{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,7]],"date-time":"2026-10-07T08:52:55Z","timestamp":1791363175858,"version":"4.3.1"},"reference-count":43,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2026,9,21]],"date-time":"2026-09-21T00:00:00Z","timestamp":1789948800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,9,21]],"date-time":"2026-09-21T00:00:00Z","timestamp":1789948800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2021ZD0201300"],"award-info":[{"award-number":["2021ZD0201300"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Cogn Neurodyn"],"published-print":{"date-parts":[[2026,12]]},"DOI":"10.1007\/s11571-026-10555-8","type":"journal-article","created":{"date-parts":[[2026,9,21]],"date-time":"2026-09-21T04:32:30Z","timestamp":1789965150000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Emergent attractor-like dynamics support stable and generalizable path integration in recurrent neural networks"],"prefix":"10.1007","volume":"20","author":[{"given":"Yunxiang","family":"Chen","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Tao","family":"Wang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Wei","family":"Wang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9560-8518","authenticated-orcid":false,"given":"Feng","family":"Liu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,9,21]]},"reference":[{"issue":"52","key":"10555_CR1","doi-asserted-by":"publisher","first-page":"18428","DOI":"10.1073\/pnas.1421056111","volume":"111","author":"CB Alme","year":"2014","unstructured":"Alme CB, Miao C, Jezek K, Treves A, Moser EI, Moser MB (2014) Place cells in the hippocampus: eleven maps for eleven rooms. Proc Natl Acad Sci USA 111(52):18428\u201318435","journal-title":"Proc Natl Acad Sci USA"},{"issue":"7705","key":"10555_CR2","doi-asserted-by":"publisher","first-page":"429","DOI":"10.1038\/s41586-018-0102-6","volume":"557","author":"A Banino","year":"2018","unstructured":"Banino A et al (2018) Vector-based navigation using grid-like representations in artificial agents. Nature 557(7705):429\u2013433","journal-title":"Nature"},{"issue":"2","key":"10555_CR3","doi-asserted-by":"publisher","DOI":"10.1371\/journal.pcbi.1000291","volume":"5","author":"Y Burak","year":"2009","unstructured":"Burak Y, Fiete IR (2009) Accurate path integration in continuous attractor network models of grid cells. PLoS Comput Biol 5(2):e1000291","journal-title":"PLoS Comput Biol"},{"key":"10555_CR4","doi-asserted-by":"publisher","first-page":"381243","DOI":"10.1155\/2008\/381243","volume":"2008","author":"CB Canto","year":"2008","unstructured":"Canto CB, Wouterlood FG, Witter MP (2008) What does the anatomical organization of the entorhinal cortex tell us? Neural Plast 2008:381243","journal-title":"Neural Plast"},{"issue":"9","key":"10555_CR5","doi-asserted-by":"publisher","first-page":"1512","DOI":"10.1038\/s41593-019-0460-x","volume":"22","author":"R Chaudhuri","year":"2019","unstructured":"Chaudhuri R, Ger\u00e7ek B, Pandey B, Peyrache A, Fiete I (2019) The intrinsic attractor manifold and population dynamics of a canonical cognitive circuit across waking and sleep. Nat Neurosci 22(9):1512\u20131520","journal-title":"Nat Neurosci"},{"key":"10555_CR6","first-page":"101975","volume":"38","author":"T Chu","year":"2025","unstructured":"Chu T, Wu Y, Burgess N, Ji Z, Wu S (2025) Unfolding the black box of recurrent neural networks for path integration. Adv Neural Inf Process Syst 38:101975\u2013102008","journal-title":"Adv Neural Inf Process Syst"},{"issue":"3","key":"10555_CR7","doi-asserted-by":"publisher","first-page":"318","DOI":"10.1038\/nn.3310","volume":"16","author":"JJ Couey","year":"2013","unstructured":"Couey JJ et al (2013) Recurrent inhibitory circuitry as a mechanism for grid formation. Nat Neurosci 16(3):318\u2013324","journal-title":"Nat Neurosci"},{"key":"10555_CR8","unstructured":"Cueva CJ, Wei XX (2018) Emergence of grid-like representations by training recurrent neural networks to perform spatial localization. arXiv:1803.07770."