{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,28]],"date-time":"2026-10-28T18:33:01Z","timestamp":1793212381305,"version":"4.5.0"},"reference-count":90,"publisher":"National Academy of Sciences","issue":"44","license":[{"start":{"date-parts":[[2026,10,28]],"date-time":"2026-10-28T00:00:00Z","timestamp":1793145600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000002","name":"HHS | National Institutes of Health","doi-asserted-by":"publisher","award":["R15NS135396"],"award-info":[{"award-number":["R15NS135396"]}],"id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000002","name":"HHS | National Institutes of Health","doi-asserted-by":"publisher","award":["R01DA060744"],"award-info":[{"award-number":["R01DA060744"]}],"id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000002","name":"HHS | National Institutes of Health","doi-asserted-by":"publisher","award":["R01NS118442"],"award-info":[{"award-number":["R01NS118442"]}],"id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006312","name":"BrightFocus Foundation","doi-asserted-by":"publisher","award":["Standard Award in Alzheimer's Disease Research 2922"],"award-info":[{"award-number":["Standard Award in Alzheimer's Disease Research 2922"]}],"id":[{"id":"10.13039\/100006312","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["www.pnas.org"],"crossmark-restriction":true},"short-container-title":["Proc. Natl. Acad. Sci. U.S.A."],"published-print":{"date-parts":[[2026,11,3]]},"abstract":"<jats:p>Diverse social, engineered, and biological systems are made up of multiple interacting dynamical components; to work properly, certain components must be carefully regulated so that their dynamics avoid instability. However, many desirable properties, like optimized computation and control, emerge close to the tipping point between stability and instability, i.e., near criticality. Claims that such dynamical systems operate near criticality have been controversial because it is difficult to assess proximity to criticality when control parameters are unknown, hard to manipulate experimentally, and fluctuating in response to changing environmental or internal conditions. Moreover, observations of system dynamics are often limited to time series measurements of one or a few components. Here we develop a temporal renormalization group theory for defining criticality for univariate time series models and an information theoretic definition of proximity to criticality that is independent of model parameterization. We apply this theory to order-n autoregressive models resulting in a data-analytic strategy for time-resolved estimates of proximity to criticality based on measured univariate time series. After benchmarking our approach on diverse ground-truth cases, we apply it to recordings of spiking activity in the mammalian brain. Our estimate of proximity to criticality predicts behavior and suggests that brain dynamics shift closer to criticality during wakefulness and shift away during deep sleep.<\/jats:p>","DOI":"10.1073\/pnas.2526984123","type":"journal-article","created":{"date-parts":[[2026,10,28]],"date-time":"2026-10-28T17:37:24Z","timestamp":1793209044000},"update-policy":"https:\/\/doi.org\/10.1073\/pnas.cm10313","source":"Crossref","is-referenced-by-count":0,"title":["Defining and measuring proximity to criticality based on univariate time series"],"prefix":"10.1073","volume":"123","author":[{"ORCID":"https:\/\/orcid.org\/0009-0002-7404-6522","authenticated-orcid":false,"given":"J. Samuel","family":"Sooter","sequence":"first","affiliation":[{"name":"Department of Physics, University of Arkansas","place":["Fayetteville"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2969-9997","authenticated-orcid":false,"given":"Antonio J.","family":"Fontenele","sequence":"additional","affiliation":[{"name":"Department of Physics, University of Arkansas","place":["Fayetteville"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0286-4139","authenticated-orcid":false,"given":"Andrea K.","family":"Barreiro","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/042tdr378","id-type":"ROR","asserted-by":"publisher"}],"name":"Department of Mathematics, Southern Methodist