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A new criterion of delay-dependent asymptotic stability for Hopfield neural networks with time delay

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Abstract In this brief, the problem of global asymptotic stability for delayed Hopfield neural networks (HNNs) is investigated. A new criterion of asymptotic stability is derived by introducing a new kind of Lyapunov-Krasovskii functional and is
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  TitleA new criterion of delay-dependent asymptotic stability forHopfield neural networks with time delayAuthor(s)Mou, S; Gao, H; Lam, J; Qiang, WCitationIeee Transactions On Neural Networks, 2008, v. 19 n. 3,p. 532-535Issue Date2008URLhttp://hdl.handle.net/10722/57195Rights ©2008 IEEE. Personal use of this material is permitted.However, permission to reprint/republish this material foradvertising or promotional purposes or for creating newcollective works for resale or redistribution to servers orlists, or to reuse any copyrighted component of this workin other works must be obtained from the IEEE.  532 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 19, NO. 3, MARCH 2008 A New Criterion of Delay-Dependent Asymptotic Stabilityfor Hopfield Neural Networks With Time Delay Shaoshuai Mou, Huijun Gao, James Lam, and Wenyi Qiang  Abstract— In this brief, the problem of global asymptotic stability fordelayed Hopfield neural networks (HNNs) is investigated. A new criterionof asymptotic stability is derived by introducing a new kind of Lya-punov–Krasovskii functional and is formulated in terms of a linear matrixinequality (LMI), which can be readily solved via standard software. Thisnewcriterionbasedonadelayfractioningapproachprovestobemuchlessconservative and the conservatism could be notably reduced by thinningthe delay fractioning. An example is provided to show the effectivenessand the advantage of the proposed result.  Index Terms— Global asymptotic stability, Hopfield neural network(HNN), linear matrix inequality (LMI), Lyapunov functional. I. I NTRODUCTION In recent years, Hopfield neural networks (HNNs) have found manyapplications in a broad range of areas such as associative memory,repetitive learning, classification of patterns, and optimization prob-lems. Thus, considerable attention has been devoted to the research onHNNs [2], [6]. Inparticular, the stability analysis of HNNs has becomeatopicofboththeoreticalandpracticalimportancesincestabilityisoneof the most important issues related to their dynamic behaviors. Therehave been extensive results on global asymptotic stability in the litera-ture (see [3], [12]–[14], [20], and the references therein).In practice, due to the finite switching speed of amplifiers or finitespeed of information processing, time delays are often encounteredin hardware implementation [1], [8], [9], which may be a source of oscillation, divergence, and instability in HNNs. Therefore, the sta-bility problems of HNNs with time delay have gained great researchinterest. Based on different assumptions and different approaches, agreatnumberofstabilitycriteriafordelayedHNNshavebeenproposed(see [16], [17], [19], and [23]–[25]). Among these results, Xu and Lamobtained an improved stability condition in [22] over the existing cri-teria in [21] and [23].In this brief, we revisit the problem of stability analysis for delayedHNNs. It is shown that the result in [22] can be further improved byconstructing a new Lyapunov–Krasovskii functional with the idea of delay fractioning. The condition is formulated in the form of a linearmatrix inequality (LMI) and proves to be much less conservative,shown via a numerical example. The rest of this brief is organized asfollows. In Section II, the problem of asymptotic stability analysis fordelayed HNNs is formulated. Section III presents our main result. Anumerical example is provided in Section IV, and we conclude thebrief in Section V.  