### Journal Menu >>

 Applied Mathematics, 2011, 2, 533-540 doi:10.4236/am.2011.25070 Published Online May 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM Estimate of Multiple Attracing Domains for Cohen-Grossberg Neural Network with Distributed Delays Zhenkun Huang1, Zongyue Wang2 1School of Science, Jimei University, Xiamen, China 2Computer Engineering College, Jimei University, Xiamen, China E-mail: hzk974226@jmu.edu.cn Received January 21, 2011; revised March 17, 2011; accepted Marc h 20, 2011 Abstract In this paper, we present multiplicity results of exponential stability and attracting domains for Cohen- Grossberg neural network (CGNN) with distributed delays. We establish new criteria for the coexistence of equilibrium points and estimate their attracting domains. Moreover, we base our criteria on coefficients of the networks and the derivative of activation functions within the attracting domains. It is shown that our results are new and complement corresponding results existing in the previous literature. 2N Keywords: Cohen-Grossberg Networks, Distributed Delays, Exponential Stability, Attracting Domains 1. Introduction Cohen-Grossberg neural network (see [1,2]) is usually described by the following differential equations system  1,Niiiiiijj jjut autdutbgutt dd where , is the number of neurons in the network; describes the state vari-able of neuron i at time t; i:1,2,,iiaN2Nut represents an amplifica-tion function and the function can include a con-stant term indicating a fixed input to the network; idijbt weights the strength of the j unit on the unit at time t; the activation function thijg shows how the neurons react to the input. CGNN not only has a wide range of applications in pattern recognition, associative memory and combinatorial optimization but also includes a num-ber of models from neurobiology, population biology and evolution theory. Hence studies on stability of CGNN with or without delays have been vigorously done and many criteria have been obtained so far [3-14]. In the applications of neural network to associative memory storage or pattern recognitions, the coexistence of multiple stable equilibrium points is an important fea-ture [15-19,20-21]. However, few papers focus on the existence of multiple equilibrium points of CGNN and their complex convergence analysis. Hence, we should consider multistability of the following CGNN with dis-tributed delays 1,iut ii iitNij jijjjaut duttbgk t su ssdd (1.1) where the delay kern el function is assumed to be where dijkt ng piecewise continuous and satisfyitt 0,d0,e d,sij ijij ijktksslks s   is a positive constant. In this paper, we not only de new criteria for the existence of 2riveN equi-librium points of CGNN (1.1) but also estimate atacting domains for these equilibrium points. When we relax our conditions to be common assumptions, our results im-prove corresponding results in [12]. Moreover, our re-sults can extend the corresponding results in [3-13] to local exponential stability of multiple equilibrium points of Cohen-Grossberg networks. It is shown that our re-sults are new and complement the existing results in the literature. The resttr of this paper is organized as follows. In Sec-tion 2, we should make some preparations by giving some notations, assumptions and a basic lemma. Mean-while, we discuss the existence of 2N equilibrium points of CGNN (1.1). In Section 3, we only discuss local exponential stability of 2notN equilibrium points of CGNN (1.1) but also compareur results with existing o Z. K. HUANG ET AL. 534 . Coexistence of Equilibrium Points this paper, we denote by ones in the literature. In Section 4, two examples are given to illustrate the new results. Finally, concluding re- marks are given in Section 5. 