},{"key":"10555_CR9","doi-asserted-by":"publisher","first-page":"128","DOI":"10.1016\/j.neunet.2021.04.031","volume":"142","author":"S Dang","year":"2021","unstructured":"Dang S, Wu Y, Yan R, Tang H (2021) Why grid cells function as a metric for space. Neural Netw 142:128\u2013137","journal-title":"Neural Netw"},{"issue":"16","key":"10555_CR10","doi-asserted-by":"publisher","first-page":"4266","DOI":"10.1523\/JNEUROSCI.4353-05.2006","volume":"26","author":"MC Fuhs","year":"2006","unstructured":"Fuhs MC, Touretzky DS (2006) A spin glass model of path integration in rat medial entorhinal cortex. J Neurosci 26(16):4266\u20134276","journal-title":"J Neurosci"},{"issue":"2021","key":"10555_CR11","first-page":"28623","volume":"34","author":"R Gao","year":"2021","unstructured":"Gao R, Xie J, Wei XX, Zhu SC, Wu YN (2021) On path integration of grid cells: group representation and isotropic scaling. Adv Neural Inf Process Syst 34(2021):28623\u201328635","journal-title":"Adv Neural Inf Process Syst"},{"issue":"7895","key":"10555_CR12","doi-asserted-by":"publisher","first-page":"123","DOI":"10.1038\/s41586-021-04268-7","volume":"602","author":"RJ Gardner","year":"2022","unstructured":"Gardner RJ et al (2022) Toroidal topology of population activity in grid cells. Nature 602(7895):123\u2013128","journal-title":"Nature"},{"issue":"3","key":"10555_CR13","doi-asserted-by":"publisher","first-page":"736","DOI":"10.1016\/j.cell.2018.08.066","volume":"175","author":"Y Gu","year":"2018","unstructured":"Gu Y et al (2018) A map-like micro-organization of grid cells in the medial entorhinal cortex. Cell 175(3):736\u2013750","journal-title":"Cell"},{"issue":"7052","key":"10555_CR14","doi-asserted-by":"publisher","first-page":"801","DOI":"10.1038\/nature03721","volume":"436","author":"T Hafting","year":"2005","unstructured":"Hafting T, Fyhn M, Molden S, Moser MB, Moser EI (2005) Microstructure of a spatial map in the entorhinal cortex. Nature 436(7052):801\u2013806","journal-title":"Nature"},{"key":"10555_CR15","doi-asserted-by":"publisher","DOI":"10.3389\/fncom.2021.616748","volume":"15","author":"L Kang","year":"2021","unstructured":"Kang L, Xu B, Morozov D (2021) Evaluating state space discovery by persistent cohomology in the spatial representation system. Front Comput Neurosci 15:616748","journal-title":"Front Comput Neurosci"},{"issue":"8057","key":"10555_CR16","doi-asserted-by":"publisher","first-page":"155","DOI":"10.1038\/s41586-024-08541-3","volume":"640","author":"M Khona","year":"2025","unstructured":"Khona M, Chandra S, Fiete I (2025) Global modules robustly emerge from local interactions and smooth gradients. Nature 640(8057):155\u2013164","journal-title":"Nature"},{"issue":"1","key":"10555_CR17","doi-asserted-by":"publisher","first-page":"105","DOI":"10.1152\/jn.2001.85.1.105","volume":"85","author":"JJ Knierim","year":"2001","unstructured":"Knierim JJ, McNaughton BL (2001) Hippocampal place-cell firing during movement in three-dimensional space. J Neurophysiol 85(1):105\u2013116","journal-title":"J Neurophysiol"},{"issue":"7561","key":"10555_CR18","doi-asserted-by":"publisher","first-page":"419","DOI":"10.1038\/nature14622","volume":"523","author":"E