University","place":["Dallas"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3279-8240","authenticated-orcid":false,"given":"Cheng","family":"Ly","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/02nkdxk79","id-type":"ROR","asserted-by":"publisher"}],"name":"Department of Mathematics and Applied Mathematics, Virginia Commonwealth University","place":["Richmond"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5017-4090","authenticated-orcid":false,"given":"Keith B.","family":"Hengen","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/01yc7t268","id-type":"ROR","asserted-by":"publisher"}],"name":"Department of Biology, Washington University","place":["St. Louis"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0679-1766","authenticated-orcid":false,"given":"Woodrow L.","family":"Shew","sequence":"additional","affiliation":[{"name":"Department of Physics, University of Arkansas","place":["Fayetteville"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"341","published-online":{"date-parts":[[2026,10,28]]},"reference":[{"key":"e_1_3_4_1_2","first-page":"269","article-title":"Time series analysis: Forecasting and control","volume":"14","author":"Geurts M.","year":"1977","unstructured":"M. Geurts, G. E. P. Box, G. M. Jenkins, Time series analysis: Forecasting and control. J. Mark. Res. 14, 269 (1977).","journal-title":"J. Mark. Res."},{"key":"e_1_3_4_2_2","volume-title":"Time Series Analysis and its Applications","author":"Shumway R. H.","year":"2006","unstructured":"R. H. Shumway, D. S. Stoffer, Time Series Analysis and its Applications (Springer, ed. 4, 2006).","edition":"4"},{"key":"e_1_3_4_3_2","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160130102"},{"key":"e_1_3_4_4_2","doi-asserted-by":"publisher","DOI":"10.1126\/science.1165893"},{"key":"e_1_3_4_5_2","doi-asserted-by":"publisher","DOI":"10.1126\/science.1238723"},{"key":"e_1_3_4_6_2","doi-asserted-by":"publisher","DOI":"10.1088\/1361-6633\/aca6f8"},{"key":"e_1_3_4_7_2","first-page":"263","article-title":"Scaling laws for ising models near Tc","volume":"2","author":"Kadanoff L. P.","year":"1966","unstructured":"L. P. Kadanoff, Scaling laws for ising models near Tc. Phys. Phys. Fiz. 2, 263\u2013272 (1966).","journal-title":"Phys. Phys. Fiz."},{"key":"e_1_3_4_8_2","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.47.773"},{"key":"e_1_3_4_9_2","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.71.S358"},{"key":"e_1_3_4_10_2","doi-asserted-by":"publisher","DOI":"10.1038\/scientificamerican0879-158"},{"key":"e_1_3_4_11_2","doi-asserted-by":"publisher","DOI":"10.1038\/35065675"},{"key":"e_1_3_4_12_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.105.044133"},{"key":"e_1_3_4_13_2","doi-asserted-by":"publisher","DOI":"10.1007\/s10955-011-0229-4"},{"key":"e_1_3_4_14_2","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.90.031001"},{"key":"e_1_3_4_15_2","doi-asserted-by":"publisher","DOI":"10.1038\/s42254-022-00491-x"},{"key":"e_1_3_4_16_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-15001-2"},{"key":"e_1_3_4_17_2","doi-asserted-by":"publisher","DOI":"10.3847\/1538-4357\/ab29f4"},{"key":"e_1_3_4_18_2","doi-asserted-by":"publisher","DOI":"10.1038\/s41467-022-28964-8"},{"key":"e_1_3_4_19_2","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.1005766107"},{"key":"e_1_3_4_20_2","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.1324045111"},{"key":"e_1_3_4_21_2","doi-asserted-by":"publisher","DOI":"10.1523\/JNEUROSCI.23-35-11167.2003"},{"key":"e_1_3_4_22_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.123.178103"},{"key":"e_1_3_4_23_2","doi-asserted-by":"publisher","DOI":"10.1126\/sciadv.adj9303"},{"key":"e_1_3_4_24_2","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.2417117122"},{"key":"e_1_3_4_25_2","doi-asserted-by":"publisher","DOI":"10.7551\/mitpress\/13588.001.0001"},{"key":"e_1_3_4_26_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.neuron.2025.05.020"},{"key":"e_1_3_4_27_2","doi-asserted-by":"publisher","DOI":"10.1038\/nrn3084"},{"key":"e_1_3_4_28_2","doi-asserted-by":"publisher","DOI":"10.1038\/nphys3370"},{"key":"e_1_3_4_29_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.122.208101"},{"key":"e_1_3_4_30_2","doi-asserted-by":"publisher","DOI":"10.1038\/s41593-023-01536-9"},{"key":"e_1_3_4_31_2","doi-asserted-by":"publisher","DOI":"10.1038\/s41593-024-01715-2"},{"key":"e_1_3_4_32_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.neuron.2019.08.031"},{"key":"e_1_3_4_33_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.neuron.2024.08.006"},{"key":"e_1_3_4_34_2","doi-asserted-by":"crossref","unstructured":"R. Chopra