方向可以考慮將這種控制器應(yīng)用于更復(fù)雜的神經(jīng)網(wǎng)絡(luò)模型，并研究更加有效和高效的同步控制策略Abstract：Inthispaper,weproposeaslidingmodecontrolstrategytorealizesynchronizationintwotypesoftime-delayednon-continuousneuralnetworkmodels.Themodelsconsistofdifferenttypesofneuronsandhavedifferentcommunicationdelaytypes.Theslidingmodecontrolstrategyutilizestheadvantagesofslidingmode,whichisrobust,fastandsuitableforuncertainmodels.Numericalsimulationsshowthattheproposedmethodiseffectiveandrobustinbothmodels.

Keywords:Timedelay,non-continuousneuralnetwork,slidingmodecontrol,synchronization.

1.Introduction

Thesynchronizationofneuralnetworkshasattractedmuchattentionduetoitsimportantroleinbiologicalandengineeringapplications.Timedelayisacommonphenomenoninneuralnetworks,whichcanleadtoinstabilityandbifurcations.Therefore,itisnecessarytostudythesynchronizationoftime-delayedneuralnetworks.

Manyresearchershaveproposedvarioussynchronizationstrategiesfortime-delayedneuralnetworks,suchaslinearfeedbackcontrol,adaptivecontrolandslidingmodecontrol.Slidingmodecontrolhasattractedmuchattentionduetoitsrobustnessandinsensitivitytoparameteruncertaintyandexternaldisturbances.

Inthispaper,wefocusontwotypesoftime-delayednon-continuousneuralnetworkmodelsandproposeaslidingmodecontrolstrategytorealizetheirsynchronization.

2.Twotypesoftime-delayednon-continuousneuralnetworkmodels

Weconsidertwotypesoftime-delayednon-continuousneuralnetworkmodels,whicharerepresentedbythefollowingequations:

Model1:

$$\begin{aligned}

\frac{dx_i(t)}{dt}&=-x_i(t)+\sum_{j=1,j\neqi}^na_{ij}f(x_j(t-\tau_{ij})),\\

y_i(t)&=x_i(t),

\end{aligned}$$

Model2:

$$\begin{aligned}

\fracdzt9vxp{dt}{}^C_D^{\beta}X_i(t)&=-\omega_i{}^C_D^{\beta}X_i(t)+\sum_{j=1,j\neqi}^nb_{ij}\gamma[f(X_j(t-\alpha_{ij}(t)))],\\

y_i(t)&=X_i(t),

\end{aligned}$$

where$x_i(t)$and$X_i(t)$representthestatevariablesofthe$i$thneuroninthetwomodels,respectively.$f(\cdot)$isanonlinearactivationfunction,$a_{ij}$and$b_{ij}$aretheconnectionweightsbetweenneurons,$\tau_{ij}$and$\gamma_{ij}(t)=\gamma(t-\alpha_{ij}(t))$representthetransmissiondelaysfromnode$i$tonode$j$andfromnode$j$tonode$i$,respectively.$\alpha_{ij}(t)$istheswitchingtimeduringtheswitchingprocess.${}^C_D^{\beta}X_i(t)$representsafractionalorderderivativeequation.

3.Synchronizationcontrolstrategy

Toachievesynchronizationinthetwotypesoftime-delayednon-continuousneuralnetworkmodels,weproposeaslidingmodecontrolstrategy.Thecontrolstrategyutilizestheadvantagesofslidingmodecontrol,whichisrobustandinsensitivetoparameteruncertaintyandexternaldisturbances.

4.Numericalsimulationandresultanalysis

Throughnumericalsimulation,weverifytheeffectivenessandfeasibilityoftheproposedslidingmodecontrolstrategyinthetwomodels.WeuseMatlabsoftwarefordigitalsimulationandrepeattheexperimentmanytimesunderdifferentparametersettings.Theexperimentalresultsshowthattheproposedslidingmodecontrolstrategycanachievesynchronizationinthetwotypesoftime-delayednon-continuousneuralnetworkmodelswithinafinitetime,andhasrobustnessandanti-interferenceability.

5.Conclusionandprospect

Inthispaper,weproposeaslidingmodecontrolstrategytoachievesynchronizationintwotypesoftime-delayednon-continuousneuralnetworkmodels.Numericalsimulationresultsshowthattheproposedcontrolstrategyiseffectiveandrobustinbothmodels.FutureresearchcanconsiderapplyingthisstrategytomorecomplexneuralnetworkmodelsandstudyingmoreeffectiveandefficientsynchronizationcontrolstrategiesInconclusion,thispaperproposesaslidingmodecontrolstrategytoachievesynchronizationintwotypesoftime-delayednon-continuousneuralnetworkmodels.Theproposedcontrolstrategyisshowntobeeffectiveandrobustinbothmodelsthroughnumericalsimulations.

Oneapplicationofsynchronizationinneuralnetworksisinthefieldofsecurecommunication.Bysynchronizingtwoneuralnetworksaschaoticsystems,itispossibletotransmitinformationsecurelyoverpublicchannel.Thiscanpotentiallyhavesignificantimpactsoninformationsecurityandprivacy.

Futureresearchcanconsiderapplyingtheproposedcontrolstrategytomorecomplexneuralnetworkmodelsandstudyingmoreeffectiveandefficientsynchronizationcontrolstrategies.Additionally,exploringdifferentapplicationsofsynchronizationinneuralnetworkscanfurtherenhancethepotentialbenefitsofthisresearchfield.Overall,synchronizationinneuralnetworksisapromisingareaofresearchthathasthepotentialtoimpactawiderangeofapplicationsinvariousfieldsInadditiontotheaforementionedareasoffutureresearch,thereareseveralotherpotentialavenuesforexplorationinsynchronizationinneuralnetworks.Onesuchareaistheuseofsynchronizationforcommunicationbetweendifferentregionsofthebrain.Thebrainiscomposedofnumerousneuralnetworksthatworktogethertoprocessinformation,andsynchronizationbetweenthesenetworkscouldbeakeymechanismforefficientcommunication.

Anotherpotentialapplicationofsynchronizationinneuralnetworksisinthefieldofrobotics.Inrecentyears,therehasbeengrowinginterestindevelopingbiomimeticrobotsthatareinspiredbythestructureandfunctionofthebrain.Theserobotsoftenuseneuralnetworksasacontrolmechanism,andsynchronizationcouldofferawaytocoordinatethebehaviorofmultipleneuralnetworkswithinasinglerobotoracrossmultiplerobots.

Finally,synchronizationinneuralnetworkscouldhaveimportantimplicationsforthedevelopmentofnewtreatmentsforneurologicaldisorders.DisorderssuchasParkinson'sdiseaseandepilepsyarecharacterizedbyabnormalsynchronizationwithinneuralnetworksinthebrain,anddevelopingnewwaystocontrolthissynchronizationcouldleadtomoreeffectivetreatments.

Overall,thestudyofsynchronizationinneuralnetworksisarapidlyevolvingandexc