Kropff","year":"2015","unstructured":"Kropff E, Carmichael JE, Moser MB, Moser EI (2015) Speed cells in the medial entorhinal cortex. Nature 523(7561):419\u2013424","journal-title":"Nature"},{"issue":"12","key":"10555_CR19","doi-asserted-by":"publisher","first-page":"1256","DOI":"10.1002\/hipo.20520","volume":"18","author":"E Kropff","year":"2008","unstructured":"Kropff E, Treves A (2008) The emergence of grid cells: intelligent design or just adaptation? Hippocampus 18(12):1256\u20131269","journal-title":"Hippocampus"},{"issue":"5985","key":"10555_CR20","doi-asserted-by":"publisher","first-page":"1576","DOI":"10.1126\/science.1188210","volume":"328","author":"RF Langston","year":"2010","unstructured":"Langston RF et al (2010) Development of the spatial representation system in the rat. Science 328(5985):1576\u20131580","journal-title":"Science"},{"issue":"8","key":"10555_CR21","doi-asserted-by":"publisher","first-page":"663","DOI":"10.1038\/nrn1932","volume":"7","author":"BL McNaughton","year":"2006","unstructured":"McNaughton BL, Battaglia FP, Jensen O, Moser EI, Moser MB (2006) Path integration and the neural basis of the \u2018cognitive map\u2019. Nat Rev Neurosci 7(8):663\u2013678","journal-title":"Nat Rev Neurosci"},{"issue":"11","key":"10555_CR22","doi-asserted-by":"publisher","first-page":"1448","DOI":"10.1038\/nn.4653","volume":"20","author":"EI Moser","year":"2017","unstructured":"Moser EI, Moser MB, McNaughton BL (2017) Spatial representation in the hippocampal formation: a history. Nat Neurosci 20(11):1448\u20131464","journal-title":"Nat Neurosci"},{"issue":"7","key":"10555_CR23","doi-asserted-by":"publisher","first-page":"466","DOI":"10.1038\/nrn3766","volume":"15","author":"EI Moser","year":"2014","unstructured":"Moser EI, Roudi Y, Witter MP, Kentros C, Bonhoeffer T, Moser MB (2014) Grid cells and cortical representation. Nat Rev Neurosci 15(7):466\u2013481","journal-title":"Nat Rev Neurosci"},{"key":"10555_CR24","doi-asserted-by":"publisher","DOI":"10.7554\/eLife.48494","volume":"8","author":"N Mosheiff","year":"2019","unstructured":"Mosheiff N, Burak Y (2019) Velocity coupling of grid cell modules enables stable embedding of a low dimensional variable in a high dimensional neural attractor. eLife 8:e48494","journal-title":"eLife"},{"issue":"1","key":"10555_CR25","doi-asserted-by":"publisher","first-page":"141","DOI":"10.1016\/j.neuron.2012.11.032","volume":"77","author":"H Pastoll","year":"2013","unstructured":"Pastoll H, Solanka L, van Rossum MCW, Nolan MF (2013) Feedback inhibition enables theta-nested gamma oscillations and grid firing fields. Neuron 77(1):141\u2013154","journal-title":"Neuron"},{"issue":"7767","key":"10555_CR26","doi-asserted-by":"publisher","first-page":"106","DOI":"10.1038\/s41586-019-1424-8","volume":"572","author":"J Pei","year":"2019","unstructured":"Pei J et al (2019) Towards artificial general intelligence with hybrid Tianjic chip architecture. Nature 572(7767):106\u2013111","journal-title":"Nature"},{"issue":"6173","key":"10555_CR27","doi-asserted-by":"publisher","first-page":"891","DOI":"10.1126\/science.1243028","volume":"343","author":"S Ray","year":"2014","unstructured":"Ray S, Naumann R, Burgalossi A, Tang Q, Schmidt H, Brecht M (2014) Grid-layout and theta-modulation of layer 2 pyramidal neurons in medial entorhinal cortex. Science 343(6173):891\u2013896","journal-title":"Science"},{"issue":"5774","key":"10555_CR28","doi-asserted-by":"publisher","first-page":"758","DOI":"10.1126\/science.1125572","volume":"312","author":"F Sargolini","year":"2006","unstructured":"Sargolini