{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,28]],"date-time":"2025-11-28T12:39:09Z","timestamp":1764333549600,"version":"build-2065373602"},"reference-count":62,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2025,4,25]],"date-time":"2025-04-25T00:00:00Z","timestamp":1745539200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The present study proposes a multiscale analysis of the simplicial complex approximate entropy (MS-SCAE) applied to cardiac interbeat series. The MS-SCAE method is based on quantifying the changes in the simplicial complex associated with the time series when a coarse-grained procedure is performed. Our findings are consistent with those of previously reported studies, which indicate that the complexity of healthy interbeat dynamics remains relatively stable over different scales. However, these dynamics undergo changes in the presence of certain cardiac pathologies, such as congestive heart failure and atrial fibrillation. The method we present here allows for effective differentiation between different dynamics and is robust in its ability to characterize both real and simulated sequences. This makes it a suitable candidate for application to a variety of complex signals.<\/jats:p>","DOI":"10.3390\/e27050467","type":"journal-article","created":{"date-parts":[[2025,4,25]],"date-time":"2025-04-25T08:02:57Z","timestamp":1745568177000},"page":"467","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Multiscale Simplicial Complex Entropy Analysis of Heartbeat Dynamics"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2578-7810","authenticated-orcid":false,"given":"Alvaro","family":"Zabaleta-Ortega","sequence":"first","affiliation":[{"name":"Laboratorio de Sistemas Complejos, Unidad Interdisciplinaria en Ingenier\u00eda y Tecnolog\u00edas Avanzadas, Instituto Polit\u00e9cnico Nacional, Av. IPN No. 2580, L. Ticom\u00e1n, Ciudad de M\u00e9xico 07340, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2474-5264","authenticated-orcid":false,"given":"Carlos","family":"Carrizales-Velazquez","sequence":"additional","affiliation":[{"name":"Laboratorio de Sistemas Complejos, Unidad Interdisciplinaria en Ingenier\u00eda y Tecnolog\u00edas Avanzadas, Instituto Polit\u00e9cnico Nacional, Av. IPN No. 2580, L. Ticom\u00e1n, Ciudad de M\u00e9xico 07340, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3079-3166","authenticated-orcid":false,"given":"Bibiana","family":"Obreg\u00f3n-Quintana","sequence":"additional","affiliation":[{"name":"Facultad de Ciencias, Universidad Nacional Aut\u00f3noma de M\u00e9xico, Ciudad de M\u00e9xico 04510, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3098-8796","authenticated-orcid":false,"given":"Lev","family":"Guzm\u00e1n-Vargas","sequence":"additional","affiliation":[{"name":"Laboratorio de Sistemas Complejos, Unidad Interdisciplinaria en Ingenier\u00eda y Tecnolog\u00edas Avanzadas, Instituto Polit\u00e9cnico Nacional, Av. IPN No. 2580, L. Ticom\u00e1n, Ciudad de M\u00e9xico 07340, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,4,25]]},"reference":[{"key":"ref_1","unstructured":"Bassingthwaighte, J.B., Liebovitch, L.S., and West, B.J. (2013). Fractal Physiology, Springer."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"617","DOI":"10.1103\/RevModPhys.57.617","article-title":"Ergodic theory of chaos and strange attractors","volume":"57","author":"Eckmann","year":"1985","journal-title":"Rev. Mod. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2297","DOI":"10.1073\/pnas.88.6.2297","article-title":"Approximate entropy as a measure of system complexity","volume":"88","author":"Pincus","year":"1991","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"H2039","DOI":"10.1152\/ajpheart.2000.278.6.H2039","article-title":"Physiological time-series analysis using approximate entropy and sample entropy","volume":"278","author":"Richman","year":"2000","journal-title":"Am. J. Physiol.-Heart Circ. Physiol."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"110","DOI":"10.1063\/1.166092","article-title":"Approximate entropy (ApEn) as a complexity measure","volume":"5","author":"Pincus","year":"1995","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"021906","DOI":"10.1103\/PhysRevE.71.021906","article-title":"Multiscale entropy analysis of biological signals","volume":"71","author":"Costa","year":"2005","journal-title":"Phys. Rev. E"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"R789","DOI":"10.1152\/ajpregu.00069.2002","article-title":"Sample entropy analysis of neonatal heart rate variability","volume":"283","author":"Lake","year":"2002","journal-title":"Am. J. Physiol.-Regul. Integr. Comp. Physiol."