{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,6]],"date-time":"2026-02-06T23:41:33Z","timestamp":1770421293238,"version":"3.49.0"},"reference-count":61,"publisher":"Public Library of Science (PLoS)","issue":"1","license":[{"start":{"date-parts":[[2015,1,24]],"date-time":"2015-01-24T00:00:00Z","timestamp":1422057600000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["www.ploscompbiol.org"],"crossmark-restriction":false},"short-container-title":["PLoS Comput Biol"],"DOI":"10.1371\/journal.pcbi.1004039","type":"journal-article","created":{"date-parts":[[2015,1,24]],"date-time":"2015-01-24T19:13:34Z","timestamp":1422126814000},"page":"e1004039","update-policy":"https:\/\/doi.org\/10.1371\/journal.pcbi.corrections_policy","source":"Crossref","is-referenced-by-count":16,"title":["Computation in Dynamically Bounded Asymmetric Systems"],"prefix":"10.1371","volume":"11","author":[{"given":"Ueli","family":"Rutishauser","sequence":"first","affiliation":[]},{"given":"Jean-Jacques","family":"Slotine","sequence":"additional","affiliation":[]},{"given":"Rodney","family":"Douglas","sequence":"additional","affiliation":[]}],"member":"340","published-online":{"date-parts":[[2015,1,24]]},"reference":[{"key":"ref1","first-page":"345","article-title":"On computable numbers, with an application to the entscheidungsproblem","volume":"58","author":"AM Turing","year":"1936","journal-title":"J of Math"},{"key":"ref2","article-title":"Computation: finite and infinite machines","author":"L Minksy","year":"1967"},{"key":"ref3","article-title":"On undecidable propositions of formal mathematical systems","author":"K G\u00f6del","year":"1934"},{"key":"ref4","doi-asserted-by":"crossref","DOI":"10.1145\/1940721.1944826","article-title":"Ubiquity symposium: Biological computation","author":"M Mitchell","year":"2011"},{"key":"ref5","doi-asserted-by":"crossref","DOI":"10.7551\/mitpress\/9108.001.0001","article-title":"Principles of brain dynamics: global state interactions","author":"MI Rabinovich","year":"2012"},{"key":"ref6","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4899-0718-9_31","article-title":"The architecture of complexity","author":"HA Simon","year":"1991"},{"key":"ref7","article-title":"Feynman lectures on computation","author":"RP Feynman","year":"1998"},{"key":"ref8","doi-asserted-by":"crossref","first-page":"815","DOI":"10.1109\/TSMC.1983.6313075","article-title":"Absolute stability of global pattern-formation and parallel memory storage by competitive neural networks","volume":"13","author":"MA Cohen","year":"1983","journal-title":"IEEE Transactions on Systems Man and Cybernetics"},{"key":"ref9","doi-asserted-by":"crossref","first-page":"3088","DOI":"10.1073\/pnas.81.10.3088","article-title":"Neurons with graded response have collective computational properties like those of two-state neurons","volume":"81","author":"JJ Hopfield","year":"1984","journal-title":"Proceedings of the National Academy of Sciences"},{"key":"ref10","doi-asserted-by":"crossref","first-page":"2554","DOI":"10.1073\/pnas.79.8.2554","article-title":"Neural networks and physical systems with emergent collective computational abilities","volume":"79","author":"JJ Hopfield","year":"1982","journal-title":"Proceedings of the National Academy of Sciences"},{"key":"ref11","doi-asserted-by":"crossref","DOI":"10.1063\/1.2810360","article-title":"Introduction to the Theory of Neural Computation","author":"J Hertz","year":"1991"},{"key":"ref12","first-page":"329","article-title":"Minimax and hamiltonian dynamics of excitatory-inhibitory networks","author":"HS Seung","year":"1998"},{"key":"ref13","first-page":"667","article-title":"Attractor network models","volume":"volume 1","author":"X Wang","year":"2008"},{"key":"ref14","doi-asserted-by":"crossref","first-page":"3844","DOI":"10.1073\/pnas.92.9.3844","article-title":"Theory of orientation tuning in visual cortex","volume":"92","author":"R Ben-Yishai","year":"1995","journal-title":"Proceedings of the National