{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,29]],"date-time":"2026-07-29T12:21:37Z","timestamp":1785327697861,"version":"3.55.0"},"reference-count":55,"publisher":"MIT Press - Journals","issue":"8","content-domain":{"domain":["direct.mit.edu"],"crossmark-restriction":true},"short-container-title":[],"published-print":{"date-parts":[[2021,7,26]]},"abstract":"<jats:p>While neural networks are highly effective at learning task-relevant representations from data, they typically do not learn representations with the kind of symbolic structure that is hypothesized to support high-level cognitive processes, nor do they naturally model such structures within problem domains that are continuous in space and time. To fill these gaps, this work exploits a method for defining vector representations that bind discrete (symbol-like) entities to points in continuous topological spaces in order to simulate and predict the behavior of a range of dynamical systems. These vector representations are spatial semantic pointers (SSPs), and we demonstrate that they can (1) be used to model dynamical systems involving multiple objects represented in a symbol-like manner and (2) be integrated with deep neural networks to predict the future of physical trajectories. These results help unify what have traditionally appeared to be disparate approaches in machine learning.<\/jats:p>","DOI":"10.1162\/neco_a_01410","type":"journal-article","created":{"date-parts":[[2021,6,30]],"date-time":"2021-06-30T21:34:05Z","timestamp":1625088845000},"page":"2033-2067","update-policy":"https:\/\/doi.org\/10.1162\/mitpressjournals.corrections.policy","source":"Crossref","is-referenced-by-count":17,"title":["Simulating and Predicting Dynamical Systems With Spatial Semantic Pointers"],"prefix":"10.1162","volume":"33","author":[{"given":"Aaron R.","family":"Voelker","sequence":"first","affiliation":[{"name":"Applied Brain Research, Waterloo, ON N2L 3G1, Canada arvoelke@uwaterloo.ca"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Peter","family":"Blouw","sequence":"additional","affiliation":[{"name":"Applied Brain Research, Waterloo, ON N2L 3G1, Canada peter.blouw@appliedbrainresearch.com"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xuan","family":"Choo","sequence":"additional","affiliation":[{"name":"Applied Brain Research, Waterloo, ON N2L 3G1, Canada xuan.choo@appliedbrainresearch.com"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nicole Sandra-Yaffa","family":"Dumont","sequence":"additional","affiliation":[{"name":"Cheriton School of Computer Science, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada ns2dumont@uwaterloo.ca"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Terrence C.","family":"Stewart","sequence":"additional","affiliation":[{"name":"National Research Council of Canada, University of Waterloo Collaboration Centre, Waterloo, ON N2L 3G1 Canada terrence.stewart@nrc-cnrc.gc.ca"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Chris","family":"Eliasmith","sequence":"additional","affiliation":[{"name":"Centre for Theoretical Neuroscience, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada celiasmith@uwaterloo.ca"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"281","published-online":{"date-parts":[[2021,7,26]]},"reference":[{"key":"2021072618250451200_B1","first-page":"265","article-title":"TensorFlow: A system for large-scale machine learning","author":"Abadi","year":"2016","journal-title":"Proceedings of the 12th USENIX Symposium on Operating Systems Design and Implementation"},{"key":"2021072618250451200_B2","article-title":"Nengo spa","author":"Applied Brain Research","year":"2020"},{"key":"2021072618250451200_B3","doi-asserted-by":"crossref","DOI":"10.3389\/fninf.2013.00048","article-title":"Nengo: A Python tool for building large-scale functional brain models","volume":"7","author":"Bekolay","year":"2014","journal-title":"Frontiers in Neuroinformatics"},{"issue":"5","key":"2021072618250451200_B4","doi-asserted-by":"crossref","first-page":"1128","DOI":"10.1111\/cogs.12265","article-title":"Concepts as semantic pointers: A framework and computational model","volume":"40","author":"Blouw","year":"2016","journal-title":"Cognitive Science"},{"issue":"2","key":"2021072618250451200_B5","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1007\/s10994-013-5335-x","article-title":"From