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"659","DOI":"10.1113\/expphysiol.2007.037150","article-title":"Cross-sample entropy statistic as a measure of complexity and regularity of renal sympathetic nerve activity in the rat","volume":"92","author":"Zhang","year":"2007","journal-title":"Exp. Physiol."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1483","DOI":"10.1007\/s10439-010-9933-5","article-title":"Fuzzy approximate entropy analysis of chaotic and natural complex systems: Detecting muscle fatigue using electromyography signals","volume":"38","author":"Xie","year":"2010","journal-title":"Ann. Biomed. Eng."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Faust, O., and Bairy, M.G. (2012). Nonlinear analysis of physiological signals: A review. J. Mech. Med. Biol., 12.","DOI":"10.1142\/S0219519412400155"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1188","DOI":"10.1016\/j.medengphy.2012.12.010","article-title":"Bivariate piecewise stationary segmentation; improved pre-treatment for synchronization measures used on non-stationary biological signals","volume":"35","author":"Terrien","year":"2013","journal-title":"Med. Eng. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Liang, Z., Wang, Y., Sun, X., Li, D., Voss, L.J., Sleigh, J.W., Hagihira, S., and Li, X. (2015). EEG entropy measures in anesthesia. Front. Comput. Neurosci., 9.","DOI":"10.3389\/fncom.2015.00016"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Aguilar-Vel\u00e1zquez, D., and Guzm\u00e1n-Vargas, L. (2019). Critical synchronization and 1\/f noise in inhibitory\/excitatory rich-club neural networks. Sci. Rep., 9.","DOI":"10.1038\/s41598-018-37920-w"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Delgado-Bonal, A., and Marshak, A. (2019). Approximate Entropy and Sample Entropy: A Comprehensive Tutorial. Entropy, 21.","DOI":"10.3390\/e21060541"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Jamin, A., Duval, G., Annweiler, C., Abraham, P., and Humeau-Heurtier, A. (2020, January 9\u201312). Study of the influence of Age: Use of Sample Entropy and CEEMDAN on Navigation Data Acquired from a Bike Simulator. Proceedings of the 2020 Tenth International Conference on Image Processing Theory, Tools and Applications (IPTA), Paris, France.","DOI":"10.1109\/IPTA50016.2020.9286648"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"420","DOI":"10.1037\/cou0000383","article-title":"Physiological synchronization in the clinical process: A research primer","volume":"67","author":"Kleinbub","year":"2020","journal-title":"J. Couns. Psychol."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1343","DOI":"10.1103\/PhysRevLett.70.1343","article-title":"Long-range anticorrelations and non-Gaussian behavior of the heartbeat","volume":"70","author":"Peng","year":"1993","journal-title":"Phys. Rev. Lett."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Ivanov, P.C. (2003). Long-range dependence in heartbeat dynamics. Processes with Long-Range Correlations: Theory and Applications, Springer.","DOI":"10.1007\/3-540-44832-2_19"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1247","DOI":"10.1007\/s00249-007-0254-z","article-title":"Correlation properties of heartbeat dynamics","volume":"37","author":"Platisa","year":"2008","journal-title":"Eur. Biophys. J."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1197","DOI":"10.3390\/e17031197","article-title":"Generalized Multiscale Entropy Analysis: Application to Quantifying the Complex Volatility of Human Heartbeat Time Series","volume":"17","author":"Costa","year":"2015","journal-title":"Entropy"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"052901","DOI":"10.1103\/PhysRevE.67.052901","article-title":"Simple model of the aging effect in heart interbeat time series","volume":"67","year":"2003","journal-title":"Phys. Rev. E"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"304","DOI":"10.1016\/j.physa.2004.09.019","article-title":"Influence of the loss of time-constants repertoire in pathologic heartbeat dynamics","volume":"348","year":"2005","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"6026","DOI":"10.1103\/PhysRevLett.86.6026","article-title":"Behavioral-independent features of complex heartbeat dynamics","volume":"86","author":"Amaral","year":"2001","journal-title":"Phys. Rev. Lett."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1038\/20924","article-title":"Multifractality in human heartbeat dynamics","volume":"399","author":"Ivanov","year":"1999","journal-title":"Nature"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Nardelli, M., Lanata, A., Bertschy, G., Scilingo, E.P., and Valenza, G. (2017). Heartbeat complexity modulation in bipolar disorder during daytime and nighttime. Sci. Rep., 7.","DOI":"10.1038\/s41598-017-18036-z"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"021906","DOI":"10.1103\/PhysRevE.85.021906","article-title":"Modified permutation-entropy analysis of heartbeat dynamics","volume":"85","author":"Bian","year":"2012","journal-title":"Phys. Rev. E\u2014Stat. Nonlinear Soft Matter Phys."