Academy of Sciences"},{"key":"ref15","article-title":"Theoretical Neuroscience","author":"P Dayan","year":"2001"},{"key":"ref16","doi-asserted-by":"crossref","first-page":"947","DOI":"10.1038\/35016072","article-title":"Digital selection and analogue amplification coexist in a cortex-inspired silicon circuit","volume":"405","author":"R Hahnloser","year":"2000","journal-title":"Nature"},{"key":"ref17","doi-asserted-by":"crossref","DOI":"10.1016\/S0005-1098(98)00019-3","article-title":"On contraction analysis for nonlinear systems","volume":"34","author":"W Lohmiller","year":"1998","journal-title":"Automatica"},{"key":"ref18","first-page":"115","article-title":"A logical calculus of the ideas immanent in nervous activity","volume":"5","author":"W McCulloch","year":"1943","journal-title":"Bulletin of Mathematical Biology"},{"key":"ref19","article-title":"Introduction to VLSI systems","volume":"volume 1080","author":"C Mead","year":"1980"},{"key":"ref20","doi-asserted-by":"crossref","first-page":"2554","DOI":"10.1073\/pnas.79.8.2554","article-title":"Neural networks and physical systems with emergent collective computational abilities","volume":"79","author":"J Hopfield","year":"1982","journal-title":"Proceedings of the National Academy of Sciences of the United States of America"},{"key":"ref21","article-title":"The Computer and the Brain","author":"J von Neumann","year":"1958"},{"key":"ref22","first-page":"643","article-title":"Characteristics of random nets of analog neuron-like elements","author":"SI Amari","year":"1972","journal-title":"IEEE Transactions on Systems, Man and Cybernetics SMC-2"},{"key":"ref23","doi-asserted-by":"crossref","DOI":"10.1162\/neco.2008.03-08-734","article-title":"State-dependent computation using coupled recurrent networks","volume":"21","author":"U Rutishauser","year":"2009","journal-title":"Neural Computation"},{"key":"ref24","doi-asserted-by":"crossref","DOI":"10.1162\/NECO_a_00091","article-title":"Collective stability of networks of winner-take-all circuits","volume":"23","author":"U Rutishauser","year":"2011","journal-title":"Neural computation"},{"key":"ref25","doi-asserted-by":"crossref","first-page":"621","DOI":"10.1162\/089976603321192103","article-title":"Permitted and forbidden sets in symmetric threshold-linear networks","volume":"15","author":"R Hahnloser","year":"2003","journal-title":"Neural Comput"},{"key":"ref26","doi-asserted-by":"crossref","first-page":"3088","DOI":"10.1073\/pnas.81.10.3088","article-title":"Neurons with graded response have collective computational properties like those of 2-state neurons","volume":"81","author":"JJ Hopfield","year":"1984","journal-title":"Proceedings of the National Academy of Sciences of the United States of America-Biological Sciences"},{"key":"ref27","doi-asserted-by":"crossref","first-page":"L25","DOI":"10.1088\/0305-4470\/37\/3\/L01","article-title":"Potential in stochastic differential equations: novel construction","volume":"37","author":"P Ao","year":"2004","journal-title":"Journal of physics A: mathematical and general"},{"key":"ref28","doi-asserted-by":"crossref","first-page":"14272","DOI":"10.1073\/pnas.0605792103","article-title":"Helmholtz decomposition coupling rotational to irrotational ow of a viscous uid","volume":"103","author":"DD Joseph","year":"2006","journal-title":"Proceedings of the National Academy of Sciences"},{"key":"ref29","doi-asserted-by":"crossref","first-page":"E3468","DOI":"10.1073\/pnas.1212083110","article-title":"Synthesizing cognition in neuromorphic electronic systems","volume":"110","author":"E Neftci","year":"2013","journal-title":"Proceedings of the National Academy of Sciences"},{"key":"ref30","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1016\/S0893-6080(00)00089-7","article-title":"Modularity, evolution, and the binding problem: a view from stability theory","volume":"14","author":"J Slotine","year":"2001","journal-title":"Neural Networks"},{"key":"ref31","article-title":"A discipline