2 In,0 ,NC pings from the set of all continuous and bounded map,0 to N equipped with p-norm 1pp defined by 01sup ,pNpisis  Where12,,, ,0,NNC, For any given ,0 ,NC, we denote by ;ut the solutio with  us s all n of CGNN (1.1) for,0s . Given any 0, we define ;su us; for all ,0 , s then  ;,0,NuC . Throughout this paper we always assume that 1S inedFo,. For each and there r each i, ia is a continuous function def on . Meil assume that 010av anwhe, weiii v2Sexist i, , idia, , i 2igC constants †idg suc that h   †0,lim,0limsup 0,0, 0,ii iiivii iivvii iddvd gvggv gvgv gvvgvdvv v    where i and l networks [52iiiiblark 2.1.limvidv . Hopfield-type neuRem For ra,8-10], we have 1iav, iidv dv, tanhigvv, where 0id is at. y, w that dvd, 0idv , constanObviousle can checkiilim tanh1vv, ''1 sup1limvtanhvv0, tanh0v, nce, 2tanh'' 0vv. Hetanhvtanh v'1 and TSW a consta2Se hold. say nt vector 1,,NN) if for each i, uuu is an eqConsider uilibrium point of CGNN (1.1N1iiijjjjdut bgut  iiiiiiFvdvbgut  where . Then it follows ma 2 Assund the following assumption v, iLem .1.me 12SS a1,AiiiHd bisupiiivl gv ihold. For each , there exist only twpoints and with o 1iv 2iv120iivv such that Fv0ii and2sgn 0,iiiFvv v1v vwhere v a 1,2.vv nd,i Proof. We get from A and t 1H2S tha0i00 0iii iiiFdblg  and ††lim 0iiviiiiiiFvb gdl dv  as . It follows from that v 2S0ivF v which implies that iFv is strictly increasing on ,0 ore, there0andly decreasing on  . exist only two points 1iv and v with 12iiv is strict0, Theref2iv such that 0iiFv and  n 0,v vv12iiisgFv v and 2.,1,ivvvwhere The proof is com-plete. It follows from 1AH0 for each iiib that . Now, we considwier the follong additional assumption: 21,1,2,NAii ijjHFvbg i1,jji  in 2AH, it is easy for us to get that Take 110,ijjvbgi1,ii jjiFN (2.1) Due to iFv as Lthe continuity of v,emma 2.1 and iFv, there exists a unique with †v1i11iiv†v such that †2Niiv1,†221,0,0, forall,ijjjjiNiijjiijjiF bgFvbg vvv (2.2) Take 2 in 2AHere exists a unique ,by similar argument, we de-rive that t with h†v2i22ii†vv such that †11,†111,0,0, forall,1ii ijjjjiNii ijjiijjiFv bgNFvbg vvv (2.3) Let 121111,iiNNiijj iijjjv dbgvdbgj  Due to the monotonicity of id, it is easy for us to Copyright © 2011 SciRes. AM Z. K. HUANG ET AL.535 ch 2 Feck that ††11 20ii ivv vvor each ii., de-fine 1i  . Hence, wcon-struc††112 2,, ,iii iivv vv  2t 2e can N subsets 1122NNN  , where  12,,,N with  i1, 2, ie : . following theoremWith these notations, whe have tTheorem 2.1. Under the assumptions 12SSAA and 12HH, there exist at least 2N e point.1). Proof. For each ,quilibriumNs of CGNN (112,,  with 1, 2i, i, we define ,N12,,FFFF h :tWiNF by 1iiijFdu12,,, TNuu u,jjijjglu 1Nbre whe, we have . It is obvioforuus that any u12,iiiFvvu for each ich i. For ea, i,s for us to fur discussion. Case I: If 1i there ar e two caserthe, †, then from i.e., 1iiuv1AH and (2.2) we get 1111,NiijjijjNiii iiijd bgludbglvIf 2i†ij jjibg†(2.4) C11iiv ijFuase II: , i.e. , uv†2ii, then from 1A Hand (2.3) we get 1111,NiijjijjNiiiiii jbgludbglviiiF††22iiv ijHence ijjjibg(2.5) Fu du, that is y Brouwer's Futheory, f for or eall  . Bfixed point ach u, ist at least one u such that there exFuu. Therefore, there exist at least 2N equilibriu CGNN (1.1). T he pro of is compe. Next we should make some preparations for the com-inm points ofletg section. For each i, we