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Shi, B., Zhang, Y., Yuan, C., Wang, S., and Li, P. (2017). Entropy analysis of short-term heartbeat interval time series during regular walking. Entropy, 19.","DOI":"10.3390\/e19100568"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1007\/s11517-014-1216-0","article-title":"Assessing the complexity of short-term heartbeat interval series by distribution entropy","volume":"53","author":"Li","year":"2015","journal-title":"Med. Biol. Eng. Comput."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1109\/TBME.2005.859782","article-title":"Renyi entropy measures of heart rate Gaussianity","volume":"53","author":"Lake","year":"2005","journal-title":"IEEE Trans. Biomed. Eng."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1282","DOI":"10.1109\/10.959324","article-title":"Entropy, entropy rate, and pattern classification as tools to typify complexity in short heart period variability series","volume":"48","author":"Porta","year":"2002","journal-title":"IEEE Trans. Biomed. Eng."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"068102","DOI":"10.1103\/PhysRevLett.89.068102","article-title":"Multiscale Entropy Analysis of Complex Physiologic Time Series","volume":"89","author":"Costa","year":"2002","journal-title":"Phys. Rev. Lett."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"855","DOI":"10.5194\/nhess-8-855-2008","article-title":"Multiscale entropy analysis of electroseismic time series","volume":"8","year":"2008","journal-title":"Nat. Hazards Earth Syst. Sci."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1250033","DOI":"10.1142\/S0219477512500332","article-title":"Multiscale entropy analysis of financial time series","volume":"11","author":"Xia","year":"2012","journal-title":"Fluct. Noise Lett."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"224","DOI":"10.1016\/j.sigpro.2018.02.004","article-title":"Two-dimensional multiscale entropy analysis: Applications to image texture evaluation","volume":"147","author":"Silva","year":"2018","journal-title":"Signal Process."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"10713","DOI":"10.1002\/pamm.200810713","article-title":"Vertical Vibrations of a Vehicle Excited by Real Road Profiles","volume":"Volume 8","author":"Borowiec","year":"2008","journal-title":"PAMM: Proceedings in Applied Mathematics and Mechanics"},{"key":"ref_36","unstructured":"Costa, M., Goldberger, A., and Peng, C.K. (2002, January 22\u201325). Multiscale entropy to distinguish physiologic and synthetic RR time series. Proceedings of the Computers in Cardiology, Memphis, TN, USA."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1069","DOI":"10.3390\/e15031069","article-title":"Time Series Analysis Using Composite Multiscale Entropy","volume":"15","author":"Wu","year":"2013","journal-title":"Entropy"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"2202","DOI":"10.1109\/TBME.2009.2021986","article-title":"Refined multiscale entropy: Application to 24-h holter recordings of heart period variability in healthy and aortic stenosis subjects","volume":"56","author":"Valencia","year":"2009","journal-title":"IEEE Trans. Biomed. Eng."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"975","DOI":"10.1016\/j.physa.2017.08.047","article-title":"Refined generalized multiscale entropy analysis for physiological signals","volume":"490","author":"Liu","year":"2018","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Lee, D.Y., and Choi, Y.S. (2018). Multiscale Distribution Entropy Analysis of Short-Term Heart Rate Variability. Entropy, 20.","DOI":"10.3390\/e20120952"},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Shi, M., Shi, Y., Lin, Y., and Qi, X. (2024). Modified multiscale Renyi distribution entropy for short-term heart rate variability analysis. BMC Med. Inform. Decis. Mak., 24.","DOI":"10.1186\/s12911-024-02763-1"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"3110","DOI":"10.3390\/e17053110","article-title":"The Multiscale Entropy Algorithm and Its Variants: A Review","volume":"17","year":"2015","journal-title":"Entropy"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"5865","DOI":"10.1016\/j.physa.2013.07.075","article-title":"Modified multiscale entropy for short-term time series analysis","volume":"392","author":"Wu","year":"2013","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Guzman-Vargas, L., Zabaleta-Ortega, A., and Guzman-Saenz, A. (2023). Simplicial complex entropy for time series analysis. Sci. Rep., 13.","DOI":"10.1038\/s41598-023-49958-6"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"511","DOI":"10.1007\/s00454-002-2885-2","article-title":"Topological Persistence and Simplification","volume":"28","author":"Edelsbrunner","year":"2002","journal-title":"Discret. Comput. Geom."