machine learning to machine reasoning: An essay","volume":"94","author":"Bottou","year":"2014","journal-title":"Machine Learning"},{"key":"2021072618250451200_B6","author":"Choo","year":"2010","journal-title":"The ordinal serial encoding model: Serial memory in spiking neurons"},{"key":"2021072618250451200_B7","author":"Choo","year":"2018","journal-title":"Spaun 2.0: Extending the World's Largest Functional Brain Model"},{"key":"2021072618250451200_B8","author":"Co-Reyes","year":"2018","journal-title":"Self-consistent trajectory autoencoder: Hierarchical reinforcement learning with trajectory embeddings"},{"issue":"2","key":"2021072618250451200_B9","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1007\/s10827-005-6558-z","article-title":"A controlled attractor network model of path integration in the rat","volume":"18","author":"Conklin","year":"2005","journal-title":"Journal of Computational Neuroscience"},{"key":"2021072618250451200_B10","doi-asserted-by":"crossref","first-page":"782","DOI":"10.1111\/cogs.12261","article-title":"Biologically plausible, human- scale knowledge representation","volume":"40","author":"Crawford","year":"2015","journal-title":"Cognitive Science"},{"key":"2021072618250451200_B11","article-title":"Accurate representation for spatial cognition using grid cells","author":"Dumont","year":"2020","journal-title":"Proceedings of the 42nd Annual Meeting of the Cognitive Science Society"},{"issue":"6","key":"2021072618250451200_B12","doi-asserted-by":"crossref","first-page":"1276","DOI":"10.1162\/0899766053630332","article-title":"A unified approach to building and controlling spiking attractor networks","volume":"17","author":"Eliasmith","year":"2005","journal-title":"Neural Computation"},{"key":"2021072618250451200_B13","doi-asserted-by":"crossref","DOI":"10.1093\/acprof:oso\/9780199794546.001.0001","author":"Eliasmith","year":"2013","journal-title":"How to build a brain: A neural architecture for biological cognition"},{"key":"2021072618250451200_B14","author":"Eliasmith","year":"2003","journal-title":"Neural engineering: Computation, representation, and dynamics in neurobiological systems"},{"key":"2021072618250451200_B15","doi-asserted-by":"crossref","first-page":"1202","DOI":"10.1126\/science.1225266","article-title":"A large-scale model of the functioning brain","volume":"338","author":"Eliasmith","year":"2012","journal-title":"Science"},{"issue":"1\u20132","key":"2021072618250451200_B16","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1016\/0010-0277(88)90031-5","article-title":"Connectionism and cognitive architecture: A critical analysis","volume":"28","author":"Fodor","year":"1988","journal-title":"Cognition"},{"key":"2021072618250451200_B17","article-title":"Sparse coding in the primate cortex.","author":"Foldiak","year":"2003","journal-title":"The handbook of brain theory and neural networks"},{"key":"2021072618250451200_B18","author":"Frady","year":"2020","journal-title":"Variable binding for sparse distributed representations: Theory and applications"},{"key":"2021072618250451200_B19","article-title":"A framework for linking computations and rhythm-based timing patterns in neural firing, such as phase precession in hippocampal place cells","author":"Frady","year":"2018","journal-title":"Proceedings of the Conference on Cognitive Computational Neuroscience"},{"key":"2021072618250451200_B20","author":"Gayler","year":"2004","journal-title":"Vector symbolic architectures answer Jackendoff's challenges for cognitive neuroscience"},{"key":"2021072618250451200_B21","author":"Goodfellow","year":"2016","journal-title":"Deep learning"},{"key":"2021072618250451200_B22","author":"Gosmann","year":"2018","journal-title":"An integrated model of context, short-term, and long-term memory"},{"issue":"5","key":"2021072618250451200_B23","doi-asserted-by":"crossref","first-page":"849","DOI":"10.1162\/neco_a_01179","article-title":"Vector-derived transformation binding: An improved binding operation for deep symbol-like processing in neural networks","volume":"31","author":"Gosmann","year":"2019","journal-title":"Neural Computation"},{"key":"2021072618250451200_B24","doi-asserted-by":"crossref","first-page":"104","DOI":"10.1037\/rev0000250","article-title":"CUE: A unified spiking neuron model of short-term and long-term memory","volume":"128","author":"Gosmann","year":"2020","journal-title":"Psych. Review"},{"key":"2021072618250451200_B25","article-title":"A spiking independent accumulator model for winner-take-all computation","author":"Gosmann","year":"2017","journal-title":"Proceedings of the 39th Annual Conference of the Cognitive Science Society"},{"key":"2021072618250451200_B26","doi-asserted-by":"crossref","first-page":"510","DOI":"10.1162\/neco.2008.07-07-572","article-title":"The problems of rapid variable creation","volume":"21","author":"Hadley","year":"2009","journal-title":"Neural Computation"},{"key":"2021072618250451200_B27","first-page":"47","article-title":"Memoire sur l'it\u00e9ration des fonctions rationnelles","volume":"8","author":"Julia","year":"1918","journal-title":"J. Math. Pures Appl."},{"issue":"2","key":"2021072618250451200_B28","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1007\/s12559-009-9009-8","article-title":"Hyperdimensional computing: An introduction to computing in distributed representation with high-dimensional random vectors","volume":"1","author":"Kanerva","year":"2009","journal-title":"Cognitive Computation"},{"key":"2021072618250451200_B29","author":"Komer","year":"2020","journal-title":"Biologically inspired spatial representation"},{"key":"2021072618250451200_B30","article-title":"Efficient navigation using a scalable, biologically inspired spatial representation","author":"Komer","year":"2020","journal-title":"Proceedings of the 42nd Annual Meeting of the Cognitive Science Society"},{"key":"2021072618250451200_B31","article-title":"A neural representation of continuous space using fractional binding","author":"Komer","year":"2019","journal-title":"Proceedings of the 41st Annual Meeting of the Cognitive Science Society"},{"issue":"7553","key":"2021072618250451200_B32","doi-asserted-by":"crossref","first-page":"436","DOI":"10.1038\/nature14539","article-title":"Deep learning","volume":"521","author":"LeCun","year":"2015","journal-title":"Nature"},{"key":"2021072618250451200_B33","article-title":"Representing spatial relations with fractional binding","author":"Lu","year":"2019","journal-title":"Proceedings of the 41st Annual Meeting of the Cognitive Science Society"},{"key":"2021072618250451200_B34","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1006\/cogp.1998.0694","article-title":"Rethinking eliminative connectionism","volume":"37","author":"Marcus","year":"1998","journal-title":"Cognitive Psychology"},{"key":"2021072618250451200_B35","author":"Marcus","year":"2019","journal-title":"The algebraic mind: Integrating connectionism and cognitive science"},{"issue":"8","key":"2021072618250451200_B36","doi-asserted-by":"crossref","first-page":"348","DOI":"10.1016\/j.tics.2010.06.002","article-title":"Letting structure emerge: Connectionist and dynamical systems approaches to cognitive modelling","volume":"14","author":"McClelland","year":"2010","journal-title":"Trends in Cognitive Sciences"},{"key":"2021072618250451200_B37","first-page":"1","article-title":"Analyzing the capacity of distributed vector representations to encode spatial information","author":"Mirus","year":"2020","journal-title":"Proceedings of the 2020 International Joint Conference on Neural Networks"},{"key":"2021072618250451200_B38","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1146\/annurev.neuro.31.061307.090723","article-title":"Place cells, grid cells, and the brain's spatial representation system","volume":"31","author":"Moser","year":"2008","journal-title":"Annu. Rev. Neurosci."},{"issue":"1\u20132","key":"2021072618250451200_B39","first-page":"19","article-title":"A note on a method for generating points uniformly on n-dimensional spheres","volume":"2","author":"Muller","year":"1959","journal-title":"Comm. Assoc. Comput. Mach."