{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,13]],"date-time":"2026-10-13T19:09:59Z","timestamp":1791918599798,"version":"4.3.4"},"reference-count":65,"publisher":"Frontiers Media SA","license":[{"start":{"date-parts":[[2024,7,15]],"date-time":"2024-07-15T00:00:00Z","timestamp":1721001600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["frontiersin.org"],"crossmark-restriction":true},"short-container-title":["Front. Comput. Neurosci."],"abstract":"<jats:p>Scale-free brain activity, linked with learning, the integration of different time scales, and the formation of mental models, is correlated with a metastable cognitive basis. The spectral slope, a key aspect of scale-free dynamics, was proposed as a potential indicator to distinguish between different sleep stages. Studies suggest that brain networks maintain a consistent scale-free structure across wakefulness, anesthesia, and recovery. Although differences in anesthetic sensitivity between the sexes are recognized, these variations are not evident in clinical electroencephalographic recordings of the cortex. Recently, changes in the slope of the power law exponent of neural activity were found to correlate with changes in R\u00e9nyi entropy, an extended concept of Shannon's information entropy. These findings establish quantifiers as a promising tool for the study of scale-free dynamics in the brain. Our study presents a novel visual representation called the R\u00e9nyi entropy-complexity causality space, which encapsulates complexity, permutation entropy, and the R\u00e9nyi parameter q. The main goal of this study is to define this space for classical dynamical systems within theoretical bounds. In addition, the study aims to investigate how well different time series mimicking scale-free activity can be discriminated. Finally, this tool is used to detect dynamic features in intracranial electroencephalography (iEEG) signals. To achieve these goals, the study implementse the Bandt and Pompe method for ordinal patterns. In this process, each signal is associated with a probability distribution, and the causal measures of R\u00e9nyi entropy and complexity are computed based on the parameter q. This method is a valuable tool for analyzing simulated time series. It effectively distinguishes elements of correlated noise and provides a straightforward means of examining differences in behaviors, characteristics, and classifications. For the iEEG experimental data, the REM state showed a greater number of significant sex-based differences, while the supramarginal gyrus region showed the most variation across different modes and analyzes. Exploring scale-free brain activity with this framework could provide valuable insights into cognition and neurological disorders. The results may have implications for understanding differences in brain function between the sexes and their possible relevance to neurological disorders.<\/jats:p>","DOI":"10.3389\/fncom.2024.1342985","type":"journal-article","created":{"date-parts":[[2024,7,15]],"date-time":"2024-07-15T05:02:59Z","timestamp":1721019779000},"update-policy":"https:\/\/doi.org\/10.3389\/crossmark-policy","source":"Crossref","is-referenced-by-count":13,"title":["R\u00e9nyi entropy-complexity causality space: a novel neurocomputational tool for detecting scale-free features in EEG\/iEEG data"],"prefix":"10.3389","volume":"18","author":[{"given":"Natal\u00ed","family":"Guisande","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Fernando","family":"Montani","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1965","published-online":{"date-parts":[[2024,7,15]]},"reference":[{"key":"B1","unstructured":"R\u00e9nyi entropy in measuring information levels in vorono\u00ef tessellation cells with application in digital image analysis7795Theor. Appl. Math. Comput. Sci62016"},{"key":"B2","doi-asserted-by":"publisher","first-page":"167263","DOI":"10.1063\/5.0167263","article-title":"Ordinal methods: concepts, applications, new developments, and challenges\u2014in memory of karsten keller (1961\u20132022)","volume":"33","author":"Amig\u00f3","year":"2023","journal-title":"Chaos"},{"key":"B3","doi-asserted-by":"crossref","first-page":"348","DOI":"10.1016\/S0375-9601(96)00756-6","article-title":"Some features of the l\u00f3pez-ruiz-mancini-calbet (lmc) statistical measure of