},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Zomorodian, A., and Carlsson, G. (2004, January 8\u201311). Computing Persistent Homology. Proceedings of the Twentieth Annual Symposium on Computational Geometry, New York, NY, USA.","DOI":"10.1145\/997817.997870"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1142\/S0218654305000761","article-title":"Persistence barcodes for shapes","volume":"11","author":"Carlsson","year":"2005","journal-title":"Int. J. Shape Model."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1090\/conm\/453\/08802","article-title":"Persistent homology-a survey","volume":"453","author":"Edelsbrunner","year":"2008","journal-title":"Contemp. Math."},{"key":"ref_49","unstructured":"Edelsbrunner, H., and Morozov, D. (2013). Persistent Homology: Theory and Practice, California Digital Library."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1090\/S0273-0979-09-01249-X","article-title":"Topology and data","volume":"46","author":"Carlsson","year":"2009","journal-title":"Bull. Am. Math. Soc."},{"key":"ref_51","first-page":"39","article-title":"Topological Data Analysis","volume":"70","author":"Zomorodian","year":"2012","journal-title":"Adv. Appl. Comput. Topol."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"501","DOI":"10.1146\/annurev-statistics-031017-100045","article-title":"Topological Data Analysis","volume":"5","author":"Wasserman","year":"2018","journal-title":"Annu. Rev. Stat. Its Appl."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"3061","DOI":"10.1080\/03610929208830963","article-title":"Approximate entropy: Statistical properties and applications","volume":"21","author":"Pincus","year":"1992","journal-title":"Commun. Stat.\u2014Theory Methods"},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"Dey, T.K., and Wang, Y. (2022). Computational Topology for Data Analysis, Cambridge University Press.","DOI":"10.1017\/9781009099950"},{"key":"ref_55","unstructured":"Munkres, J. (1984). Elements of Algebraic Topology, Library of Congress Cataloguing in Publication Data, Addison-Wesley Publishing Company. [2nd ed.]."},{"key":"ref_56","unstructured":"Hatcher, A. (2001). Algebraic Topology, Cornell University Press. [1st ed.]."},{"key":"ref_57","unstructured":"Munkres, J. (2014). Topology, British Library Cataloguing-in-Publication Data, Pearson Education Limited. [2nd ed.]."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"e215","DOI":"10.1161\/01.CIR.101.23.e215","article-title":"PhysioBank, PhysioToolkit, and PhysioNet: Components of a new research resource for complex physiologic signals","volume":"101","author":"Goldberger","year":"2000","journal-title":"Circulation"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"661","DOI":"10.1016\/S0735-1097(86)80478-8","article-title":"Survival of patients with severe congestive heart failure treated with oral milrinone","volume":"7","author":"Baim","year":"1986","journal-title":"J. Am. Coll. Cardiol."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"378","DOI":"10.1136\/heart.88.4.378","article-title":"The pNNx files: Re-examining a widely used heart rate variability measure","volume":"88","author":"Mietus","year":"2002","journal-title":"Heart"},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"1101","DOI":"10.1016\/S0002-9149(97)00616-4","article-title":"Long-term carvedilol therapy increases parasympathetic nervous system activity in chronic congestive heart failure","volume":"80","author":"Goldsmith","year":"1997","journal-title":"Am. J. Cardiol."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"38008","DOI":"10.1209\/0295-5075\/89\/38008","article-title":"Scaling properties of excursions in heartbeat dynamics","volume":"89","year":"2010","journal-title":"Europhys. Lett."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/5\/467\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:21:32Z","timestamp":1760030492000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/5\/467"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,25]]},"references-count":62,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2025,5]]}},"alternative-id":["e27050467"],"URL":"https:\/\/doi.org\/10.3390\/e27050467","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2025,4,25]]}}}