define the following subsets of ,0 ,C as 12,0 ,iNiN12††ii,0 ,HCsvHCsv  we have construct 2 Hence,N subsets 11HH 22,0N,NNHC . GiveHn any  ,N12,, with 1, 2i, we defisubsets for each ne semi-close i,  1†.iOv v 111iivFor any H, we1212 NNOOOO  get,0 . nd stssumesfor all ility aimation of Attracting Domains Th that assumptions3. Stab E eorem 3.1. A 12SS an 12AAHHd hold. For each , if H, then ;tuH for all t. Proof. Fix 12,,,N. Fo any  rH, we that should prove;tuH for all t. For eachi, we onlye 2i consider the cas, i.e., is †2iv for all ,0 . s rt that, fo sufficien- all We asser anytly sm†220vv, theion ii solut;itu †2iv hold0t. If this is not true, there ex-ists a *ts for all such that 0iut †2iv, *iut 0 andiut †2iv for *,tt. Due to 1 from CGAH, (2.3) and thetonicity ofrivNN mono ig, we dee(1.1) that *diutt†2†21†2††2211d0iitNii iiijij jjiiNiiiiiiijjjNiiij jjavdv gktsussbgavdv glvbgav bg††22iiiiiibbFvd      (3.1) which leads to a contradiction. Since the choice of is arbitrary, for , if †ieach i2isv for all ,0s , then †2;iiut v holds for all t. When 1i0, similar aent can ormed to show tif rgumbe perfhat i†1isv for all ,0s , n the;iiut†1v holds for all 0t. Hence, for any H, we have that ; for 0. The lete. ition 3.1. Let tuHall tproof is compDefin,Nu u  be aneuilib-rium point of CGN12,,uu.1) and  qN (1,0 ,NYuuC be a neighborhood of . If there exist 0 and Copyright © 2011 SciRes. AM Z. K. HUANG ET AL. 536 1M such that for aTYuny 12,,,N and all 0t, 11NN,011sup;epppptiisiiuts uiiu M  Where is the solu lo-cally exponentnthe attracting domistence of equilibrium points of CGor T12;;,;,,;Ntutututution of CGNN (1.1), then u is said to beially stable and Yu is contained iain of u. N The exNN (1.1) follows from Theorem 2.1. Fany given 2 , let ;ii ixtut u with H  andu, where i. Then CGNN (1.1) bcane written where as   1ˆd,iNiiiijjijjtxt dxtbgktssx ˆˆtijaxs(3.2) ˆˆ ,ˆ,ˆddiiiiiiiiii iitjij jtjijjjijjaxtaxtudxtdxtudugktsxssgktsxssglu Theorem 3.2. Assume that assumptions 12SS , if there exist and hold. For each and 12AAHHpositive constants ii such that for each i   011iNsp10110infsupe d1supe djiiivOiijj ijjvONjsjij ijijvOidvbgvksspbgvkssp (3.3) Then is locally exponentially stable and u H is containehe attracting domain of Proof. Fix d in tu. 1,,N T. Let be a solution of CGNN  1;,, ;Ntututu (1.1) with H and u is an equilibrium point of CGin . By TheoreNN (1.1) m 3.1, we know that;tuH for all tuO and ut O for 0. It is obvious that 0t. For each iall , let ;ii ixt uut and let etiiXtxt. In view of (3.2), we obtain 01e,tii iitjjjtx tg k1diiNiijijjDX tXdbtsx ss+ i.e., 011d.iiiiitNiijjjij jjDX tdX tbgk tsXss (3.4) where infiii ivOddv and supjjj jvOggvsovskii function . Now v-Kravwe define a LyapunoV as follows  11iNNijVt tVtbg t   1211110,eddNpiiiiij jtspij jtsVtVtVXtksX wws (3.5) From (3.5), it is easy for us to estimate   sup .piXs110,001 edNjsiijjjjiiisVbgksss (3.6) From (3.4), (3.5) and by simple calculations, we ob-tain     0111 1()NppX tVtb g110110ededNpiii iiiiiijjjsij jNsiijj ijjppjjDV tpXtdks Xtssbgk sXt Xts s (3.7) By using the basic inequality 11ppppa bpab  and (3.7), we obtain that Copyright © 2011 SciRes. AM Z. K. HUANG ET AL.537     piXt011101101ededNiiiiNsiijj ijjNjsijj jjijiDV tpdpbgkssbgkss (3.8) It follows from (3.3) that . Hence for any 0DV twhich leads to  0Vt V0t  1111Njiijiibg0,0eedsup .pptiiisjjjipisxtk sssXs That is, N ,011esupNN.ppptiisiixt xs where  110max max 1ed.minNjjsiij ijjiijiiibgk sss Therefore, we have 11;eNpptiipiut uMu  where pM. That is u is locally exponentiallyst able and H 1is containe