},{"key":"2021072618250451200_B40","author":"Plate","year":"2003","journal-title":"Holographic Reduced Representation: Distributed Representation for Cognitive Structures"},{"issue":"4","key":"2021072618250451200_B41","doi-asserted-by":"crossref","first-page":"611","DOI":"10.1007\/s12021-019-09424-z","article-title":"NengoDL: Combining deep learning and neuromorphic modelling methods","volume":"17","author":"Rasmussen","year":"2019","journal-title":"Neuroinformatics"},{"issue":"1","key":"2021072618250451200_B42","doi-asserted-by":"crossref","first-page":"140","DOI":"10.1111\/j.1756-8765.2010.01127.x","article-title":"A neural model of rule generation in inductive reasoning","volume":"3","author":"Rasmussen","year":"2011","journal-title":"Topics in Cognitive Science"},{"key":"2021072618250451200_B43","author":"Schlegel","year":"2020","journal-title":"A comparison of vector symbolic architectures"},{"key":"2021072618250451200_B44","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1016\/j.neunet.2014.09.003","article-title":"Deep learning in neural networks: An overview","volume":"61","author":"Schmidhuber","year":"2015","journal-title":"Neural Networks"},{"key":"2021072618250451200_B45","first-page":"1084","article-title":"Embodied cognition, neural field models of.","author":"Sch\u00f6ner","year":"2014","journal-title":"Encyclopedia of computational neuroscience"},{"issue":"1\u20132","key":"2021072618250451200_B46","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1016\/0004-3702(90)90007-M","article-title":"Tensor product variable binding and the representation of symbolic structures in connectionist systems","volume":"46","author":"Smolensky","year":"1990","journal-title":"Artificial Intelligence"},{"key":"2021072618250451200_B47","author":"Smolensky","year":"2006","journal-title":"The harmonic mind: From neural computation to optimality-theoretic grammar"},{"key":"2021072618250451200_B48","first-page":"10003","volume-title":"Advances in neural information processing systems","author":"Sorscher","year":"2019"},{"key":"2021072618250451200_B49","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1080\/09540091.2011.571761","article-title":"Neural representations of compositional structures: Representing and manipulating vector spaces with spiking neurons","volume":"23","author":"Stewart","year":"2011","journal-title":"Connection Science"},{"key":"2021072618250451200_B50","first-page":"1533","article-title":"Sentence processing in spiking neurons: A biologically plausible left-corner parser","author":"Stewart","year":"2014","journal-title":"Proceedings of the 36th Annual Conference of the Cognitive Science Society"},{"issue":"2","key":"2021072618250451200_B51","doi-asserted-by":"crossref","first-page":"84","DOI":"10.1016\/j.cogsys.2010.06.006","article-title":"A biologically realistic cleanup memory: Autoassociation in spiking neurons","volume":"12","author":"Stewart","year":"2011","journal-title":"Cognitive Systems Research"},{"key":"2021072618250451200_B52","author":"Voelker","year":"2019","journal-title":"Dynamical systems in spiking neuromorphic hardware"},{"key":"2021072618250451200_B53","author":"Voelker","year":"2020","journal-title":"A short letter on the dot product between rotated Fourier transforms"},{"key":"2021072618250451200_B54","first-page":"15544","volume-title":"Advances in neural information processing systems","author":"Voelker","year":"2019"},{"key":"2021072618250451200_B55","article-title":"A neural architecture for representing and reasoning about spatial relationships","author":"Weiss","year":"2016","journal-title":"Proceedings of the International Conference on Learning Representations\u2014Workshop Track"}],"container-title":["Neural Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/direct.mit.edu\/neco\/article-pdf\/33\/8\/2033\/1930867\/neco_a_01410.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"http:\/\/direct.mit.edu\/neco\/article-pdf\/33\/8\/2033\/1930867\/neco_a_01410.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,7,26]],"date-time":"2021-07-26T18:26:11Z","timestamp":1627323971000},"score":1,"resource":{"primary":{"URL":"https:\/\/direct.mit.edu\/neco\/article\/33\/8\/2033\/102625\/Simulating-and-Predicting-Dynamical-Systems-With"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,26]]},"references-count":55,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2021,7,26]]},"published-print":{"date-parts":[[2021,7,26]]}},"URL":"https:\/\/doi.org\/10.1162\/neco_a_01410","relation":{},"ISSN":["0899-7667","1530-888X"],"issn-type":[{"value":"0899-7667","type":"print"},{"value":"1530-888X","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2021,8]]},"published":{"date-parts":[[2021,7,26]]}}}