of programming","volume":"volume 4","author":"EW Dijkstra","year":"1976"},{"key":"ref32","doi-asserted-by":"crossref","first-page":"2519","DOI":"10.1162\/089976600300014827","article-title":"On the computational power of winner-take-all","volume":"12","author":"W Maass","year":"2000","journal-title":"Neural Computation"},{"key":"ref33","doi-asserted-by":"crossref","first-page":"746","DOI":"10.1038\/11219","article-title":"Feedback interactions between neuronal pointers and maps for attentional processing","volume":"2","author":"R Hahnloser","year":"1999","journal-title":"Nat Neurosci"},{"key":"ref34","doi-asserted-by":"crossref","first-page":"R496","DOI":"10.1016\/j.cub.2007.04.024","article-title":"Recurrent neuronal circuits in the neocortex","volume":"17","author":"R Douglas","year":"2007","journal-title":"Curr Biol"},{"key":"ref35","first-page":"1228","article-title":"Winner-take-all networks","author":"A Yuille","year":"2003"},{"key":"ref36","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1016\/0893-6080(92)90004-3","article-title":"Complex dynamics in winner-take-all neural nets with slow inhibition","volume":"5","author":"B Ermentrout","year":"1992","journal-title":"Neural Networks"},{"key":"ref37","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1162\/089976601300014574","article-title":"A competitive-layer model for feature binding and sensory segmentation","volume":"13","author":"H Wersing","year":"2001","journal-title":"Neural Computation"},{"key":"ref38","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1017\/S0033583500003024","article-title":"Decoding neuronal firing and modelling neural networks","volume":"27","author":"L Abbott","year":"1994","journal-title":"Q Rev Biophys"},{"key":"ref39","doi-asserted-by":"crossref","first-page":"2033","DOI":"10.1162\/NECO_a_00304","article-title":"Competition through selective inhibitory synchrony","volume":"24","author":"U Rutishauser","year":"2012","journal-title":"Neural computation"},{"key":"ref40","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-0701-6","article-title":"Complexity and real computation","author":"L Blum","year":"1998"},{"key":"ref41","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1090\/S0273-0979-1989-15750-9","article-title":"On a theory of computation and complexity over the real numbers: <italic>np<\/italic>-completeness, recursive functions and universal machines","volume":"21","author":"L Blum","year":"1989","journal-title":"Bulletin (New Series) of the American Mathematical Society"},{"key":"ref42","doi-asserted-by":"crossref","first-page":"214","DOI":"10.1016\/S0167-2789(98)00057-8","article-title":"Analog computation with dynamical systems","volume":"120","author":"HT Siegelmann","year":"1998","journal-title":"Physica D: Nonlinear Phenomena"},{"key":"ref43","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1006\/jcss.1995.1013","article-title":"On the computational power of neural nets","volume":"50","author":"HT Siegelmann","year":"1995","journal-title":"Journal of computer and system sciences"},{"key":"ref44","doi-asserted-by":"crossref","first-page":"e1000092","DOI":"10.1371\/journal.pcbi.1000092","article-title":"The dynamic brain: from spiking neurons to neural masses and cortical fields","volume":"4","author":"G Deco","year":"2008","journal-title":"PLoS computational biology"},{"key":"ref45","doi-asserted-by":"crossref","first-page":"255","DOI":"10.3389\/neuro.01.031.2008","article-title":"Simple substrates for complex cognition","volume":"2","author":"P Dayan","year":"2008","journal-title":"Frontiers in neuroscience"},{"key":"ref46","doi-asserted-by":"crossref","first-page":"1432","DOI":"10.1038\/nn1790","article-title":"Bayesian inference with probabilistic population codes","volume":"9","author":"WJ Ma","year":"2006","journal-title":"Nature neuroscience"},{"key":"ref47","first-page":"274","article-title":"Computational differences between asymmetrical and symmetrical networks","author":"Z