F et al (2006) Conjunctive representation of position, direction, and velocity in entorhinal cortex. Science 312(5774):758\u2013762","journal-title":"Science"},{"key":"10555_CR29","doi-asserted-by":"publisher","first-page":"16052","DOI":"10.52202\/068431-1168","volume":"35","author":"R Schaeffer","year":"2022","unstructured":"Schaeffer R, Khona M, Fiete I (2022) No free lunch from deep learning in neuroscience: a case study through models of the entorhinal-hippocampal circuit. Adv Neural Inf Process Syst 35:16052\u201316067","journal-title":"Adv Neural Inf Process Syst"},{"issue":"4","key":"10555_CR30","doi-asserted-by":"publisher","DOI":"10.1371\/journal.pcbi.1003558","volume":"10","author":"B Si","year":"2014","unstructured":"Si B, Romani S, Tsodyks M (2014) Continuous attractor network model for conjunctive position-by-velocity tuning of grid cells. PLoS Comput Biol 10(4):e1003558","journal-title":"PLoS Comput Biol"},{"issue":"1","key":"10555_CR31","doi-asserted-by":"publisher","first-page":"121","DOI":"10.1016\/j.neuron.2022.10.003","volume":"111","author":"B Sorscher","year":"2023","unstructured":"Sorscher B, Mel GC, Ocko SA, Giocomo LM, Ganguli S (2023) A unified theory for the computational and mechanistic origins of grid cells. Neuron 111(1):121\u2013137","journal-title":"Neuron"},{"issue":"11","key":"10555_CR32","doi-asserted-by":"publisher","first-page":"1643","DOI":"10.1038\/nn.4650","volume":"20","author":"KL Stachenfeld","year":"2017","unstructured":"Stachenfeld KL, Botvinick MM, Gershman SJ (2017) The hippocampus as a predictive map. Nat Neurosci 20(11):1643\u20131653","journal-title":"Nat Neurosci"},{"issue":"11","key":"10555_CR33","doi-asserted-by":"publisher","DOI":"10.1126\/science.1500816","volume":"1","author":"M Stemmler","year":"2015","unstructured":"Stemmler M, Mathis A, Herz AVM (2015) Connecting multiple spatial scales to decode the population activity of grid cells. Sci Adv 1(11):e1500816","journal-title":"Sci Adv"},{"issue":"7427","key":"10555_CR34","doi-asserted-by":"publisher","first-page":"72","DOI":"10.1038\/nature11649","volume":"492","author":"H Stensola","year":"2012","unstructured":"Stensola H, Stensola T, Solstad T, Fr\u00f8land K, Moser MB, Moser EI (2012) The entorhinal grid map is discretized. Nature 492(7427):72\u201378","journal-title":"Nature"},{"issue":"2","key":"10555_CR35","doi-asserted-by":"publisher","first-page":"420","DOI":"10.1523\/JNEUROSCI.10-02-00420.1990","volume":"10","author":"JS Taube","year":"1990","unstructured":"Taube JS, Muller RU, Ranck JB Jr (1990a) Head-direction cells recorded from the postsubiculum in freely moving rats. I. Description and quantitative analysis. J Neurosci 10(2):420\u2013435","journal-title":"J Neurosci"},{"issue":"2","key":"10555_CR36","doi-asserted-by":"publisher","first-page":"436","DOI":"10.1523\/JNEUROSCI.10-02-00436.1990","volume":"10","author":"JS Taube","year":"1990","unstructured":"Taube JS, Muller RU, Ranck JB Jr (1990b) Head-direction cells recorded from the postsubiculum in freely moving rats. II. Effects of environmental manipulations. J Neurosci 10(2):436\u2013447","journal-title":"J Neurosci"},{"issue":"11","key":"10555_CR37","doi-asserted-by":"publisher","first-page":"1204","DOI":"10.1002\/hipo.22765","volume":"27","author":"E Urdapilleta","year":"2017","unstructured":"Urdapilleta E, Si B, Treves A (2017) Selforganization of modular activity of grid cells. Hippocampus 27(11):1204\u20131213","journal-title":"Hippocampus"},{"key":"10555_CR38","doi-asserted-by":"crossref","unstructured":"Wang T, Yang F, Wang Z, Zhang B, Wang W, Liu F (2021) Modularization of grid cells constrained by the pyramidal patch lattice. iScience 24(4):102301","DOI":"10.1016\/j.isci.2021.102301"},{"issue":"4","key":"10555_CR39","doi-asserted-by":"publisher","DOI":"10.1016\/j.isci.2023.106351","volume":"26","author":"Z Wang","year":"2023","unstructured":"Wang Z, Wang T, Yang F, Liu F, Wang W (2023) Intrinsic theta oscillation in the attractor network of grid cells. iScience 26(4):106351","journal-title":"iScience"},{"issue":"5985","key":"10555_CR40","doi-asserted-by":"publisher","first-page":"1573","DOI":"10.1126\/science.1188224","volume":"328","author":"TJ Wills","year":"2010","unstructured":"Wills TJ, Cacucci F, Burgess N, O\u2019Keefe J (2010) Development of the hippocampal cognitive map in preweanling rats. Science 328(5985):1573\u20131576","journal-title":"Science"},{"issue":"15","key":"10555_CR41","doi-asserted-by":"publisher","first-page":"2503","DOI":"10.1016\/j.neuron.2022.05.010","volume":"110","author":"C Xue","year":"2022","unstructured":"Xue C, Kramer LE, Cohen MR (2022) Dynamic task-belief is an integral part of decision-making. Neuron 110(15):2503\u20132511","journal-title":"Neuron"},{"issue":"2","key":"10555_CR42","doi-asserted-by":"publisher","first-page":"297","DOI":"10.1038\/s41593-018-0310-2","volume":"22","author":"GR Yang","year":"2019","unstructured":"Yang GR, Joglekar MR, Song HF, Newsome WT, Wang XJ (2019) Task representations in neural networks trained to perform many cognitive tasks. Nat Neurosci 22(2):297\u2013306","journal-title":"Nat Neurosci"},{"issue":"1","key":"10555_CR43","doi-asserted-by":"publisher","first-page":"3701","DOI":"10.1038\/s41467-018-06104-5","volume":"9","author":"I Zutshi","year":"2018","unstructured":"Zutshi I, Fu ML, Lilascharoen V, Leutgeb JK, Lim BK, Leutgeb S (2018) Recurrent circuits within medial entorhinal cortex superficial layers support grid cell firing. Nat Commun 9:3701","journal-title":"Nat Commun"}],"container-title":["Cognitive Neurodynamics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11571-026-10555-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11571-026-10555-8","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11571-026-10555-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,10,7]],"date-time":"2026-10-07T08:31:27Z","timestamp":1791361887000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11571-026-10555-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,9,21]]},"references-count":43,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,12]]}},"alternative-id":["10555"],"URL":"https:\/\/doi.org\/10.1007\/s11571-026-10555-8","relation":{},"ISSN":["1871-4080","1871-4099"],"issn-type":[{"value":"1871-4080","type":"print"},{"value":"1871-4099","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,9,21]]},"assertion":[{"value":"13 May 2026","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 September 2026","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 September 2026","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 September 2026","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 September 2026","order":6,"name":"change_date","label":"Change Date","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"Update","order":7,"name":"change_type","label":"Change Type","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The high quality figures have been updated.","order":8,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare no competing interests.","order":1,"name":"Ethics","label":"Competing interests","group":{"name":"EthicsHeading","label":"Declarations"}}],"article-number":"179"}}