Notation: The notation used throughout this brief is fairly stan-dard. n  denotes the n  -dimensional Euclidean space and the notation P>  0(    0)  means that P  is real symmetric and positive definite Manuscript received April 12, 2007; revised August 14, 2007; accepted Oc-tober 12, 2007. This work was supported in part by the National Natural Sci-ence Foundationof ChinaunderGrant60504008,the Programfor NewCenturyExcellent Talents in University, China, and the Research Grant Council underGrant HKU 7031/06P.S. Mou, H. Gao, and W. Qiang are with the Department of ControlScience and Engineering, Harbin Institute of Technology, Harbin, Hei-longjiang Province 150001, China (e-mail: shaoshuaimou@gmail.com;hjgao@hit.edu.cn; qiangwy@hit.edu.cn).J. Lam is with the Department of Mechanical Engineering, University of Hong Kong, Hong Kong 00852, China (e-mail: james.lam@hku.hk).Digital Object Identifier 10.1109/TNN.2007.912593 (semidefinite). I  and 0 denote the identity matrix and zero matrix withcompatibledimensions,and diag  f  ...  g  standsforablock-diagonalma-trix. Matrices are assumed to be compatible for algebraic operations if their dimensions are not explicitly stated.II. P ROBLEM F ORMULATION Consider the following continuous HNN with time delay: _  x  i  (  t  )=  0  a  i  x  i  (  t  )+  n j  =1  b  ij  f  j  (  x  j  (  t  0    ))+  c  i  (1)or, equivalently _  x  (  t  )=  0  Ax  (  t  )+  Bf  (  x  (  t  0    ))+  c:  (2)Here, x  (  t  )=[  x  1  (  t  )  ;x  2  (  t  )  ;  ...  ;x  n  (  t  )]  T  is the neural state vector; f  (  x  (  t  0    ))=[  f  1  (  x  1  (  t  0    ))  ;f  2  (  x  2  (  t  0    ))  ;  ...  ;f  n  (  x  n  (  t  0    ))]  T  denotes the activation function; c  =[  c  1  ;c  2  ;  ...  ;c  n  ]  is the constant ex-ternalinput; and A  =diag  f  a  1  ;a  2  ;  ...  ;a  n  g  >  0  2  n  2  n  is a positivediagonal matrix. The scalar >  0  is a constant time delay. The inter-connection matrix B  represents the delayed weight coefficients of theneurons.Furthermore, weassume that f  j  ;j  =1  ;    ;  ...  ;n  , satisfiesthefollowing assumption [22].  Assumption 1: The activation function f  (  x  )  is continuous andbounded and it satisfies the following inequality: 0    f  j  (  s  1  )  0  f  j  (  s  2  )  s  1  0  s  2    l  j  ;j  =1  ;    ;  ...  ;n:  for all s  1  ;s  2  2  ;s  1  6 =  s  2  .It should be pointed out that Assumption 1 guarantees there is anequilibrium point for HNN (2). This can be easily verified by em-ploying the well-known Brouwer’s fixed point theorem. Let x  3  = [  x  3  1  ;x  3  2  ;  ...  ;x  3  n  ]  be the equilibrium point. Then, in order to simplifythe equation, we make the following transformation by the change of variables: y  (  t  )=  x  (  t  )  0  x  3  :  Under this transformation, HNN (2) is rewritten as _  y  (  t  )=  0  Ay  (  t  )+  Bg  (  y  (  t  0    ))  (3)where g  j  (  y  j  (  t  ))=  f  j  (  y  j  (  t  )+  x  3  j  )  0  f  j  (  x  3  j  )  :  (4)By (4) and Assumption 1, it is not difficult to verify that g  j  (0)=0  ;  0    g  j  (  y  j  )  y  j    l  j  8  y  j  6 =0  ;j  =1  ;    ;  ...  ;n:  (5)III. M AIN R ESULT In this section, we present our new delay-dependent asymptotic sta-bility criterion for delayed HNNs. Theorem 1: Given an integer m    1  ;  the srcin of the delayedHNN in (2) is the unique equilibrium point and it is globally asymptot-ically stable, if there exist positive–definite matrices P  2  n  2  n  ;Q  2  mn  2  mn  ;R  2  n  2  n  ;  and S  =diag  f  s  1  ;s  2  ;  ...  ;s  n  g  such that 2=  W  T P    PW  P  +  W  T R    RW  R  +  W  T Q    QW  Q  +  W  T S    SW  S  <  0  (6) 1045-9227/$25.00 © 2008 IEEE Authorized licensed use limited to: The University of Hong Kong. Downloaded on June 8, 2009 at 21:30 from IEEE Xplore. Restrictions apply.  IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 19, NO. 3, MARCH 2008 533 where   P  = 0  P P  0   R  =  R  0 0  0  R    Q  =  Q  0 0  0  Q    S  = 0  S S  0  S W  R  =  0   m A  0  n;mn   m B  0  m  I  n  m  I  n  0  n;mn  W  P  =  0  A  0  n;mn  B I  n  0  n;mn  0  n  W  S  = 0  n;mn  1  p    L  0  n  0  n;mn  0  n  p    I  n  W  Q  =  I  mn  0  mn;n  0  mn;n  0  mn;n  I  mn  0  mn;n  L  =diag  f  l  1  ;l  2  ;  ...  ;l  n  g  :  Proof: The uniqueness of the equilibrium point can be proved bythe contradiction method similar in [22]. Now, we are in the positionto show that the equilibrium point is globally asymptotically stable. At fi rst, we choose a Lyapunov – Krasovskii functional candidate as V  (  t  )=  V  1  (  t  )+  V  2  (  t  )+  V  3  (  t  )  (7)where V  1  (  t  )=  y  (  t  )  T  Py  (  t  )  ; V  2  (  t  )=  0  0  t t  +    _  y  T  (  !  )  R  _  y  (  !  )  d!d V  3  (  t  )=  t t  0  7(  !  )  T  Q  7(  !  )  d!  and 7(  !  )=  y  (  !  )  y!  0  1  m   ... y!  0  m  0  1  m  :  The derivatives of  V  i  (  t  )  ;i  =1  ;    ;  3  ;  are given by _  V  1  (  t  )=  y  (  t  )  T  P  _  y  (  t  ) _  V  2  (  t  )=   m  _  y  T  (  t  )  R  _  y  (  t  )  0  t t  0  _  y  T  (  !  )  R  _  y  (  !  )  d!  _  V  3  (  t  )=7(  t  )  T  Q  7(  t  )  0  7  t  0   m  T  Q  7  t  0   m :  (8)From Jensen ’ s inequality, we can easily get 0  t t  0  _  y  T  (  !  )  R  _  y  (  !  )  d!  0  m   2  y  (  t  )  0  yt  0   m  T  Ry  (  t  )  0  yt  0   m :  (9)From (5), for any scalar s  j    0  , it is clear that   n j  =1  s  j  g  j  (  y  j  (  t  0    ))[  l  j  y  j  (  t  0    )  0  g  j  (  y  j  (  t  0    ))]    0  or, equivalently   g  (  y  (  t  0    ))  T  SLy  (  t  0    )  0    g  (  y  (  t  0    ))  T  Sg  (  y  (  t  0    )    0  :  (10)Using (3), (8), (9), and (10), we have _  V  (  t  )      y  (  t  )  T  P  [  0  Ay  (  t  )+  Bg  (  y  (  t  0    ))] +   m  [  0  Ay  (  t  )+  Bg  (  y  (  t  0    ))]  T  R  [  0  Ay  (  t  )+  Bg  (  y  (  t  0    ))]  0  m  y  (  t  )  0  yt  0   m  T  Ry  (  t  )  0  yt  0   m  +7(  t  )  T  Q  7(  t  )  0  7  t  0   m  T  Q  7  t  0   m  +  g  (  y  (  t  0    ))  T  SLy  (  t  0    )  0    g  (  y  (  t  0    ))  T  Sg  (  y  (  t  0    ) =    (  t  )  T  2    (  t  )  where   (  t  )= 7(  t  )  y  (  t  0    )  g  (  y  (  t  0    ))  (11)and 2  is de fi ned in (6).On the other hand, condition (6) implies that there exists a positive "  such that 2  <  diag  f0  "I;  0  ;  ...  ;  0  ;  0  g  :  We pre- and postmultiply this inequality by   T  (  t  )  and   (  t  )  , respec-tively. Then, one can easily achieve _  V  (  t  )      (  t  )  T  2    (  t  )  <  0  "  k  y  (  t  )  k  2  whichshows that thedelayed HNN in (2)is asymptotically stable. Thiscompletes the proof.  Remark 1: Theorem 1 presents a new delay-dependent sta-bility criterion for delayed HNNs by using a more general Lya-punov – Krasovskii functional in (7). Note that, even for m  =1  , theproposed result still demonstrates superiority over the main classicalresults in the literature. An illustrative example will be provided toshow this in Section IV.  Remark 2: The reduced conservatism of Theorem 1 bene fi ts fromtheconstructionofthenewLyapunov – Krasovskiifunctionalin(7).Themainideaistofractionthedelay,whichconstitutesthemajordifferencefrom most existing results in the literature. Moreover, the conservatismreduction increases as the delay fractioning becomes thinner.  