Li","year":"1999"},{"key":"ref48","doi-asserted-by":"crossref","first-page":"621","DOI":"10.1162\/089976603321192103","article-title":"Permitted and forbidden sets in symmetric thresholdlinear networks","volume":"15","author":"RH Hahnloser","year":"2003","journal-title":"Neural Computation"},{"key":"ref49","doi-asserted-by":"crossref","DOI":"10.1002\/0471221546","article-title":"Kalman filtering and neural networks","author":"SS Haykin","year":"2001"},{"key":"ref50","first-page":"95","article-title":"New results in linear filtering and prediction theory","volume":"83","author":"RE Kalman","year":"1961","journal-title":"Journal of Fluids Engineering"},{"key":"ref51","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0004-3702(80)90032-6","article-title":"Constraintsa language for expressing almost-hierarchical descriptions","volume":"14","author":"GJ Sussman","year":"1980","journal-title":"Artificial intelligence"},{"key":"ref52","first-page":"1","article-title":"The art of the propagator","author":"A Radul","year":"2009"},{"key":"ref53","first-page":"1979","article-title":"Nonclassical computation: a dynamical systems perspective","author":"S Stepney","year":"2012"},{"key":"ref54","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1146\/annurev.neuro.21.1.25","article-title":"From biophysics to models of network function","volume":"21","author":"E Marder","year":"1998","journal-title":"Annual Review of Neuroscience"},{"key":"ref55","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1146\/annurev.ne.12.030189.001153","article-title":"Emerging principles governing the operation of neural networks","volume":"12","author":"PA Getting","year":"1989","journal-title":"Annual Review of Neuroscience"},{"key":"ref56","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1126\/science.284.5411.96","article-title":"Complexity and the Nervous System","volume":"284","author":"C Koch","year":"1999","journal-title":"Science"},{"key":"ref57","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1016\/j.neunet.2007.04.001","article-title":"Special issue on echo state networks and liquid state machines","volume":"20","author":"H Jaeger","year":"2007","journal-title":"Neural Networks"},{"key":"ref58","doi-asserted-by":"crossref","first-page":"208102","DOI":"10.1103\/PhysRevLett.109.208102","article-title":"Computing with competition in biochemical networks","volume":"109","author":"AJ Genot","year":"2012","journal-title":"Physical review letters"},{"key":"ref59","doi-asserted-by":"crossref","first-page":"588","DOI":"10.1002\/aic.690460317","article-title":"Nonlinear process control using contraction theory","volume":"46","author":"W Lohmiller","year":"2000","journal-title":"AIChE Journal"},{"key":"ref60","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1002\/acs.754","article-title":"Modular stability tools for distributed computation and control","volume":"17","author":"J Slotine","year":"2003","journal-title":"International Journal of Adaptive Control and Signal Processing"},{"key":"ref61","article-title":"On partial contraction analysis for coupled nonlinear oscillators","volume":"91","author":"W Wang","year":"2005","journal-title":"Biological Cybernetics"}],"container-title":["PLOS Computational Biology"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/dx.plos.org\/10.1371\/journal.pcbi.1004039","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,4,27]],"date-time":"2022-04-27T15:52:10Z","timestamp":1651074730000},"score":1,"resource":{"primary":{"URL":"https:\/\/dx.plos.org\/10.1371\/journal.pcbi.1004039"}},"subtitle":[],"editor":[{"given":"Olaf","family":"Sporns","sequence":"first","affiliation":[]}],"short-title":[],"issued":{"date-parts":[[2015,1,24]]},"references-count":61,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2015,1,24]]}},"URL":"https:\/\/doi.org\/10.1371\/journal.pcbi.1004039","relation":{},"ISSN":["1553-7358"],"issn-type":[{"value":"1553-7358","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,1,24]]}}}