{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,27]],"date-time":"2026-10-27T18:35:56Z","timestamp":1793126156038,"version":"4.5.0"},"update-to":[{"DOI":"10.1371\/journal.pcbi.1014755","type":"new_version","label":"New version","source":"publisher","updated":{"date-parts":[[2026,10,27]],"date-time":"2026-10-27T00:00:00Z","timestamp":1793059200000}}],"reference-count":59,"publisher":"Public Library of Science (PLoS)","issue":"10","license":[{"start":{"date-parts":[[2026,10,15]],"date-time":"2026-10-15T00:00:00Z","timestamp":1792022400000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000026","name":"National Institute on Drug Abuse","doi-asserted-by":"publisher","award":["NIH BRAIN\/NIDA RF1 DA055667"],"award-info":[{"award-number":["NIH BRAIN\/NIDA RF1 DA055667"]}],"id":[{"id":"10.13039\/100000026","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000065","name":"National Institute of Neurological Disorders and Stroke","doi-asserted-by":"publisher","award":["NIH NINDS\/OD DP2 NS127291"],"award-info":[{"award-number":["NIH NINDS\/OD DP2 NS127291"]}],"id":[{"id":"10.13039\/100000065","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000065","name":"National Institute of Neurological Disorders and Stroke","doi-asserted-by":"publisher","award":["NIH NINDS 5F32MH132175 (CV)"],"award-info":[{"award-number":["NIH NINDS 5F32MH132175 (CV)"]}],"id":[{"id":"10.13039\/100000065","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000893","name":"Simons Foundation","doi-asserted-by":"publisher","award":["543049"],"award-info":[{"award-number":["543049"]}],"id":[{"id":"10.13039\/100000893","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000893","name":"Simons Foundation","doi-asserted-by":"publisher","award":["697092"],"award-info":[{"award-number":["697092"]}],"id":[{"id":"10.13039\/100000893","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000893","name":"Simons Foundation","doi-asserted-by":"publisher","award":["Transition to Independence Award 1155867"],"award-info":[{"award-number":["Transition to Independence Award 1155867"]}],"id":[{"id":"10.13039\/100000893","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100023581","name":"National Science Foundation Graduate Research Fellowship Program","doi-asserted-by":"publisher","award":["NSF GRFP 2141064"],"award-info":[{"award-number":["NSF GRFP 2141064"]}],"id":[{"id":"10.13039\/100023581","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100014373","name":"Wu Tsai Neurosciences Institute, Stanford University","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100014373","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003151","name":"Fonds de recherche du Qu\u00e9bec \u2013 Nature et technologies","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100003151","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["www.ploscompbiol.org"],"crossmark-restriction":false},"short-container-title":["PLoS Comput Biol"],"abstract":"<jats:p>A primary goal of systems neuroscience is to discover how ensembles of neurons transform inputs into goal-directed behavior, a process known as neural computation. A powerful framework for understanding neural computation uses neural dynamics \u2013 the rules that govern how neural activity evolves over time \u2013 to explain how goal-directed input-output transformations occur. As dynamical rules are not directly observable, we need computational models that can infer neural dynamics from recorded neural activity. We typically validate such models using synthetic datasets with known ground-truth dynamics, but unfortunately existing synthetic datasets don\u2019t reflect fundamental features of neural computation and may therefore be poor proxies for neural systems. Further, the field lacks validated metrics for quantifying the accuracy of the dynamics inferred by models. The Computation-through-Dynamics Toolkit (CtDToolkit) addresses these critical gaps by providing: 1) synthetic datasets that reflect computational properties of biological neural circuits, 2) interpretable metrics for quantifying model performance, and 3) a standardized pipeline for training and evaluating models with or without known external inputs. In this manuscript, we demonstrate how CtDToolkit can help guide the development, tuning, and troubleshooting of neural dynamics models. In summary, CtDToolkit provides a necessary framework for model developers to better understand and characterize neural computation through the lens of dynamics.