Remark 3: The delay fractioning idea can be extended to the time-varying delay case. We just need to modify our Lyapunov – Krasovskiifunctional by replacing   with   (  t  )(0      (  t  )    h;  _        )  to getLMI-based suf  fi cient condition for HNN with time-varying delay.If we choose the Lyapunov – Krasovskii functional as V  (  t  )=  y  (  t  )  T  Py  (  t  )+  t t  0  7(  !  )  T  Q  7(  !  )  d!  (12)then it is easy to obtain the following delay-independent stability con-dition. Corollary 1: Given an integer m    1  ;  the srcin of the delayedHNN in (2) is the unique equilibrium point and it is delay-independentglobally asymptotically stable, if there exist positive – de fi nite matrices P    n  2  n  ;Q    mn  2  mn  ;  and S  =diag  f  s  1  ;s  2  ;  ...  ;s  n  g  such that =  W  T P    PW  P  +  W  T Q    QW  Q  +  W  T S    SW  S  <  0  (13) Authorized licensed use limited to: The University of Hong Kong. Downloaded on June 8, 2009 at 21:30 from IEEE Xplore. Restrictions apply.  534 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 19, NO. 3, MARCH 2008 where   P  = 0  P P  0   Q  =  Q  0 0  0  Q    S  = 0  S S  0  S L  =diag  f  l  1  ;l  2  ;  ...  ;l  n  g  ; W  P  =  0  A  0  n;mn  B I  n  0  n;mn  0  n  W  S  = 0  n;mn  1  p  2  L  0  n  0  n;mn  0  n  p  2  I  n  W  Q  =  I  mn  0  mn;n  0  mn;n  0  mn;n  I  mn  0  mn;n  :  Proof: The derivative of (12) is _  V  1  (  t  )=2  y  (  t  )  T  P  _  y  (  t  )+7(  t  )  T  Q  7(  t  )  0  7  t  0   m  T  Q  7  t  0   m    2  y  (  t  )  T  P  [  0  Ay  (  t  )+  Bg  (  y  (  t  0    ))] +7(  t  )  T  Q  7(  t  )  0  7  t  0   m  T  Q  7  t  0   m  +2  g  (  y  (  t  0    ))  T  SLy  (  t  0    )  0  2  g  (  y  (  t  0    ))  T  Sg  (  y  (  t  0    ))      (  t  )  T      (  t  )  where   (  t  )  is as de fi ned in (11) and   in (13). Then, using similarmethod in Theorem 1, we complete the proof.  Remark 4: Though not in ordinary form of LMI, Theorem 1 andCollary 1 are indeed in the standard LMI form, which can be easilysolved by the standard software. Moreover, this form simpli fi ed as W  T  1  XW  1  +  W  T  2  YW  2  <  0  is more laconic. It expresses the LMI inseveral parts, each of which has a symmetric structure with the matrixvariable to be determined in center.IV. I LLUSTRATIVE E XAMPLES In this section, an example is provided to illustrate the advantageof Theorem 1 by comparing it with recently reported results on delay-dependent asymptotic stability of delayed HNNs.  Example 1: Consider the following third-order delayed HNN: A  = 4  :  198900 00  :  1600 001  :  9985  B  =  0  0  :  1052  0  0  :  5069  0  0  :  1121  0  0  :  025  0  0  :  28080  :  0212 0  :  1205  0  0  :  21530  :  1315  with L  =diag  f  0  :  4129  ;  3  :  8993  ;  1  :  0160  g  :  Our purpose is to fi nd the maximum allowable delay   max  such thatthe delayed HNN in (2) is globally asymptotically stable. Computa-tional results are shown in Table I, which summarizes the obtainedmaximum allowable delays by using the previously published methodsand our new result for various fractionings.Table I shows that for no fractioning (  m  =1)  , the new criteriongiven in Theorem 1 is still less conservative than the previous results,which corresponds to Remark 1. Moreover, for m>  1  , the conser-vatism reduction proves to be more obvious. However, it should benotedthatalthoughconservatismisreducedasthefractioningbecomesthinner, there is no signi fi cant improvement after m  =5  . TABLE IM AXIMUM A LLOWABLE D ELAYS C OMPARISON V. C ONCLUSION By de fi ning a new Lyapunov – Krasovskii functional, an improveddelay-dependent