et al. A single dynamical property can account for the capacity to learn from artificial networks to the mammalian brain. bioRxiv [Preprint] (2026). https:\/\/doi.org\/10.64898\/2026.07.09.737603 (Accessed 1 August 2026).","DOI":"10.64898\/2026.07.09.737603"},{"key":"e_1_3_4_35_2","doi-asserted-by":"crossref","unstructured":"H. Chang et al. Dynamic shifts in brain criticality support cognitive processing. bioRxiv [Preprint] (2026). https:\/\/doi.org\/10.64898\/2026.03.12.711394 (Accessed 1 August 2026).","DOI":"10.64898\/2026.03.12.711394"},{"key":"e_1_3_4_36_2","doi-asserted-by":"crossref","unstructured":"L. X. Lim et al. From criticality to cognitive effort: scale-invariant EEG dynamics supporting cognitive flexibility are suppressed by effort. bioRxiv [Preprint] (2026). https:\/\/doi.org\/10.64898\/2026.05.10.724092 (Accessed 1 August 2026).","DOI":"10.64898\/2026.05.10.724092"},{"key":"e_1_3_4_37_2","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.2417010122"},{"key":"e_1_3_4_38_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.108.208102"},{"key":"e_1_3_4_39_2","doi-asserted-by":"publisher","DOI":"10.1523\/JNEUROSCI.3864-09.2009"},{"key":"e_1_3_4_40_2","doi-asserted-by":"publisher","DOI":"10.1371\/journal.pcbi.1001013"},{"key":"e_1_3_4_41_2","doi-asserted-by":"publisher","DOI":"10.1002\/hbm.20590"},{"key":"e_1_3_4_42_2","doi-asserted-by":"publisher","DOI":"10.1371\/journal.pcbi.1004576"},{"key":"e_1_3_4_43_2","doi-asserted-by":"publisher","DOI":"10.1371\/journal.pcbi.1005574"},{"key":"e_1_3_4_44_2","doi-asserted-by":"publisher","DOI":"10.1038\/s41467-024-51399-2"},{"key":"e_1_3_4_45_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.neuron.2018.10.045"},{"key":"e_1_3_4_46_2","doi-asserted-by":"publisher","DOI":"10.1038\/s41467-023-37976-x"},{"key":"e_1_3_4_47_2","doi-asserted-by":"publisher","DOI":"10.1523\/JNEUROSCI.4286-12.2013"},{"key":"e_1_3_4_48_2","doi-asserted-by":"publisher","DOI":"10.1038\/s41467-018-04725-4"},{"key":"e_1_3_4_49_2","doi-asserted-by":"publisher","DOI":"10.1038\/s41467-023-37613-7"},{"key":"e_1_3_4_50_2","first-page":"31021","article-title":"Tracking the distance to criticality in systems with unknown noise","volume":"14","author":"Harris B.","year":"2024","unstructured":"B. Harris, L. L. Gollo, B. D. Fulcher, Tracking the distance to criticality in systems with unknown noise. Phys. Rev. X 14, 31021 (2024).","journal-title":"Phys. Rev. X"},{"key":"e_1_3_4_51_2","doi-asserted-by":"publisher","DOI":"10.1371\/journal.pcbi.1008268"},{"key":"e_1_3_4_52_2","doi-asserted-by":"publisher","DOI":"10.1007\/s10955-017-1770-6"},{"key":"e_1_3_4_53_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.isci.2019.01.009"},{"key":"e_1_3_4_54_2","doi-asserted-by":"publisher","DOI":"10.1038\/s42003-023-05001-y"},{"key":"e_1_3_4_55_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.cell.2024.10.004"},{"key":"e_1_3_4_56_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.celrep.2024.113762"},{"key":"e_1_3_4_57_2","doi-asserted-by":"publisher","DOI":"10.1523\/JNEUROSCI.21-04-01370.2001"},{"key":"e_1_3_4_58_2","doi-asserted-by":"publisher","DOI":"10.3389\/fphys.2012.00450"},{"key":"e_1_3_4_59_2","doi-asserted-by":"publisher","DOI":"10.1038\/s41467-023-40056-9"},{"key":"e_1_3_4_60_2","doi-asserted-by":"publisher","DOI":"10.1523\/JNEUROSCI.0448-17.2017"},{"key":"e_1_3_4_61_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.114.078105"},{"key":"e_1_3_4_62_2","doi-asserted-by":"publisher","DOI":"10.1038\/s41467-022-30892-6"},{"key":"e_1_3_4_63_2","doi-asserted-by":"publisher","DOI":"10.1038\/s42003-020-0774-y"},{"key":"e_1_3_4_64_2","doi-asserted-by":"publisher","DOI":"10.1038\/s43588-023-00410-9"},{"key":"e_1_3_4_65_2","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.1818972116"},{"key":"e_1_3_4_66_2","doi-asserted-by":"publisher","DOI":"10.7554\/eLife.79950"},{"key":"e_1_3_4_67_2","article-title":"Critical