{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,8,7]],"date-time":"2024-08-07T01:38:42Z","timestamp":1722994722495},"reference-count":35,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2022,1,10]],"date-time":"2022-01-10T00:00:00Z","timestamp":1641772800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2022,1,10]],"date-time":"2022-01-10T00:00:00Z","timestamp":1641772800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Neural Process Lett"],"published-print":{"date-parts":[[2022,4]]},"DOI":"10.1007\/s11063-021-10682-9","type":"journal-article","created":{"date-parts":[[2022,1,10]],"date-time":"2022-01-10T00:02:40Z","timestamp":1641772960000},"page":"1293-1315","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Continuous Recurrent Neural Networks Based on Function Satlins"],"prefix":"10.1007","volume":"54","author":[{"given":"Yue","family":"Huang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jiali","family":"Yu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jinsong","family":"Leng","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bisen","family":"Liu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhang","family":"Yi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,1,10]]},"reference":[{"key":"10682_CR1","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/978-1-4757-3819-3","volume-title":"Convergence analysis of recurrent neural networks","author":"Z Yi","year":"2004","unstructured":"Yi Z, Tan K (2004) Convergence analysis of recurrent neural networks. Springer, US, pp 1\u2013250"},{"issue":"9","key":"10682_CR2","doi-asserted-by":"publisher","first-page":"3268","DOI":"10.1109\/TCYB.2018.2839686","volume":"49","author":"Z Guo","year":"2019","unstructured":"Guo Z, Gong S, Wen S, Huang T (2019) Event-based synchronization control for memristive neural networks with time-varying delay. IEEE Trans. Cybern. 49(9):3268\u20133277","journal-title":"IEEE Trans. Cybern."},{"issue":"6","key":"10682_CR3","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s40815-016-0203-z","volume":"18","author":"J Yu","year":"2016","unstructured":"Yu J, Yi Z (2016) Stability analysis and fuzzy control for uncertain delayed t-s nonlinear systems. Int. J. Fuzzy Syst. 18(6):1\u20138","journal-title":"Int. J. Fuzzy Syst."},{"key":"10682_CR4","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.neucom.2020.08.036","volume":"419","author":"W Chen","year":"2021","unstructured":"Chen W, Yu J, Yi Z, Hong Q (2021) The effect of coefficients on the continuous attractors in coupled highway neural networks. Neurocomputing 419:1\u20138","journal-title":"Neurocomputing"},{"key":"10682_CR5","doi-asserted-by":"publisher","first-page":"25","DOI":"10.1016\/j.neucom.2020.11.047","volume":"429","author":"W Xiang","year":"2021","unstructured":"Xiang W, Yu J, Yi Z, Wang C, Gao Q, Liao Y (2021) Coexistence of continuous attractors with different dimensions for neural networks - sciencedirect. Neurocomputing 429:25\u201332","journal-title":"Neurocomputing"},{"key":"10682_CR6","unstructured":"Chen T, Rubanova Y, Bettencourt J, DD (2018) Neural ordinary differential equations. In: 32nd conference on neural information processing systems"},{"key":"10682_CR7","doi-asserted-by":"crossref","unstructured":"Haber E, Ruthotto L (2017) Stable architectures for deep neural networks. Inverse Prob 34(1):014004","DOI":"10.1088\/1361-6420\/aa9a90"},{"issue":"6","key":"10682_CR8","doi-asserted-by":"publisher","first-page":"104","DOI":"10.1016\/j.neunet.2020.01.020","volume":"125","author":"J Gou","year":"2020","unstructured":"Gou J, Wang L, Yi Z, Yuan Y, Ou