asymptotic stability criterion has been obtained for aclass of delayed HNNs. The proposed condition is given in terms of LMIs and thus can be readily solved via standard numerical software.The merit of the proposed condition lies in its reduced conservatism,which is based on a time delay fractioning approach. The result provesto become less conservative as the fractioning goes thinner. Finally, anumerical example has been provided to demonstrate the effectivenessoftheproposedcriterion.Themethodisexpectedtobefurtherextendedto other neural networks or complex networks [4], [5], [7], [10], [11],[15], [18], which is under our investigation. R EFERENCES[1] E.-K.BoukasandN.F.Al-Muthairi, “ Delay-dependentstabilizationof singularlinearsystemswithdelays, ”  Int.J.Innov.Comput.Inf.Control ,vol. 2, no. 2, pp. 283 – 291, 2006.[2] J. Cao, “ An estimation of the domain of attraction and convergencerateforHop fi eldcontinuousfeedbackneuralnetworks, ” PhysicaA ,vol.325, no. 5 – 6, pp. 370 – 374, 2004.[3] J. Cao and J. Wang, “ Global asymptotic stability of a general classof recurrent neural networks with time-varying delays, ” IEEE Trans.Circuits Syst. I, Fundam. Theory Appl. , vol. 50, no. 1, pp. 34 – 44, Jan.2003.[4] W. Chen and X. 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Liu, K. L. Teo, and B. Xu, “ Exponential stability of impulsive high-order Hop fi eld-type neural networks with time-varying delays, ” IEEE Trans. Neural Netw. , vol. 16, no. 6, pp. 1329 – 1339, Jun. 2005.[14] X. Y. Lou and B. Cui, “ New LMI conditions for delay-dependentasymptotic stability of delayed Hop fi eld neural networks, ” Neurocom- puting , vol. 69, no. 16 – 18, pp. 2374 – 2378, 2006.[15] V. Singh, “ A generalized LMI-based approach to the global asymp-totic stability ofdelayed cellular neural networks, ” IEEE Trans. Neural Netw. , vol. 15, no. 1, pp. 223 – 225, Jan. 2004.[16] Z. Wang, Y. Liu, K. Fraser, and X. Liu, “ Stochastic stability of un-certain Hop fi eld neural networks with discrete and distributed delays, ” Phys. Lett. A , vol. 354, no. 4, pp. 288 – 297, 2006.[17] Z. Wang, Y. Liu, and X. Liu, “ On global asymptotic stability of neuralnetworks with discrete and distributed delays, ” Phys. Lett. A , vol. 345,no. 4 – 6, pp. 299 – 308, 2005.[18] Z. Wang, Y. Liu, and X. Liu, “ Stability analysis for stochasticCohen-Grossberg neural networks with mixed time delays, ” IEEE Trans. Neural Netw. , vol. 17, no. 3, pp. 814 – 820, May 2006.[19] Z. Wang, H. Shu, J. Fang, and X. Liu, “ Robust stability for stochasticHop fi eld neural networks with time delays, ” Nonlinear Anal.: RealWorld Appl. , vol. 7, no. 5, pp. 1119 – 1128, 2006.[20] B. Xu, X. Liu, and X. Liao, “ Global exponential stability of high orderHop fi eld type neural networks, ” Appl. Math. Comput. , vol. 174, no. 1,pp. 98 – 116, 2006.[21] S. Xu, J. Lam, D. W. C. Ho, and Y. Zou, “ Novel global asymptoticstability criteria for delayed cellular neural networks, ” IEEE Trans.Circuits Syst. II, Exp. Briefs , vol. 52, no. 6, pp. 349 – 353, Jun.2005.[22] S. Xu, J. Lam, and D. W. C. Ho, “ A new LMI condition for delay-dependent asymptotic stability of delayed Hop fi eld neural networks, ”  IEEE Trans. Circuits Syst. II. Exp. Briefs , vol. 53, no. 3, pp. 230 – 234,Mar. 2006.[23] H. Ye, A. Michel, and K. Wang, “ Global stability and local stabilityof Hop fi eld neural networks with delays, ” Phys. Rev. E, Stat. Phys.Plasmas Fluids Relat. Interdiscip. Top. , vol. 50, pp. 4026 – 4213,1994.[24] F. Zhang and H. Huo, “ Global stability of delayed Hop fi eld neural net-works under dynamical thresholds, ” Discrete Dyn. Nature Soc. , vol. 1,pp. 1 – 17, 2005, 2005.