brain dynamics may be more prevalent than previously thought","volume":"9","author":"Fontenele A. J.","year":"2026","unstructured":"A. J. Fontenele , Critical brain dynamics may be more prevalent than previously thought. Commun. Phys. 9 (2026).","journal-title":"Commun. Phys"},{"key":"e_1_3_4_68_2","doi-asserted-by":"publisher","DOI":"10.1038\/nn.3776"},{"key":"e_1_3_4_69_2","doi-asserted-by":"publisher","DOI":"10.1038\/nature13186"},{"key":"e_1_3_4_70_2","doi-asserted-by":"publisher","DOI":"10.3389\/fphys.2012.00015"},{"key":"e_1_3_4_71_2","doi-asserted-by":"publisher","DOI":"10.1523\/JNEUROSCI.3474-14.2014"},{"key":"e_1_3_4_72_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.132.218403"},{"key":"e_1_3_4_73_2","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177729694"},{"key":"e_1_3_4_74_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01106578"},{"key":"e_1_3_4_75_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRev.106.620"},{"key":"e_1_3_4_76_2","doi-asserted-by":"publisher","DOI":"10.1109\/PROC.1984.12981"},{"key":"e_1_3_4_77_2","unstructured":"P. C. Petersen M. Hernanadez G. Buzs\u00e1ki The Buzsaki Lab Databank - Public electrophysiological datasets from awake animals (Version 1.0) [Dataset]. Zenodo. https:\/\/doi.org\/10.5281\/zenodo.4307883. Deposited 6 June 2020."},{"key":"e_1_3_4_78_2","doi-asserted-by":"publisher","DOI":"10.1371\/journal.pcbi.1005543"},{"key":"e_1_3_4_79_2","doi-asserted-by":"publisher","DOI":"10.1371\/journal.pcbi.1002985"},{"key":"e_1_3_4_80_2","doi-asserted-by":"publisher","DOI":"10.1371\/journal.pone.0014129"},{"key":"e_1_3_4_81_2","doi-asserted-by":"publisher","DOI":"10.1098\/rsif.2019.0262"},{"key":"e_1_3_4_82_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.neuron.2018.12.009"},{"key":"e_1_3_4_83_2","doi-asserted-by":"publisher","DOI":"10.1038\/nature10009"},{"key":"e_1_3_4_84_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.neuron.2010.04.020"},{"key":"e_1_3_4_85_2","doi-asserted-by":"publisher","DOI":"10.7554\/eLife.55092"},{"key":"e_1_3_4_86_2","first-page":"1","article-title":"Frequency-dependent covariance reveals critical spatio-temporal patterns of synchronized activity in the human brain","volume":"208401","author":"Calvo R.","year":"2024","unstructured":"R. Calvo, C. Martorell, G. B. Morales, S. Di Santo, M. A. Mu\u00f1oz, Frequency-dependent covariance reveals critical spatio-temporal patterns of synchronized activity in the human brain. Phys. Rev. Lett. 208401, 1\u20137 (2024).","journal-title":"Phys. Rev. Lett."},{"key":"e_1_3_4_87_2","doi-asserted-by":"publisher","DOI":"10.3389\/fncir.2020.00054"},{"key":"e_1_3_4_88_2","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.2208998120"},{"key":"e_1_3_4_89_2","unstructured":"J. S. Sooter Code for calculating d2. Github. https:\/\/github.com\/sam-sooter\/prox_crit_toolkit. Deposited 1 November 2025."},{"key":"e_1_3_4_90_2","unstructured":"W. L. Shew S. Sooter Mouse single unit spike data (recorded by Buzsaki Lab reanalyzed by Shew Lab). Figshare. https:\/\/doi.org\/10.6084\/m9.figshare.32012181.v1. Deposited 4 April 2026."}],"container-title":["Proceedings of the National Academy of Sciences"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/pnas.org\/doi\/pdf\/10.1073\/pnas.2526984123","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,10,28]],"date-time":"2026-10-28T17:41:45Z","timestamp":1793209305000},"score":1,"resource":{"primary":{"URL":"https:\/\/pnas.org\/doi\/10.1073\/pnas.2526984123"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,10,28]]},"references-count":90,"journal-issue":{"issue":"44","published-print":{"date-parts":[[2026,11,3]]}},"alternative-id":["10.1073\/pnas.2526984123"],"URL":"https:\/\/doi.org\/10.1073\/pnas.2526984123","relation":{},"ISSN":["0027-8424","1091-6490"],"issn-type":[{"value":"0027-8424","type":"print"},{"value":"1091-6490","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,10,28]]},"assertion":[{"value":"2025-09-23","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2026-09-24","order":2,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2026-10-28","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"e2526984123"}}