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

兩類時滯不連續(xù)神經(jīng)網(wǎng)絡(luò)模型的同步性研究摘要：本文研究了兩類時滯不連續(xù)神經(jīng)網(wǎng)絡(luò)模型的同步性。第一類模型是具有混沌周期的切換神經(jīng)網(wǎng)絡(luò)，第二類模型是具有分段線性分?jǐn)?shù)階動力學(xué)的神經(jīng)網(wǎng)絡(luò)。在兩類模型中，我們采用了同步控制策略，通過設(shè)計適當(dāng)?shù)幕瑒幽Ｊ娇刂破鱽韺崿F(xiàn)同步性。我們對兩個模型分別進(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)階動力學(xué)；滑動模式控制器。

1.引言

時滯不連續(xù)神經(jīng)網(wǎng)絡(luò)模型是當(dāng)前熱門研究領(lǐng)域之一。它們在人工智能、模式分類、模式識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)的非線性和時滯，神經(jīng)網(wǎng)絡(luò)同步性問題一直是一個挑戰(zhàn)。在實際應(yīng)用中，同步控制是神經(jīng)網(wǎng)絡(luò)模型的重要問題之一。因此，實現(xiàn)神經(jīng)網(wǎng)絡(luò)模型的同步性是非常必要和重要的。

本文研究了兩類時滯不連續(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)階動力學(xué)的神經(jīng)網(wǎng)絡(luò)。我們采用同步控制策略，通過設(shè)計適當(dāng)?shù)幕瑒幽Ｊ娇刂破鱽韺崿F(xiàn)同步性。我們對兩個模型分別進(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$q_{ij}(t)\inR^{n\timesn}$是兩個耦合矩陣，$f(\cdot)$是每個神經(jīng)元的非線性激活函數(shù)。同時，$\tau_{ij}(t)$和$\tau_{ji}(t-\alpha_{ij}(t))$分別表示從節(jié)點$i$到節(jié)點$j$和從節(jié)點$j$到節(jié)點$i$的傳輸延遲，$\alpha_{ij}(t)$表示切換過程中的切換時間。$u_i(t)$是外部輸入項。

2.2分段線性分?jǐn)?shù)階動力學(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$i$到節(jié)點$j$和從節(jié)點$j$到節(jié)點$i$的傳輸延遲，$\alpha_{ij}(t)$是切換過程中的切換時間。${{}^C_D^{\beta}}_tX_i(t)$表示分?jǐn)?shù)階微積分方程。

3.同步控制策略

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

4.數(shù)值模擬和結(jié)果分析

通過數(shù)值模擬，我們驗證了我們提出的滑動模式控制器在兩個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)多次實驗。實驗結(jié)果表明，我們提出的滑動模式控制器能夠使兩類時滯不連續(xù)神經(jīng)網(wǎng)絡(luò)模型在有限時間內(nèi)實現(xiàn)同步，同時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模式控制器來實現(xiàn)兩類時滯不連續(xù)神經(jīng)網(wǎng)絡(luò)模型的同步性。數(shù)值模擬結(jié)果表明，該控制器在兩個模型中都具有較好的效果和魯棒性。未來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}

\fracfhsymtd{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