[25] H. Zhao, “ Global asymptotic stability of Hop fi eld neural network in-volving distributed delays, ” Neural Netw. , vol. 17, no. 1, pp. 47 – 53,2004. Energy Function and Energy Evolutionon Neuronal Populations Rubin Wang, Zhikang Zhang, and Guanrong Chen  Abstract  — Basedontheprincipleofenergycoding,anenergyfunction of a variety of electric potentials of a neural population in cerebral cortex isformulated.Theenergyfunction isused todescribethe energyevolution of the neuronal population with time and the coupled relationship betweenneurons at the subthreshold and the suprathreshold states. The Hamil-tonian motion equation with the membrane potential is obtained from theneuroelectrophysiologicaldatacontaminatedbyGaussianwhitenoise.Theresultsofthisresearchshowthatthemeanmembranepotentialistheexactsolution of the motion equation of the membrane potential developed ina previously published paper. It also shows that the Hamiltonian energyfunction derived in this brief is not only correct but also effective. Particu-larly, based on the principle of energy coding, an interesting fi nding is thatin some subsets of neurons, fi ring action potentials at the suprathresholdandsomeotherssimultaneouslyperformactivitiesatthesubthresholdlevelin neural ensembles. Notably, this kind of coupling has not been found inother models of biological neural networks.  Index Terms  — Coupled neural population, energy coding, energy evolu-tion, Hamiltonian function. I. I NTRODUCTION Due to the limitations in current biophysical models of neuralcoding, research into the mechanisms of neural information processingremains a challenge [1] – [3], thereby the fundamental principles of neural information processing underlying cognitive processes in thebrain are still not completely understood today.Regarding neural information processing, the basic theory of energycodinghasreceivedstrongsupportsfrommanyneuroelectrophysiolog-ical experiments [4] – [11], [13] – [16], and a signi fi cant expansion onenergy coding appear in [7]. Two concerned issues pertaining to theaforementioned research results are considered in this brief.1) Neuronal activities at the subthreshold and the suprathresholdstates are separately described and discussed in their mathemat-ical models. The fact is that most activities of neurons at bothof these threshold states are mutually coupled, so this separatedescription does not agree with the real neuronal activities. Forthis reason, the coupled relationship between neurons at the sub-threshold and suprathreshold states is taken into account in thisbrief, and under such coupling con fi guration some quantitativeexpressions of energy coding are obtained.2) The Hamiltonian energy function describing the collective activ-ities of electric potentials of a whole neural population is derivedand analyzed, improving the previously published results devotedonly to a single neuron [5], [7]. Therefore, the results reported inthis brief are quite universal and signi fi cant. Manuscript received April 14, 2007; revised June 5, 2007; accepted October12, 2007. This work was supported by the National Natural Science Foundationof China under Grant 10672057.R.Wang andZ. Zhang are with the Institute forBrain Information ProcessingandCognitiveNeurodynamics,SchoolofInformationScienceandEngineering,EastChinaUniversityofScienceandTechnology,Shanghai200237,P.R.China(e-mail: rbwang@163.com).G. Chen is with the Department of Electronic Engineering, City Universityof Hong Kong, Kowloon, Hong Kong SAR, P. R. China.Digital Object Identi fi er 10.1109/TNN.2007.9141771045-9227/$25.00 © 2008 IEEE Authorized licensed use limited to: The University of Hong Kong. Downloaded on June 8, 2009 at 21:30 from IEEE Xplore. Restrictions apply.

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