W, Mao Q (2020) Weighted discriminative collaborative competitive representation for robust image classification. Neural Network 125(6):104\u2013120","journal-title":"Neural Network"},{"issue":"8","key":"10682_CR9","doi-asserted-by":"publisher","first-page":"7362","DOI":"10.1109\/TIE.2020.3001840","volume":"68","author":"D Wang","year":"2020","unstructured":"Wang D, Ha M, Qiao J (2020) Data-driven iterative adaptive critic control towards an urban wastewater treatment plant. IEEE Trans Ind Electron 68(8):7362\u20137369","journal-title":"IEEE Trans Ind Electron"},{"key":"10682_CR10","doi-asserted-by":"publisher","first-page":"331","DOI":"10.1016\/j.cnsns.2017.10.007","volume":"57","author":"Z Yi","year":"2018","unstructured":"Yi Z, Luo G (2018) Multistability of a three-degree-of-freedom vibro-impact system. Commun Nonlinear Ence Numer Simul 57:331\u2013341","journal-title":"Commun Nonlinear Ence Numer Simul"},{"key":"10682_CR11","doi-asserted-by":"publisher","first-page":"840","DOI":"10.1016\/j.ins.2018.06.001","volume":"507","author":"X Zhang","year":"2020","unstructured":"Zhang X, Yang J, Tang L (2020) Three-way class-specific attribute reducts from the information viewpoint. Inf Sci 507:840\u2013872","journal-title":"Inf Sci"},{"key":"10682_CR12","doi-asserted-by":"publisher","first-page":"10","DOI":"10.1016\/j.neucom.2018.01.024","volume":"284","author":"X Xie","year":"2018","unstructured":"Xie X, Wen S, Zeng Z, Huang T (2018) Memristor-based circuit implementation of pulse-coupled neural network with dynamical threshold generator. Neurocomputing 284:10\u201316","journal-title":"Neurocomputing"},{"issue":"1","key":"10682_CR13","doi-asserted-by":"publisher","first-page":"417","DOI":"10.1146\/annurev-vision-082114-035447","volume":"1","author":"N Kriegeskorte","year":"2015","unstructured":"Kriegeskorte N (2015) Deep neural networks: a new framework for modeling biological vision and brain information processing. Ann Rev Vis ence 1(1):417\u2013446","journal-title":"Ann Rev Vis ence"},{"key":"10682_CR14","doi-asserted-by":"publisher","first-page":"1272","DOI":"10.1109\/TAC.2019.2926167","volume":"65","author":"D Wang","year":"2019","unstructured":"Wang D, Ha M, Qiao J (2019) Self-learning optimal regulation for discrete-time nonlinear systems under event-driven formulation. IEEE Trans Autom Control 65:1272\u20131279","journal-title":"IEEE Trans Autom Control"},{"issue":"10","key":"10682_CR15","doi-asserted-by":"publisher","first-page":"1380","DOI":"10.4249\/scholarpedia.1380","volume":"2","author":"Ca Eliasmith","year":"2007","unstructured":"Eliasmith Ca (2007) Attractor network. Scholarpedia 2(10):1380","journal-title":"Scholarpedia"},{"issue":"10","key":"10682_CR16","doi-asserted-by":"publisher","first-page":"3088","DOI":"10.1073\/pnas.81.10.3088","volume":"81","author":"JJ Hopfield","year":"1984","unstructured":"Hopfield JJ (1984) Neurons with graded response have collective computational properties like those of two-state neurons. Proceed National Acad Sci 81(10):3088\u20133092","journal-title":"Proceed National Acad Sci"},{"key":"10682_CR17","doi-asserted-by":"publisher","first-page":"2215","DOI":"10.1162\/0899766054615626","volume":"17","author":"S Wu","year":"2005","unstructured":"Wu S, Amari S (2005) Computing with continuous attractors: stability and online aspects. Neural Comput 17:2215\u20132239","journal-title":"Neural Comput"},{"key":"10682_CR18","doi-asserted-by":"publisher","first-page":"994","DOI":"10.1162\/neco.2008.10-06-378","volume":"20","author":"S Wu","year":"2008","unstructured":"Wu S, Hamaguchi K, Amari S (2008) Dynamics