the attracting domain ofomplete. Take d in u. The proof is ci and is easy for us tve the following crollary. llary 3.1. Assume that 1p, ito haoCoro12SS, if there exand Ahold. For any given ists a A12HH constant  such that for each i,  0110infsupe diiNsii jjiijivO jvOdvbgvk ss (3.9) Then is locally exponentially stable and u H is containehe attracting domain of Rem 1 For each d in tark 3.u. j Li, if is tz continu gjglobally Lipschious with apschitz constant jL and there exists nstant 0ja co such that 1jjjdu du uv for all ,uv. It is ob- us that we have supjviojjvOgvL and inf  jjvO dvj. If we replace inf dv by jjvOj,supjjvOgv by jL in (3.3), then we get 011Np10110ed1edsii iijjijjNjsjij ijijibL k sspbLk ssp  (3.) 10Take 0, 1p and 1ijl in (3.10), [9] ved that there exists a unique equilibriumpro point thee of acon s withconfines of attracting dom can en we relax co (3.10) to be(3.3), results in [7,9,11] are not applicable for CGNN (1. resunmme sults in [9]. is easy for us to have the fol-lowing corollary. Corolla ry 3.2. Under the following basic assumptions of CGNN (1.1) which is globally asymptotically stable. It is obvi-ous that our criteria only base on parameters of net-work and the deriv ativtivation functiin the ains, theybe easily checked. However, whndition 1). It is obvious that ourlts are ew and cople-nt the corresponding reFrom above remark, it*1S. For each i, ia eis a continuou that s function on . Meanwhile, we assum010iiav i  and id is csing ontinuous increawith  10iiiu udu dv. *2S For each i, ig is globally Lipschitz continuous with a Lipschitz constant iL and there exists a constant 0ig such that iigvg for all v. If there exist positive constants ii and  such that for each i, 011NiiiijjijpbLk  10jNp (3.11) 1ed1sjss shequilibrium point oNly stable. W, we only consider mapping 10edjij ijijibLk sspT en there exists at least a unique f CGN (1.1). which is globally expone ntialProof. ith assumptions AAHH12,,,12NFFFF from to  defined by 11,NiijjjFdbgluiijju where 1Nu  and ,ii i. By Brouwer’s fixed ory, ther equilibrium point 12,ivve exists at least onepoint theˆu such that ˆˆFuu. Similarly n show that ˆuas Thewe caorem 3.2, by (3.11)  is Copyright © 2011 SciRes. AM Z. K. HUANG ET AL. Copyright © 2011 SciRes. AM 538 the unique equilibrium point of CGNN (1.1) which is globally exponentially stable. The proof is complete. Remark 3.2. Take  11 12221112 2111 221221 12221,1.4,3.1 ,60.125e ,0.25e ,e,tanh ,3.6742, 3.1458, 10.sssdut utdututbks ksks ksgvgvvbbb  , 1122aut aut and . When 1ijl1,2p, 0Corollary 3.2 leads to Theorem 3.2 and Theorem 3.3 in [17]. It is obvious that Theorem 3.2-3.5 in [9] are only ouvior of C4. Examples Consider the following Cohen-Grossberg networks with ted delays r special cases of Corollary 3.2. Furthermore, our re-sults can extend the existence of multiple equilibrium points and their attracting behaohen-Grossberg networks to [3-12]. distribu  12 2 1221222 22211dNjtNjbg kt su ssut autdu ttbdd (4.1) 1 21122 2222ddttNutgktsussbgksu ssder 111 1111 11111dtNjut aduttbgktsussdd1jExample 4.1 ConsiSince 112 2ii ibL bL and (iLi are defined in Co 3.2), most resultrollarys reorted in [7,9-11] could not applicable for CGNN (4.1) even though we take de-lay kernels into consideration. It is obvious that p12SS hold and we have  1122 112112 120011212d1, 2d11.46,3.1 10,1llkss llkss2FvvgvFvvgggv  From some computations, we get 11 1.3565v, 12v1.3565, 21 1.19v, 22 1.19v such that 0iiFv, hence 1 3.3211Fv1,522v , where 44 and 1 4.6168F. From (2.2)-( we2.3),†††.8190 1.81901.91vvv  can estimate †111221 221 .9v It is easy for us to get 120.0999, 0.0086gv gv. 1) satisfied our assumptions inTherefore, CGNN (4. Theorem 3.2 and assumptions 12AAHH hold. Let 1,1 16p and 121e calcula-check tolds. By Theorem 3.2, there exist only four locally exponentially stable equilib-rium points of CGNN (4.1) located in . Moreover, their attracting domains can be estimated b. From som hy tions, we can hat (3.3)    1,1 212121,2 212122,1 2121 22,2 21212,,0 ,1.1819,1.9,,0 ,1.1819,1.9,,0 ,1.1819,1.9,,0 ,1.1819,1.9HC ssHC ssHC ssHCss         Example 4.2 Consider We can estimate that So,     '†11 1'†22 2'' ''111'' ''2121.381.4 0.02tanh1.4,3.083.10.02tanh3.1,0.02tanh0tanh ,0.02tanh0tanh .ddv vdddv vdvd vvvbvvd vvvbv          112 211 1122 2211 1112212212 12211222 121,1.4 0.02tanh,3.1 0.02tanh,6,e ,etanh ,,83.6724, 3.6724, 10, 3.1458ssaut autdut ututdutututb ksksksksgvgvvk sbbbb 12SS 11 1.37 hold. Similarly as Example 4.1, we get v, 12 1.37v, , 21 1.21v22 1.21v such that iiFv0, hence 11Fv13.372 and  Z. K. HUANG ET AL.539 , where 22 14.633Fv 1, 2.†1.91vptions ere exist From (2.2)-(2 .3), sum and 3.2, thCGNN we can estimate ††11 121.85vvWe can check that as(3.3) hold. By Theoremequilibrium points of †21 221.851.91v 12AAHHonly four stable (4.1) located in . Moreover, their attracting domains H can be esti-mated as Example 4.1. 5. Concluding Remarks In this paper, some new cistence of 2N equilibriumalso given for each eresults are new and co[7,9-11]. Furthermore, oding ones reported in [2ity of multiple equilibrium 6. Acknowledgements This work was supportPrding of Jimei Univ ersity (gram Foundation foResearch Talents of Fujian JA10198). 7. References petitive NeSystems, Man and C[2] S. Grossberg, “NMechanisms and Archit622-634. Grossberg Neurriterived for the coex- poidomain is quilibrium point. It is shown that our mplemprevious results in ur rextend correspon- 2-23] to local exponential stabil- poof neural network. byation for Young ersity, Scientific Research Fun- ZQ201 and Training Pro- r Distinguung Scholars anducation (JA10184, Formation and l Me StoraEE Transactioonlinearal Networks: Principles, ecturNeural NetworksVy havnd TheVol. 15, No. 6, 2007, pp. doi:10.1016/j.simpat.2006.12.003 Stability of Cohen- al Networks,” etworks, Vol. 15, Delays,” Journal of Mathematical Analysis and Applica-tions, Vol. 296, No. 2, 2004, pp. 665-685. doi:10.1016/j.jmaa.2004.04.039 [7] X. F. Liao, C. G. Li and K.-W. Wong, “Criteria for Ex-ponential Stability of Cohen-Grossberg Neural Net-works,” Neural Networks, Vol. 17, No. 10, 2004, pp. 1401-1414. doi:10.1016/j.neunet.2004.08.007 [8] C. X. Huang and L. H. Huang, “Dynamics of a Class of Cohen-Grossberg Neural Networks with Time-Varying Delays,” Nonlinear Analysis: Real World Applications, Vol. 8, No. 1, 2007, pp. 40-52. doi:10.1016/j.nonrwa.2005.04.008 en-Grossberg Neural Net-elays,” Applied Mathematics and Computation, Vol. 160, No. 1, 2005, pp. 93-110. un and L. Wan, “Global Exponential Stability and ions of Cohen-Grossberg Neural Networks usly Distributed Delays,” Physica D, Vol. 208, No. 1-2, 2005, pp. 1-20. 6/S0167-2789(02)00544-4 , No. 10, 2006, pp. 6.07.006 l Academy of Sci-. 3088-3092. 17 a are derints and attracting ent the sults can eints the Found0003)ished YoHigher EdParalleorks,” IEs, Vol. Neures,” ory, onential Neural Nedofessors of Jimei Univ do[1] M. Cohen and S. Grossberg, “Absolute Stability and Global Patternmoryge by Comural Netwns on ybernetic 13, No. 5, 1983, pp. 815-825. , ol. 1, [15] J. Hopfield, “Neurons with Graded Response have Col-lective Computational Properties like Those of Two State Neurons,” Proceedings of the Nationaences, USA, Vol. 81, No. 10, 1984, ppNo. 1, 1988, pp. 17-61. doi:10.1016/0893-6080(88)90021-4 [3] Z. K. Huang and Y. H. Xia, “Exponential p-Stabilitof Second Order Cohen-Grossberg Neural Networks with Transmission Delays and Learning Beior,” Simulation Modelling Practice a[4] L. Wang and X. Zou, “ExpNo. 