complexity","volume":"223","author":"Anteneodo","year":"1996","journal-title":"Phys. Lett. A"},{"key":"B4","doi-asserted-by":"publisher","first-page":"858","DOI":"10.1089\/jwh.2019.8247","article-title":"Sex as a biological variable: a 5-year progress report and call to action","volume":"29","author":"Arnegard","year":"2020","journal-title":"J. Women's Health"},{"key":"B5","doi-asserted-by":"crossref","first-page":"1316","DOI":"10.23919\/EUSIPCO54536.2021.9616140","article-title":"\u201cMultiscale fractal analysis on eeg signals for music-induced emotion recognition,\u201d","volume-title":"2021 29th European Signal Processing Conference (EUSIPCO)","author":"Avramidis","year":"2021"},{"key":"B6","doi-asserted-by":"publisher","first-page":"174102","DOI":"10.1103\/physrevlett.88.174102","article-title":"Permutation entropy: a natural complexity measure for time series","volume":"88","author":"Bandt","year":"2002","journal-title":"Phys. Rev. Lett"},{"key":"B7","doi-asserted-by":"publisher","first-page":"165","DOI":"10.1016\/j.neures.2019.10.011","article-title":"Is learning scale-free? Chemistry learning increases eeg fractal power and changes the power law exponent","volume":"156","author":"Bongers","year":"2020","journal-title":"Neurosci. Res"},{"key":"B8","doi-asserted-by":"publisher","first-page":"510","DOI":"10.1016\/j.bja.2023.06.042","article-title":"Impact of female sex on anaesthetic awareness, depth, and emergence: a systematic review and meta-analysis","volume":"131","author":"Braithwaite","year":"2023","journal-title":"Br. J. Anaesth"},{"key":"B9","doi-asserted-by":"publisher","first-page":"589303","DOI":"10.3389\/fnins.2020.589303","article-title":"Predicting sex from EEG: validity and generalizability of deep-learning-based interpretable classifier","volume":"14","author":"Bu\u010dov\u00e1","year":"2020","journal-title":"Front. Neurosci"},{"key":"B10","doi-asserted-by":"publisher","first-page":"264","DOI":"10.1038\/nrn3687","article-title":"The log-dynamic brain: how skewed distributions affect network operations","volume":"15","author":"Buzs\u00e1ki","year":"2014","journal-title":"Nat. Rev. Neurosci"},{"key":"B11","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1016\/j.ijpsycho.2021.01.008","article-title":"Sex differences in resting EEG in healthy young adults","volume":"161","author":"Cave","year":"2021","journal-title":"Int. J. Psychophysiol"},{"key":"B12","doi-asserted-by":"publisher","first-page":"631","DOI":"10.1016\/j.neuroimage.2019.01.024","article-title":"The spectral exponent of the resting EEG indexes the presence of consciousness during unresponsiveness induced by propofol, xenon, and ketamine","volume":"189","author":"Colombo","year":"2019","journal-title":"NeuroImage"},{"key":"B13","doi-asserted-by":"publisher","first-page":"405","DOI":"10.1007\/s10827-010-0263-2","article-title":"Comparative power spectral analysis of simultaneous elecroencephalographic and magnetoencephalographic recordings in humans suggests non-resistive extracellular media","volume":"29","author":"Dehghani","year":"2010","journal-title":"J. Comput. Neurosci"},{"key":"B14","doi-asserted-by":"publisher","first-page":"739","DOI":"10.1038\/nrn1198","article-title":"The high-conductance state of neocortical neurons in vivo","volume":"4","author":"Destexhe","year":"2003","journal-title":"Nat. Rev. Neurosci"},{"key":"B15","doi-asserted-by":"publisher","first-page":"1767","DOI":"10.3389\/fphys.2018.01767","article-title":"Fractal and multifractal properties of electrographic recordings of human brain activity: toward its use as a signal feature for machine learning in clinical applications","volume":"9","author":"Fran\u00e7a","year":"2018","journal-title":"Front. Physiol"},{"key":"B16","doi-asserted-by":"publisher","first-page":"1130","DOI":"10.1093\/brain\/awy035","article-title":"Atlas of the normal intracranial electroencephalogram: neurophysiological awake activity in different cortical areas","volume":"141","author":"Frauscher","year":"","journal-title":"Brain"},{"key":"B17","doi-asserted-by":"publisher","first-page":"374","DOI":"10.1002\/ana.25304","article-title":"High-frequency oscillations in the normal human brain","volume":"84","author":"Frauscher","year":"","journal-title":"Ann. Neurol"},{"key":"B18","doi-asserted-by":"publisher","first-page":"4574","DOI":"10.1093\/cercor\/bhac363","article-title":"The fractal brain: scale-invariance in