and computation of continuous attractors. Neural Comput 20:994\u20131025","journal-title":"Neural Comput"},{"key":"10682_CR19","doi-asserted-by":"publisher","first-page":"1253","DOI":"10.1016\/S0893-6080(98)00064-1","volume":"11","author":"HS Seung","year":"1998","unstructured":"Seung HS (1998) Continuous attractors and oculomotor control. Neural Networks 11:1253\u20131258","journal-title":"Neural Networks"},{"key":"10682_CR20","doi-asserted-by":"publisher","first-page":"2268","DOI":"10.1126\/science.290.5500.2268","volume":"290","author":"HS Seung","year":"2000","unstructured":"Seung HS, Lee DD (2000) The manifold ways of perception. Science 290:2268\u20132269","journal-title":"Science"},{"issue":"15","key":"10682_CR21","doi-asserted-by":"publisher","first-page":"5900","DOI":"10.1523\/JNEUROSCI.17-15-05900.1997","volume":"17","author":"A Samsonovich","year":"1997","unstructured":"Samsonovich A, Mcnaughton BL (1997) Path integration and cognitive mapping in a continuous attractor neural network model. J Neurosci 17(15):5900\u20135920","journal-title":"J Neurosci"},{"key":"10682_CR22","doi-asserted-by":"publisher","first-page":"524","DOI":"10.1049\/iet-cta.2016.0333","volume":"11","author":"C Wang","year":"2016","unstructured":"Wang C, Wu Y, Yu J (2016) Barrier lyapunov functions-based dynamic surface control for pure-feedback systems with full state constraints. IET Control Theory Appl 11:524\u2013530","journal-title":"IET Control Theory Appl"},{"key":"10682_CR23","doi-asserted-by":"publisher","first-page":"402","DOI":"10.1109\/TIE.2019.2897542","volume":"67","author":"J Lau","year":"2020","unstructured":"Lau J, Liang W, Tan K (2020) Motion control for piezoelectric-actuator-based surgical device using neural network and extended state observer. IEEE Trans Ind Electron 67:402\u2013412","journal-title":"IEEE Trans Ind Electron"},{"key":"10682_CR24","doi-asserted-by":"publisher","first-page":"421","DOI":"10.1007\/s00211-012-0463-y","volume":"122","author":"MC Zelati","year":"2012","unstructured":"Zelati MC, Tone F (2012) Multivalued attractors and their approximation: Applications to the navier-stokes equations. Numerische Mathematik 122:421\u2013441","journal-title":"Numerische Mathematik"},{"key":"10682_CR25","doi-asserted-by":"publisher","first-page":"367","DOI":"10.1016\/j.jde.2005.06.008","volume":"223","author":"C Zhong","year":"2006","unstructured":"Zhong C, Yang M, Sun C (2006) The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations. J Diff Equ 223:367\u2013399","journal-title":"J Diff Equ"},{"key":"10682_CR26","doi-asserted-by":"publisher","first-page":"368","DOI":"10.1109\/TNN.2008.2010771","volume":"20","author":"J Yu","year":"2009","unstructured":"Yu J, Yi Z, Zhang L (2009) Representations of continuous attractors of recurrent neural networks. IEEE Trans Neural Network 20:368\u2013372","journal-title":"IEEE Trans Neural Network"},{"key":"10682_CR27","doi-asserted-by":"publisher","first-page":"175","DOI":"10.1109\/TNN.2008.2009535","volume":"20","author":"L Zou","year":"2009","unstructured":"Zou L, Tang H, Tan K, Zhang W (2009) Analysis of continuous attractors for 2-d linear threshold neural networks. IEEE Trans Neural Network 20:175\u2013180","journal-title":"IEEE Trans Neural Network"},{"issue":"3","key":"10682_CR28","doi-asserted-by":"publisher","first-page":"639","DOI":"10.1162\/089976603321192112","volume":"15","author":"Z Yi","year":"2003","unstructured":"Yi Z, Tan KK, Lee TH (2003) Multistability analysis for recurrent neural networks with unsaturating piecewise linear transfer functions. Neural Comput 