3, 2002, pp. 415-422. doi:10.1016/S0893-6080(02)00025-4 [5] T. Chen and L. Rong, “Robust Global Exponential Sta-bility of Cohen-Grossberg Neural Networks with Time Delays,” IEEE Transactions on Neural Networks, Vol. 15, No. 1, 2004, pp. 203-205. doi:10.1109/TNN.2003.822974 [6] J. D. Cao and J. L. Liang, “Boundedness and Stability forCohen-Grossberg Neural Network with Time-Var ying [9] L. Wang, “Stability of Cohworks with Distributed Ddoi:10.1016/j.amc.2003.09.014 [10] X. F. Liao and C. D. Li, “Global Attractivity of Cohen- Grossberg Model with Finite and Infinite Delays,” Jour-nal of Mathematical Analysis and Applications, Vol. 315, No. 1, 2006, pp. 244-262. doi:10.1016/j.jmaa.2005.04.076 11] J. H. S[Periodic Solutwith Continuoi:10.1016/j.physd.2005.05.009 [12] L. Wang and X. F. Zou, “Harmless Delays in Cohen- Grossberg Neural Networks,” Physica D: Nonlinear Phenomena, Vol. 170, No. 2, 2002, pp. 162-173. doi:10.101[13] K. N. Lu, D. Y. Xu and Z. C. Yang, “Global Attraction and Stability for Cohen-Grossberg Neural Networks with Delays,” Neural Networks, Vol. 191538-1549. doi:10.1016/j.neunet.200[14] Z. K. Huang, X. H. Wang and Y. H. Xia, “Exponential Stability of Impulsive Cohen-Grossberg Networks with Distributed Delays,” International Journal of Circuit Theory and Applications, Vol. 36, No. 3, 2008, pp. 345-365. doi:10.1002/cta.424 doi:10.1073/pnas.81.10.3088 [16] C. Y. Cheng, K. H. Lin and C. W. Shih, “Multistability in Recurrent Neural Networks,” SIAM Journal on Applied Mathematics, Vol. 66, No. 4, 2006, pp. 1301-1320. doi:10.1137/050632440 [17] Z. Zeng, D. S. Huang and Z. Wang, “Memory Pattern Analysis of Cellular Neural Networks,” Physics Letters A, Vol. 342, No. 1-2, 2005, pp. 114-128. doi:10.1016/j.physleta.2005.05.0[18] C. Y. Cheng, K. H. Lin and C. W. Shih, “Multistability and Convergence in Delayed Neural Networks,” Physica D, Vol. 225, No. 1, 2007, pp. 61-74. doi:10.1016/j.physd.2006.10.003 [19] L. P. Shayer and S. A. Campbell, “Stability, Bifurcation Copyright © 2011 SciRes. AM Z. K. HUANG ET AL. Copyright © 2011 SciRes. AM 540 0, pp. 673-700. andl. 57, No. 8, 2010, ppeural 0, 2010, pp. 1643-1655. s, Vol. 15, No. 7, 3-6080(02)00039-4 7-53. 7-7 and Multistability in a System of Two Coupled Neurons with Multiple Time Delays,” SIAM Journal on Applied Mathematics, Vol. 61, No. 2, 200doi:10.1137/S0036139998344015 [20] Z. K. Huang, Q. K. Song and C. H. Feng, “Multistability in Networks with Self-Excitation and High-Order Synap-tic Connectivity,” IEEE Transactions on Circuits Systems-I: Regular Papers, Vo . [22144-215. doi:10.1109/TCSI.2009.2037401 [21] Z. K. Huang, X. H. Wang and C. H. Feng, “Multiperi-odicity of Periodically Oscillated Discrete-Time Neural Networks with Transient Excitatory Self-Connections and Sigmoidal Nonlinearities,” IEEE Transactions on NNetworks, Vol. 21, No. 1doi:10.1109/TNN.2010.2067225 [22] Y. M. Chen, “Global Stability of Neural Networks with Distributed Delays,” Neural Network2002, pp. 867-871. doi:10.1016/S0893] H. Y. Zhao, “Global Asymptotic Stability of Hopfield Neural Network involving Distributed Delays,” Neural Networks, Vol. 17, No. 1, 2004, pp. 4doi:10.1016/S0893-6080(03)0007