{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:23:19Z","timestamp":1760242999117,"version":"build-2065373602"},"reference-count":21,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2015,2,26]],"date-time":"2015-02-26T00:00:00Z","timestamp":1424908800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"European Community's Seventh Framework Programme FP7\/2007-2013","award":["HEALTH-F2-2009-241526"],"award-info":[{"award-number":["HEALTH-F2-2009-241526"]}]},{"DOI":"10.13039\/501100002347","name":"BMBF","doi-asserted-by":"publisher","award":["FKZ 031A147"],"award-info":[{"award-number":["FKZ 031A147"]}],"id":[{"id":"10.13039\/501100002347","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100004875","name":"DFG","doi-asserted-by":"publisher","award":["SFB 1002"],"award-info":[{"award-number":["SFB 1002"]}],"id":[{"id":"10.13039\/100004875","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100010447","name":"DZHK","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100010447","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The characterization of spatiotemporal complexity remains a challenging task. This holds in particular for the analysis of data from fluorescence imaging (optical mapping), which allows for the measurement of membrane potential and intracellular calcium at high spatial and temporal resolutions and, therefore, allows for an investigation of cardiac dynamics. Dominant frequency maps and the analysis of phase singularities are frequently used for this type of excitable media. These methods address some important aspects of cardiac dynamics; however, they only consider very specific properties of excitable media. To extend the scope of the analysis, we present a measure based on entropy rates for determining spatiotemporal complexity patterns of excitable media. Simulated data generated by the Aliev\u2013Panfilov model and the cubic Barkley model are used to validate this method. Then, we apply it to optical mapping data from monolayers of cardiac cells from chicken embryos and compare our findings with dominant frequency maps and the analysis of phase singularities. The studies indicate that entropy rate maps provide additional information about local complexity, the origins of wave breakup and the development of patterns governing unstable wave propagation.<\/jats:p>","DOI":"10.3390\/e17030950","type":"journal-article","created":{"date-parts":[[2015,2,26]],"date-time":"2015-02-26T10:07:43Z","timestamp":1424945263000},"page":"950-967","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Entropy Rate Maps of Complex Excitable Dynamics in Cardiac Monolayers"],"prefix":"10.3390","volume":"17","author":[{"given":"Alexander","family":"Schlemmer","sequence":"first","affiliation":[{"name":"Max Planck Institute for Dynamics and Self-Organization, Am Fa\u00dfberg 17, 37077 G\u00f6ttingen, Germany"},{"name":"Institute for Nonlinear Dynamics, Georg-August-Universit\u00e4t G\u00f6ttingen, Am Fa\u00dfberg 17,37077 G\u00f6ttingen, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sebastian","family":"Berg","sequence":"additional","affiliation":[{"name":"Max Planck Institute for Dynamics and Self-Organization, Am Fa\u00dfberg 17, 37077 G\u00f6ttingen, Germany"},{"name":"Institute for Nonlinear Dynamics, Georg-August-Universit\u00e4t G\u00f6ttingen, Am Fa\u00dfberg 17,37077 G\u00f6ttingen, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"T.","family":"Shajahan","sequence":"additional","affiliation":[{"name":"Max Planck Institute for Dynamics and Self-Organization, Am Fa\u00dfberg 17, 37077 G\u00f6ttingen, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Stefan","family":"Luther","sequence":"additional","affiliation":[{"name":"Max Planck Institute for Dynamics and Self-Organization, Am Fa\u00dfberg 17, 37077 G\u00f6ttingen, Germany"},{"name":"Institute for Nonlinear Dynamics, Georg-August-Universit\u00e4t G\u00f6ttingen, Am Fa\u00dfberg 17,37077 G\u00f6ttingen, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ulrich","family":"Parlitz","sequence":"additional","affiliation":[{"name":"Max Planck Institute for Dynamics and Self-Organization, Am Fa\u00dfberg 17, 37077 G\u00f6ttingen, Germany"},{"name":"Institute for Nonlinear Dynamics, Georg-August-Universit\u00e4t G\u00f6ttingen, Am Fa\u00dfberg 17,37077 G\u00f6ttingen, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2015,2,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"125016","DOI":"10.1088\/1367-2630\/10\/12\/125016","article-title":"Visualization of spiral and scroll waves in simulated and experimental cardiac tissue","volume":"10","author":"Cherry","year":"2008","journal-title":"New J. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"057201","DOI":"10.1103\/PhysRevE.83.057201","article-title":"Synchronization patterns in transient spiral wave dynamics","volume":"83","author":"Parlitz","year":"2011","journal-title":"Phys. Rev. E"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1038\/32164","article-title":"Spatial and temporal organization during cardiac fibrillation","volume":"392","author":"Gray","year":"1998","journal-title":"Nature"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"033132","DOI":"10.1063\/1.4747709","article-title":"Pacemaker interactions induce reentrant wave dynamics in engineered cardiac culture","volume":"22","author":"Borek","year":"2012","journal-title":"Chaos"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Shajahan, T., Krinski, V., Knyazeva, S., and Luther, S. (2014, January 25\u201328). Eliminating pinned spiral waves in cardiac monolayer by far field pacing. Trento, Italy.","DOI":"10.1109\/ESGCO.2014.6847564"},{"key":"ref_6","unstructured":"Press, W.H., Teukolsky, S.A., Vetterling, W.T., and Flannery, B.P. (2007). Numerical Recipes: The Art of Scientific Computing, Cambridge University Press. [3rd]."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"849","DOI":"10.1161\/CIRCRESAHA.111.300158","article-title":"Rotors and the Dynamics of Cardiac Fibrillation","volume":"112","author":"Pandit","year":"2013","journal-title":"Circ. Res."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"051916","DOI":"10.1103\/PhysRevE.85.051916","article-title":"Reconstruction of phase maps from the configuration of phase singularities in two-dimensional manifolds","volume":"85","author":"Herlin","year":"2012","journal-title":"Phys. Rev. E"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1114\/1.1335538","article-title":"An Experimentalist\u2019s Approach to Accurate Localization of Phase Singularities during Reentry","volume":"29","author":"Iyer","year":"2001","journal-title":"Ann. Biomed. Eng."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"56","DOI":"10.1109\/TBME.2003.820341","article-title":"Combined phase singularity and wavefront analysis for optical maps of ventricular fibrillation","volume":"51","author":"Rogers","year":"2004","journal-title":"IEEE Bio-Med. Eng."},{"key":"ref_11","first-page":"18","article-title":"Computation of hilbert transform via discrete cosine transform","volume":"1","author":"Olkkonen","year":"2010","journal-title":"J. Signal Inf. Process."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"414","DOI":"10.1063\/1.166191","article-title":"Entropy estimation of symbol sequences","volume":"6","author":"Grassberger","year":"1996","journal-title":"Chaos"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Webber, C.L., and Marwan, N. (2015). Recurrence Quantification Analysis, Springer.","DOI":"10.1007\/978-3-319-07155-8"},{"key":"ref_14","unstructured":"pyEntropyRateMap A Python Algorithm for Computing Entropy Rate Maps. Available online: http:\/\/www.bmp.ds.mpg.de\/software.html."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2095","DOI":"10.1103\/PhysRevE.61.2095","article-title":"Coherent structure analysis of spatiotemporal chaos","volume":"61","author":"Jung","year":"2000","journal-title":"Phys. Rev. E"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"028105","DOI":"10.1103\/PhysRevLett.94.028105","article-title":"Global organization of dynamics in oscillatory heterogeneous excitable media","volume":"94","author":"Bub","year":"2005","journal-title":"Phys. Rev. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1016\/0960-0779(95)00089-5","article-title":"A Simple Two-variable Model of Cardiac Excitation","volume":"7","author":"Aliev","year":"1996","journal-title":"Chaos Solitons Fractals"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"013502","DOI":"10.1063\/1.1840311","article-title":"Modeling wave propagation in realistic heart geometries using the phase-field method","volume":"15","author":"Fenton","year":"2005","journal-title":"Chaos"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1016\/0167-2789(91)90194-E","article-title":"A model for fast computer simulation of waves in excitable media","volume":"49","author":"Barkley","year":"1991","journal-title":"Physica D"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Barkley, D. (2008). Barkley model. Scholarpedia, 3.","DOI":"10.4249\/scholarpedia.1877"},{"key":"ref_21","unstructured":"MultiRecorder, a Multiple-Camera Recording and Analysis Software. Available online: http:\/\/www.bmp.ds.mpg.de\/multirecorder.html."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/17\/3\/950\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T20:42:56Z","timestamp":1760215376000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/17\/3\/950"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,2,26]]},"references-count":21,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2015,3]]}},"alternative-id":["e17030950"],"URL":"https:\/\/doi.org\/10.3390\/e17030950","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2015,2,26]]}}}