<\/jats:p>","DOI":"10.1371\/journal.pcbi.1014755","type":"journal-article","created":{"date-parts":[[2026,10,15]],"date-time":"2026-10-15T17:46:50Z","timestamp":1792086410000},"page":"e1014755","update-policy":"https:\/\/doi.org\/10.1371\/journal.pcbi.corrections_policy","source":"Crossref","is-referenced-by-count":0,"title":["Computation-through-Dynamics Toolkit: Simulated datasets and quality metrics for dynamical models of neural activity"],"prefix":"10.1371","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4269-5109","authenticated-orcid":true,"given":"Christopher","family":"Versteeg","sequence":"first","affiliation":[{"name":"Emory University and Georgia Institute of Technology","place":["Atlanta, Georgia, United States of America"],"department":["Wallace H. Coulter Department of Biomedical Engineering"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jonathan D.","family":"McCart","sequence":"additional","affiliation":[{"name":"Emory University and Georgia Institute of Technology","place":["Atlanta, Georgia, United States of America"],"department":["Wallace H. Coulter Department of Biomedical Engineering"]},{"name":"Georgia Institute of Technology","place":["Atlanta, Georgia, United States of America"],"department":["Center for Machine Learning"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mitchell","family":"Ostrow","sequence":"additional","affiliation":[{"name":"Massachusetts Institute of Technology","place":["Cambridge, Massachusetts, United States of America"],"department":["Department of Brain and Cognitive Sciences"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"David M.","family":"Zoltowski","sequence":"additional","affiliation":[{"name":"Stanford University","place":["Stanford, California, United States of America"],"department":["Wu Tsai Neurosciences Institute"]},{"name":"Stanford University","place":["Stanford, California, United States of America"],"department":["Department of Statistics"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Clayton B.","family":"Washington","sequence":"additional","affiliation":[{"name":"Emory University and Georgia Institute of Technology","place":["Atlanta, Georgia, United States of America"],"department":["Wallace H. Coulter Department of Biomedical Engineering"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5247-1857","authenticated-orcid":true,"given":"Laura","family":"Driscoll","sequence":"additional","affiliation":[{"name":"Allen Institute for Neural Dynamics","place":["Seattle, Washington, United States of America"]},{"name":"University of Washington","place":["Seattle, Washington, United States of America"],"department":["Department of Neurobiology & Biophysics"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Olivier","family":"Codol","sequence":"additional","affiliation":[{"name":"Universit\u00e9 de Montr\u00e9al","place":["Montr\u00e9al, Canada"],"department":["D\u00e9partement de Neurosciences, Facult\u00e9 de M\u00e9decine"]},{"name":"Quebec Artificial Intelligence Institute","place":["Montr\u00e9al, Quebec, Canada"],"department":["MILA"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5179-3181","authenticated-orcid":true,"given":"Jonathan A.","family":"Michaels","sequence":"additional","affiliation":[{"name":"York University","place":["Toronto, Ontario, Canada"],"department":["School of Kinesiology and Health Science, Faculty of Health"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Scott W.","family":"Linderman","sequence":"additional","affiliation":[{"name":"Stanford University","place":["Stanford, California, United States of America"],"department":["Wu Tsai Neurosciences Institute"]},{"name":"Stanford University","place":["Stanford, California, United States of America"],"department":["Department of Statistics"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"David","family":"Sussillo","sequence":"additional","affiliation":[{"name":"Stanford University","place":["Stanford, California, United States of America"],"department":["Wu Tsai Neurosciences Institute"]},{"name":"Stanford University","place":["Stanford, California, United States of America"],"department":["Department of Electrical Engineering"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Chethan","family":"Pandarinath","sequence":"additional","affiliation":[{"name":"Emory