structure and dynamics","volume":"33","author":"Grosu","year":"2022","journal-title":"Cerebr. Cortex"},{"key":"B19","doi-asserted-by":"publisher","first-page":"e043111","DOI":"10.1063\/5.0142773","article-title":"Chaotic dynamics of the h\u00e9non map and neuronal input\u2013output: a comparison with neurophysiological data","volume":"33","author":"Guisande","year":"2023","journal-title":"Chaos"},{"key":"B20","doi-asserted-by":"publisher","first-page":"13786","DOI":"10.1523\/JNEUROSCI.2111-11.2011","article-title":"Scale-free properties of the functional magnetic resonance imaging signal during rest and task","volume":"31","author":"He","year":"2011","journal-title":"J. Neurosci"},{"key":"B21","doi-asserted-by":"publisher","first-page":"480","DOI":"10.1016\/j.tics.2014.04.003","article-title":"Scale-free brain activity: past, present, and future","volume":"18","author":"He","year":"2014","journal-title":"Trends Cogn. Sci"},{"key":"B22","doi-asserted-by":"publisher","first-page":"353","DOI":"10.1016\/j.neuron.2010.04.020","article-title":"The temporal structures and functional significance of scale-free brain activity","volume":"66","author":"He","year":"2010","journal-title":"Neuron"},{"key":"B23","doi-asserted-by":"publisher","first-page":"74","DOI":"10.1016\/j.physa.2018.01.026","article-title":"Characterization of time series via r\u00e9nyi complexity\u2014entropy curves","volume":"498","author":"Jauregui","year":"2018","journal-title":"Phys. A"},{"key":"B24","doi-asserted-by":"publisher","first-page":"341","DOI":"10.1063\/1.1427481","article-title":"The world according to r\u00e9nyi: thermodynamics of fractal systems","volume":"597","author":"Jizba","year":"2001","journal-title":"AIP Conf. Proc"},{"key":"B25","doi-asserted-by":"publisher","first-page":"e79950","DOI":"10.7554\/eLife.79950","article-title":"Scale-free behavioral dynamics directly linked with scale-free cortical dynamics","volume":"12","author":"Jones","year":"2023","journal-title":"eLife"},{"key":"B26","doi-asserted-by":"publisher","first-page":"2427","DOI":"10.1890\/06-1736.1","article-title":"Partitioning diversity into independent alpha and beta components","volume":"88","author":"Jost","year":"2007","journal-title":"Ecology"},{"key":"B27","doi-asserted-by":"publisher","first-page":"40","DOI":"10.3389\/fnhum.2018.00040","article-title":"Distinguishing anesthetized from awake state in patients: a new approach using one second segments of raw EEG","volume":"12","author":"Juel","year":"2018","journal-title":"Front. Hum. Neurosci"},{"key":"B28","doi-asserted-by":"publisher","first-page":"1217","DOI":"10.1080\/03640210801944898","article-title":"The pervasiveness of 1\/f scaling in speech reflects the metastable basis of cognition","volume":"32","author":"Kello","year":"2008","journal-title":"Cogn. Sci"},{"key":"B29","doi-asserted-by":"publisher","DOI":"10.1523\/eneuro.0094-22.2022","article-title":"Sources of variation in the spectral slope of the sleep EEG","author":"Kozhemiako","year":"2022","journal-title":"Eneuro"},{"key":"B30","doi-asserted-by":"publisher","first-page":"119","DOI":"10.1016\/j.physa.2003.11.005","article-title":"Intensive entropic non-triviality measure","volume":"334","author":"Lamberti","year":"2004","journal-title":"Phys. A"},{"key":"B31","doi-asserted-by":"publisher","first-page":"1081","DOI":"10.1097\/aln.0b013e3181f229b5","article-title":"Brain networks maintain a scale-free organization across consciousness, anesthesia, and recovery","volume":"113","author":"Lee","year":"2010","journal-title":"Anesthesiology"},{"key":"B32","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1016\/0167-2789(94)90016-7","article-title":"High-dimensional chaos in delayed dynamical systems","volume":"70","author":"Lepri","year":"1994","journal-title":"Phys D"},{"key":"B33","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1016\/0375-9601(95)00867-5","article-title":"A statistical measure of complexity","volume":"209","author":"L\u00f3pez-Ruiz","year":"1995","journal-title":"Phys. Lett. A"},{"key":"B34","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1016\/j.physrep.2013.11.002","article-title":"Power laws and self-organized criticality in theory and nature","volume":"536","author":"Markovi\u0107","year":"2014","journal-title":"Phys. Rep"},{"key":"B35","doi-asserted-by":"publisher","first-page":"439","DOI":"10.1016\/j.physa.2005.11.053","article-title":"Generalized