15(3):639\u2013662","journal-title":"Neural Comput"},{"key":"10682_CR29","doi-asserted-by":"publisher","first-page":"329","DOI":"10.1109\/TNN.2004.824272","volume":"15","author":"Z Yi","year":"2004","unstructured":"Yi Z, Tan K (2004) Multistability analysis of discrete-time recurrent neural networks with unsaturating piecewise linear activation functions, IEEE Trans. Neural Networks 15:329\u2013336","journal-title":"Neural Networks"},{"key":"10682_CR30","doi-asserted-by":"publisher","first-page":"691","DOI":"10.1016\/S0893-6080(98)00012-4","volume":"11","author":"RLT Hahnloser","year":"1998","unstructured":"Hahnloser RLT (1998) On the piecewise analysis of networks of linear threshold neurons. Neural Networks 11:691\u2013697","journal-title":"Neural Networks"},{"key":"10682_CR31","doi-asserted-by":"publisher","first-page":"639","DOI":"10.1162\/089976603321192112","volume":"15","author":"Z Yi","year":"2003","unstructured":"Yi Z, Tan K, Lee TH (2003) Multistability analysis for recurrent neural networks with unsaturating piecewise linear transfer functions. Neural Comput 15:639\u2013662","journal-title":"Neural Comput"},{"issue":"4","key":"10682_CR32","doi-asserted-by":"publisher","first-page":"911","DOI":"10.1109\/TNNLS.2015.2485263","volume":"28","author":"J Yu","year":"2017","unstructured":"Yu J, Mao H, Yi Z (2017) Parameter as a switch between dynamical states of a network in population decoding. Neural Network Learn Syst IEEE Trans 28(4):911\u2013916","journal-title":"Neural Network Learn Syst IEEE Trans"},{"issue":"2","key":"10682_CR33","doi-asserted-by":"publisher","first-page":"341","DOI":"10.1007\/s00009-020-1487-7","volume":"17","author":"M He","year":"2020","unstructured":"He M, Leng L, Yu J, Xu Y (2020) On the sum of k-frames in hilbert spaces. Mediterranean J Math 17(2):341\u2013366","journal-title":"Mediterranean J Math"},{"key":"10682_CR34","first-page":"1003","volume":"52","author":"K Tan","year":"2005","unstructured":"Tan K, Tang H, Zhang W (2005) Qualitative analysis for recurrent neural networks with linear threshold transfer functions. Circuit Syst 52:1003\u20131012","journal-title":"Circuit Syst"},{"issue":"1","key":"10682_CR35","doi-asserted-by":"publisher","first-page":"175","DOI":"10.1109\/TNN.2008.2009535","volume":"20","author":"L Zou","year":"2009","unstructured":"Zou L, Tang H, Tan K, Zhang W (2009) Analysis of continuous attractors for 2-d linear threshold neural networks. IEEE Trans. Neural Networks 20(1):175\u201380","journal-title":"IEEE Trans. Neural Networks"}],"container-title":["Neural Processing Letters"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11063-021-10682-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11063-021-10682-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11063-021-10682-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,4,9]],"date-time":"2022-04-09T17:11:31Z","timestamp":1649524291000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11063-021-10682-9"}},"subtitle":["Coexistence of Multiple Continuous Attractors"],"short-title":[],"issued":{"date-parts":[[2022,1,10]]},"references-count":35,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2022,4]]}},"alternative-id":["10682"],"URL":"https:\/\/doi.org\/10.1007\/s11063-021-10682-9","relation":{},"ISSN":["1370-4621","1573-773X"],"issn-type":[{"value":"1370-4621","type":"print"},{"value":"1573-773X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,1,10]]},"assertion":[{"value":"29 October 2021","order":1,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 January 2022","order":2,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}