University and Georgia Institute of Technology","place":["Atlanta, Georgia, United States of America"],"department":["Wallace H. Coulter Department of Biomedical Engineering"]},{"name":"Emory University","place":["Atlanta, Georgia, United States of America"],"department":["Department of Neurosurgery"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"340","published-online":{"date-parts":[[2026,10,15]]},"reference":[{"issue":"2","key":"pcbi.1014755.ref001","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1038\/nn.2731","article-title":"How advances in neural recording affect data analysis","volume":"14","author":"IH Stevenson","year":"2011","journal-title":"Nat Neurosci"},{"issue":"572","key":"pcbi.1014755.ref002","doi-asserted-by":"crossref","first-page":"308","DOI":"10.1098\/rspb.1911.0077","article-title":"The intrinsic factors in the act of progression in the mammal","volume":"84","author":"TG Brown","year":"1911","journal-title":"Proceedings of the Royal Society of London Series B, Containing Papers of a Biological Character"},{"issue":"8","key":"pcbi.1014755.ref003","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":"Proc Natl Acad Sci U S A"},{"issue":"4","key":"pcbi.1014755.ref004","doi-asserted-by":"crossref","first-page":"661","DOI":"10.1016\/j.neuron.2005.09.032","article-title":"Transient dynamics versus fixed points in odor representations by locust antennal lobe projection neurons","volume":"48","author":"O Mazor","year":"2005","journal-title":"Neuron"},{"issue":"3","key":"pcbi.1014755.ref005","doi-asserted-by":"crossref","first-page":"626","DOI":"10.1162\/NECO_a_00409","article-title":"Opening the black box: low-dimensional dynamics in high-dimensional recurrent neural networks","volume":"25","author":"D Sussillo","year":"2013","journal-title":"Neural Comput"},{"key":"pcbi.1014755.ref006","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1146\/annurev-neuro-092619-094115","article-title":"Computation through neural population dynamics","volume":"43","author":"S Vyas","year":"2020","journal-title":"Annual Review of Neuroscience"},{"key":"pcbi.1014755.ref007","article-title":"Linear dynamical neural population models through nonlinear embeddings","author":"Y Gao","year":"2016","journal-title":"arXiv"},{"issue":"5","key":"pcbi.1014755.ref008","doi-asserted-by":"crossref","first-page":"1293","DOI":"10.1162\/NECO_a_00953","article-title":"Variational Latent Gaussian Process for Recovering Single-Trial Dynamics from Population Spike Trains","volume":"29","author":"Y Zhao","year":"2017","journal-title":"Neural Comput"},{"key":"pcbi.1014755.ref009","unstructured":"Linderman S, Johnson M, Miller A, Adams R, Blei D, Paninski L. Bayesian learning and inference in recurrent switching linear dynamical systems. In: Artificial intelligence and statistics, 2017. 914\u201322."},{"issue":"10","key":"pcbi.1014755.ref010","doi-asserted-by":"crossref","first-page":"805","DOI":"10.1038\/s41592-018-0109-9","article-title":"Inferring single-trial neural population dynamics using sequential auto-encoders","volume":"15","author":"C Pandarinath","year":"2018","journal-title":"Nat Methods"},{"key":"pcbi.1014755.ref011","article-title":"iLQR-VAE: control-based learning of input-driven dynamics with applications to neural data","author":"M Schimel","year":"2021","journal-title":"bioRxiv"},{"key":"pcbi.1014755.ref012","unstructured":"Zoltowski D, Pillow J, Linderman S. A general recurrent state space framework for modeling neural dynamics during decision-making. In: Proceedings of the International Conference on Machine Learning, 2020. 11680\u201391."},{"key":"pcbi.1014755.ref013","article-title":"Reverse engineering recurrent neural networks with Jacobian switching linear dynamical systems","author":"JTH Smith","year":"2021","journal-title":"arXiv"},{"key":"pcbi.1014755.ref014","unstructured":"Versteeg C, Sedler AR, McCart JD, Pandarinath C. Expressive dynamics models with nonlinear injective readouts enable reliable recovery of latent features from neural activity. In: Proceedings of the 2nd NeurIPS Workshop on Symmetry and Geometry in Neural Representations, 2024. 