statistical complexity measures: geometrical and analytical properties","volume":"369","author":"Martin","year":"2006","journal-title":"Phys. A"},{"key":"B36","unstructured":"Find Periodicity Using Autocorrelation2024"},{"key":"B37","doi-asserted-by":"publisher","first-page":"21700","DOI":"10.1101\/2020.09.15.298497v3","article-title":"Complexity and 1\/f slope jointly reflect brain states","volume":"13","author":"Medel","year":"2023","journal-title":"Sci. Rep"},{"key":"B38","doi-asserted-by":"publisher","first-page":"e1000609","DOI":"10.1338\/journal.pcbi.1000609","article-title":"Power-law scaling in the brain surface electric potential","volume":"5","author":"Miller","year":"2009","journal-title":"PLoS Comput. Biol"},{"key":"B39","doi-asserted-by":"publisher","first-page":"4677","DOI":"10.3390\/e16084677","article-title":"Entropy-complexity characterization of brain development in chickens","volume":"16","author":"Montani","year":"2014","journal-title":"Entropy"},{"key":"B40","doi-asserted-by":"publisher","DOI":"10.2139\/ssrn.1911068","article-title":"Augmented dickey fuller test","author":"Mushtaq","year":"2011","journal-title":"SSRN Electr. J"},{"key":"B41","doi-asserted-by":"publisher","first-page":"1","DOI":"10.5772\/intechopen.88107","article-title":"Informational time causal planes: a tool for chaotic map dynamic visualization","volume":"2020","author":"Olivares","year":"2020","journal-title":"Nonlin. Syst. Theoret. Aspect. Recent Appl"},{"key":"B42","doi-asserted-by":"publisher","first-page":"729","DOI":"10.1016\/j.tins.2018.08.008","article-title":"Roles of brain criticality and multiscale oscillations in temporal predictions for sensorimotor processing","volume":"41","author":"Palva","year":"2018","journal-title":"Trends Neurosci"},{"key":"B43","doi-asserted-by":"publisher","first-page":"49901","DOI":"10.1063\/5.0049901","article-title":"ORDPY: a python package for data analysis with permutation entropy and ordinal network methods","volume":"31","author":"Pessa","year":"2021","journal-title":"Chaos"},{"key":"B44","doi-asserted-by":"publisher","DOI":"10.3389\/fphy.2021.639389","article-title":"Self-organized criticality in the brain","author":"Plenz","year":"2021","journal-title":"Front. Phys"},{"key":"B45","first-page":"547","article-title":"On measures of entropy and information","volume":"4","author":"R\u00e9nyi","year":"1961","journal-title":"Berkeley Symp. Math. Stat. Probabil"},{"key":"B46","doi-asserted-by":"publisher","first-page":"e062106","DOI":"10.1103\/PhysRevE.95.062106","article-title":"Characterizing time series via complexity-entropy curves","volume":"95","author":"Ribeiro","year":"2017","journal-title":"Phys. Rev. E"},{"key":"B47","doi-asserted-by":"publisher","DOI":"10.1103\/physrevlett.99.154102","article-title":"Distinguishing noise from chaos","author":"Rosso","year":"2007","journal-title":"Phys. Rev. Lett"},{"key":"B48","doi-asserted-by":"publisher","first-page":"476","DOI":"10.1007\/s10439-007-9428-1","article-title":"Estimation of breathing rate from respiratory sinus arrhythmia: comparison of various methods","volume":"36","author":"Sch\u00e4fer","year":"2008","journal-title":"Ann. Biomed. Eng"},{"key":"B49","doi-asserted-by":"publisher","first-page":"989262","DOI":"10.3389\/fninf.2022.989262","article-title":"Scale-free and oscillatory spectral measures of sleep stages in humans","volume":"16","author":"Schneider","year":"2022","journal-title":"Front. Neuroinformat"},{"key":"B50","doi-asserted-by":"publisher","first-page":"20150324","DOI":"10.1098\/rspa.2015.0324","article-title":"Dynamics of non-stationary processes that follow the maximum of the r\u00e9nyi entropy principle","volume":"472","author":"Shalymov","year":"2016","journal-title":"Proc. Royal Soc. A"},{"key":"B51","doi-asserted-by":"publisher","first-page":"2417","DOI":"10.1016\/j.neuropsychologia.2010.03.004","article-title":"Sex and individual differences in induced and evoked EEG measures of action observation","volume":"48","author":"Silas","year":"2010","journal-title":"Neuropsychologia"},{"key":"B52","doi-asserted-by":"publisher","first-page":"252","DOI":"10.1109\/JQE.2010.2078799","article-title":"Time scales of a chaotic semiconductor laser with optical feedback under the lens of a permutation information analysis","volume":"47","author":"Soriano","year":"2011","journal-title":"IEEE