255\u201378. https:\/\/proceedings.mlr.press\/v228\/versteeg24a.html"},{"issue":"11","key":"pcbi.1014755.ref015","doi-asserted-by":"crossref","first-page":"693","DOI":"10.1038\/s41583-023-00740-7","article-title":"Reconstructing computational system dynamics from neural data with recurrent neural networks","volume":"24","author":"D Durstewitz","year":"2023","journal-title":"Nat Rev Neurosci"},{"key":"pcbi.1014755.ref016","unstructured":"Brenner M, Hess F, Mikhaeil JM, Bereska LF, Monfared Z, Kuo PC, et al. Tractable Dendritic RNNs for Reconstructing Nonlinear Dynamical Systems. In: Proceedings of the 39th International Conference on Machine Learning, 2022. 2292\u2013320. https:\/\/proceedings.mlr.press\/v162\/brenner22a.html"},{"key":"pcbi.1014755.ref017","doi-asserted-by":"crossref","unstructured":"Mikhaeil J, Monfared Z, Durstewitz D. On the Difficulty of Learning Chaotic Dynamics with RNNs. In: Advances in Neural Information Processing Systems 35, 2022. 11297\u2013312. https:\/\/doi.org\/10.52202\/068431-0821","DOI":"10.52202\/068431-0821"},{"key":"pcbi.1014755.ref018","doi-asserted-by":"crossref","DOI":"10.7551\/mitpress\/9780262514620.001.0001","volume-title":"Vision: A computational investigation into the human representation and processing of visual information","author":"D Marr","year":"2010"},{"key":"pcbi.1014755.ref019","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1146\/annurev-neuro-062111-150509","article-title":"Cortical control of arm movements: a dynamical systems perspective","volume":"36","author":"KV Shenoy","year":"2013","journal-title":"Annu Rev Neurosci"},{"issue":"5","key":"pcbi.1014755.ref020","doi-asserted-by":"crossref","first-page":"978","DOI":"10.1016\/j.neuron.2017.05.025","article-title":"Neural Manifolds for the Control of Movement","volume":"94","author":"JA Gallego","year":"2017","journal-title":"Neuron"},{"key":"pcbi.1014755.ref021","article-title":"Interpreting neural computations by examining intrinsic and embedding dimensionality of neural activity","author":"M Jazayeri","year":"2021","journal-title":"arXiv"},{"issue":"3","key":"pcbi.1014755.ref022","doi-asserted-by":"crossref","first-page":"665","DOI":"10.1038\/s41593-025-01869-7","article-title":"Latent circuit inference from heterogeneous neural responses during cognitive tasks","volume":"28","author":"C Langdon","year":"2025","journal-title":"Nat Neurosci"},{"key":"pcbi.1014755.ref023","unstructured":"Kim TD, Luo TZ, Pillow JW, Brody CD. Inferring latent dynamics underlying neural population activity via neural differential equations. In: International Conference on Machine Learning, 2021. 5551\u201361."},{"key":"pcbi.1014755.ref024","doi-asserted-by":"crossref","unstructured":"Dinc F, Shai A, Schnitzer M, Tanaka H. CORNN: Convex optimization of recurrent neural networks for rapid inference of neural dynamics. In: Advances in Neural Information Processing Systems 36, 2023. 51273\u2013301. https:\/\/doi.org\/10.52202\/075280-2232","DOI":"10.52202\/075280-2232"},{"key":"pcbi.1014755.ref025","doi-asserted-by":"crossref","unstructured":"Valente A, Pillow J, Ostojic S. Extracting Computational Mechanisms from Neural Data Using Low-Rank RNNs. In: Advances in Neural Information Processing Systems 35, 2022. 24072\u201386. https:\/\/doi.org\/10.52202\/068431-1748","DOI":"10.52202\/068431-1748"},{"key":"pcbi.1014755.ref026","article-title":"Flow-field inference from neural data using deep recurrent networks","author":"TD Kim","year":"2023","journal-title":"bioRxiv"},{"issue":"1","key":"pcbi.1014755.ref027","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1038\/s41551-023-01106-1","article-title":"Dynamical flexible inference of nonlinear latent factors and structures in neural population activity","volume":"8","author":"H Abbaspourazad","year":"2024","journal-title":"Nat Biomed Eng"},{"issue":"8","key":"pcbi.1014755.ref028","doi-asserted-by":"crossref","DOI":"10.1371\/journal.pcbi.1007263","article-title":"Identifying nonlinear dynamical systems via generative recurrent neural networks with applications to fMRI","volume":"15","author":"G Koppe","year":"2019","journal-title":"PLoS Comput Biol"},{"key":"pcbi.1014755.ref029","first-page":"29379","article-title":"Targeted neural dynamical modeling.","volume-title":"Advances in Neural Information Processing Systems","author":"C Hurwitz","year":"2021"},{"key":"pcbi.1014755.ref030","doi-asserted-by":"crossref","unstructured":"Ostrow M, Eisen A, Kozachkov L, Fiete I. Beyond Geometry: Comparing the Temporal Structure of Computation in Neural Circuits with Dynamical Similarity Analysis. In: Advances in Neural Information Processing Systems 36, 2023. 33824\u201337. https:\/\/doi.org\/10.52202\/075280-1467","DOI":"10.52202\/075280-1467"},{"issue":"2","key":"pcbi.1014755.ref031","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1175\/1520-0469(1963)020<0130:DNF>2.0.CO;2","article-title":"Deterministic Nonperiodic Flow","volume":"20","author":"EN Lorenz","year":"1963","journal-title":"J Atmos Sci"},{"key":"pcbi.1014755.ref032","article-title":"Expressive architectures enhance interpretability of dynamics-based neural population models","author":"AR Sedler","year":"2022","journal-title":"arXiv"},{"issue":"11","key":"pcbi.1014755.ref033","doi-asserted-by":"crossref","first-page":"674","DOI":"10.1038\/s42256-020-00242-6","article-title":"Moving beyond generalization to accurate interpretation of flexible models","volume":"2","author":"M Genkin","year":"2020","journal-title":"Nat Mach Intell"},{"key":"pcbi.1014755.ref034","first-page":"2024","article-title":"Partial observation can induce mechanistic mismatches in data-constrained models of neural dynamics","volume":"2024","author":"W Qian","year":"2024","journal-title":"bioRxiv"},{"key":"pcbi.1014755.ref035","unstructured":"Smith JTH, Warrington A, Linderman SW. Simplified State Space Layers for Sequence Modeling. In: 2023. https:\/\/doi.org\/arXiv:2208.04933"},{"key":"pcbi.1014755.ref036","doi-asserted-by":"crossref","unstructured":"Ye J, Pandarinath C. Representation learning for neural population activity with Neural Data Transformers. In: Neurons, Behavior, Data analysis, and Theory, 2021. https:\/\/arxiv.org\/abs\/2108.01210","DOI":"10.1101\/2021.01.16.426955"},{"key":"pcbi.1014755.ref037","unstructured":"Pei F, Ye J, Zoltowski D, Wu A, Chowdhury RH, Sohn H. Neural latents benchmark \u201921: evaluating latent variable models of neural population activity. 2022. https:\/\/doi.org\/arXiv:2109.04463"},{"key":"pcbi.1014755.ref038","volume-title":"Gymnasium","author":"M Towers","year":"2023"},{"issue":"31","key":"pcbi.1014755.ref039","doi-asserted-by":"crossref","first-page":"1003","DOI":"10.21105\/joss.01003","article-title":"FixedPointFinder: A Tensorflow toolbox for identifying and characterizing fixed points in recurrent neural networks","volume":"3","author":"M Golub","year":"2018","journal-title":"JOSS"},{"key":"pcbi.1014755.ref040","doi-asserted-by":"crossref","unstructured":"Driscoll L, Shenoy K, Sussillo D. Flexible multitask computation in recurrent networks utilizes shared dynamical motifs. bioRxiv. 2022. 2022.08.15.503870. https:\/\/doi.org\/10.1101\/2022.08.15.503870","DOI":"10.1101\/2022.08.15.503870"},{"key":"pcbi.1014755.ref041","article-title":"MotorNet: a Python toolbox for controlling differentiable biomechanical effectors with artificial neural networks","author":"O Codol","year":"2024","journal-title":"eLife"},{"issue":"2","key":"pcbi.1014755.ref042","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1038\/s41593-018-0310-2","article-title":"Task representations in neural networks trained to perform many cognitive tasks","volume":"22","author":"GR Yang","year":"2019","journal-title":"Nat Neurosci"},{"issue":"6","key":"pcbi.1014755.ref043","doi-asserted-by":"crossref","first-page":"783","DOI":"10.1038\/s41593-022-01088-4","article-title":"The role of population structure in computations through neural dynamics","volume":"25","author":"A Dubreuil","year":"2022","journal-title":"Nat Neurosci"},{"issue":"4771","key":"pcbi.1014755.ref044","doi-asserted-by":"crossref","first-page":"1416","DOI":"10.1126\/science.3749885","article-title":"Neuronal population coding of movement direction","volume":"233","author":"AP Georgopoulos","year":"1986","journal-title":"Science"},{"issue":"4","key":"pcbi.1014755.ref045","doi-asserted-by":"crossref","DOI":"10.1016\/j.neuron.2018.09.030","article-title":"A Neural Population Mechanism for Rapid Learning","volume":"100","author":"MG Perich","year":"2018","journal-title":"Neuron"},{"key":"pcbi.1014755.ref046","doi-asserted-by":"crossref","DOI":"10.7554\/eLife.67256","article-title":"Rotational dynamics in motor cortex are consistent with a feedback controller","volume":"10","author":"HT Kalidindi","year":"2021","journal-title":"Elife"},{"issue":"2","key":"pcbi.1014755.ref047","doi-asserted-by":"crossref","DOI":"10.1371\/journal.pcbi.1011852","article-title":"Trained recurrent neural networks develop phase-locked limit cycles in a working memory task","volume":"20","author":"M