J. Quant. Electr"},{"key":"B53","doi-asserted-by":"publisher","first-page":"501","DOI":"10.1007\/s11571-018-9491-3","article-title":"The informational entropy endowed in cortical oscillations","volume":"12","author":"Tozzi","year":"2018","journal-title":"Cogn. Neurodyn"},{"key":"B54","doi-asserted-by":"publisher","first-page":"289","DOI":"10.1002\/ana.25651","article-title":"How the human brain sleeps: direct cortical recordings of normal brain activity","volume":"87","author":"von Ellenrieder","year":"2020","journal-title":"Ann. Neurol"},{"key":"B55","doi-asserted-by":"publisher","first-page":"2312913120","DOI":"10.1073\/pnas.2312913120","article-title":"Hormonal basis of sex differences in anesthetic sensitivity","volume":"121","author":"Wasilczuk","year":"2024","journal-title":"Proc. Natl. Acad. Sci. U. S. A"},{"key":"B56","doi-asserted-by":"publisher","first-page":"696","DOI":"10.1038\/s42005-021-00696-z","article-title":"Ordinal patterns-based methodologies for distinguishing chaos from noise in discrete time series","volume":"4","author":"Zanin","year":"2021","journal-title":"Commun. Phys"},{"key":"B57","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s11128-022-03570-1","article-title":"Relations between the observational entropy and r\u00e9nyi information measures","volume":"21","author":"Zhou","year":"2022","journal-title":"Quant. Inform. Process"},{"key":"B58","volume-title":"ERF and Scale-Free Analyses of Source-Reconstructed MEG Brain Signals During a Multisensory Learning Paradigm","author":"Zilber","year":"2014"},{"key":"B59","doi-asserted-by":"crossref","first-page":"998","DOI":"10.1109\/ISBI.2013.6556645","article-title":"\u201cLearning-induced modulation of scale-free properties of brain activity measured with MEG,\u201d","volume-title":"2013 IEEE 10th International Symposium on Biomedical Imaging","author":"Zilber","year":"2013"},{"key":"B60","doi-asserted-by":"publisher","first-page":"135","DOI":"10.1016\/j.camwa.2013.01.017","article-title":"Entropy of fractal systems","volume":"66","author":"Zmeskal","year":"2013","journal-title":"Comput. Math. Appl"},{"key":"B61","doi-asserted-by":"crossref","first-page":"24","DOI":"10.1109\/IRANIANCEE.2010.5507110","article-title":"\u201cAnalyzing autocorrelation fluctuation of EEG signal for estimating depth of anesthesia,\u201d","volume-title":"2010 18th Iranian Conference on Electrical Engineering","author":"Zoughi","year":"2010"},{"key":"B62","doi-asserted-by":"publisher","first-page":"e045310","DOI":"10.1103\/PhysRevE.105.045310","article-title":"Permutation jensen-shannon distance: a versatile and fast symbolic tool for complex time-series analysis","volume":"105","author":"Zunino","year":"2022","journal-title":"Phys. Rev. E"},{"key":"B63","doi-asserted-by":"publisher","first-page":"e046210","DOI":"10.1103\/PhysRevE.86.046210","article-title":"Distinguishing chaotic and stochastic dynamics from time series by using a multiscale symbolic approach","volume":"86","author":"Zunino","year":"2012","journal-title":"Phys. Rev. E"},{"key":"B64","unstructured":"ZuninoL.\n            SorianoM. C.\n            FischerI.\n            RossoO. A.\n            MirassoC. R.\n          Delay System Identification Using Permutation Entropy and Statistical Complexity: Resonance-Like Behavior in a Noise Environment"},{"key":"B65","doi-asserted-by":"publisher","first-page":"e046212","DOI":"10.1103\/PhysRevE.82.046212","article-title":"Permutation-information-theory approach to unveil delay dynamics from time-series analysis","volume":"82","author":"Zunino","year":"","journal-title":"Phys. Rev. E"}],"container-title":["Frontiers in Computational Neuroscience"],"original-title":[],"link":[{"URL":"https:\/\/www.frontiersin.org\/articles\/10.3389\/fncom.2024.1342985\/full","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,7,15]],"date-time":"2024-07-15T05:03:07Z","timestamp":1721019787000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.frontiersin.org\/articles\/10.3389\/fncom.2024.1342985\/full"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,7,15]]},"references-count":65,"alternative-id":["10.3389\/fncom.2024.1342985"],"URL":"https:\/\/doi.org\/10.3389\/fncom.2024.1342985","relation":{},"ISSN":["1662-5188"],"issn-type":[{"value":"1662-5188","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,7,15]]},"article-number":"1342985"}}