Pals","year":"2024","journal-title":"PLoS Comput Biol"},{"issue":"7","key":"pcbi.1014755.ref048","doi-asserted-by":"crossref","first-page":"925","DOI":"10.1038\/nn.3405","article-title":"Robust timing and motor patterns by taming chaos in recurrent neural networks","volume":"16","author":"R Laje","year":"2013","journal-title":"Nat Neurosci"},{"issue":"4","key":"pcbi.1014755.ref049","doi-asserted-by":"crossref","first-page":"544","DOI":"10.1016\/j.neuron.2009.07.018","article-title":"Generating coherent patterns of activity from chaotic neural networks","volume":"63","author":"D Sussillo","year":"2009","journal-title":"Neuron"},{"key":"pcbi.1014755.ref050","doi-asserted-by":"crossref","DOI":"10.7554\/eLife.20899","article-title":"Biologically plausible learning in recurrent neural networks reproduces neural dynamics observed during cognitive tasks","volume":"6","author":"T Miconi","year":"2017","journal-title":"Elife"},{"key":"pcbi.1014755.ref051","unstructured":"Roy O, Vetterli M. The effective rank: A measure of effective dimensionality. In: 2007 15th European signal processing conference, 2007. 606\u201310."},{"issue":"2","key":"pcbi.1014755.ref052","doi-asserted-by":"crossref","first-page":"023143","DOI":"10.1063\/5.0131787","article-title":"Using machine learning to anticipate tipping points and extrapolate to post-tipping dynamics of non-stationary dynamical systems","volume":"33","author":"D Patel","year":"2023","journal-title":"Chaos"},{"issue":"1","key":"pcbi.1014755.ref053","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1038\/s41467-017-00030-8","article-title":"Chaos as an intermittently forced linear system","volume":"8","author":"SL Brunton","year":"2017","journal-title":"Nat Commun"},{"key":"pcbi.1014755.ref054","doi-asserted-by":"crossref","first-page":"94","DOI":"10.1016\/j.neunet.2019.04.020","article-title":"Dynamic mode decomposition in vector-valued reproducing kernel Hilbert spaces for extracting dynamical structure among observables","volume":"117","author":"K Fujii","year":"2019","journal-title":"Neural Netw"},{"key":"pcbi.1014755.ref055","unstructured":"Williams AH, Kunz E, Kornblith S, Linderman S. Generalized Shape Metrics on Neural Representations. In: Advances in Neural Information Processing Systems, 2021. 4738\u201350."},{"key":"pcbi.1014755.ref056","volume-title":"Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering","author":"S Strogatz","edition":"2"},{"key":"pcbi.1014755.ref057","doi-asserted-by":"crossref","unstructured":"Eisen AJ, Ostrow M, Chandra S, Kozachkov L, Miller EK, Fiete IR. Characterizing control between interacting subsystems with deep Jacobian estimation. 2025. https:\/\/arxiv.org\/abs\/2507.01946","DOI":"10.52202\/085713-1711"},{"key":"pcbi.1014755.ref058","unstructured":"Chen RTQ, Rubanova Y, Bettencourt J, Duvenaud D. Neural Ordinary Differential Equations. Advances in Neural Information Processing Systems. 2018. p. 31\u201360."},{"issue":"10","key":"pcbi.1014755.ref059","doi-asserted-by":"crossref","first-page":"1742","DOI":"10.1038\/s42256-025-01127-2","article-title":"Single-unit activations confer inductive biases for emergent circuit solutions to cognitive tasks","volume":"7","author":"P Tolmachev","year":"2025","journal-title":"Nat Mach Intell"}],"updated-by":[{"DOI":"10.1371\/journal.pcbi.1014755","type":"new_version","label":"New version","source":"publisher","updated":{"date-parts":[[2026,10,27]],"date-time":"2026-10-27T00:00:00Z","timestamp":1793059200000}}],"container-title":["PLOS Computational Biology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dx.plos.org\/10.1371\/journal.pcbi.1014755","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,10,27]],"date-time":"2026-10-27T18:14:58Z","timestamp":1793124898000},"score":1,"resource":{"primary":{"URL":"https:\/\/dx.plos.org\/10.1371\/journal.pcbi.1014755"}},"subtitle":[],"editor":[{"given":"Abigail","family":"Morrison","sequence":"first","affiliation":[{"name":"Research Center J\u00fclich","place":["GERMANY"]}],"role":[{"vocabulary":"crossref","role":"editor"}]}],"short-title":[],"issued":{"date-parts":[[2026,10,15]]},"references-count":59,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2026,10,15]]}},"URL":"https:\/\/doi.org\/10.1371\/journal.pcbi.1014755","relation":{},"ISSN":["1553-7358"],"issn-type":[{"value":"1553-7358","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,10,15]]}}}