{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,16]],"date-time":"2026-09-16T07:09:44Z","timestamp":1789542584629,"version":"build-2803163510"},"reference-count":61,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2015,1,12]],"date-time":"2015-01-12T00:00:00Z","timestamp":1421020800000},"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>In the framework of information dynamics, the temporal evolution of coupled systems can be studied by decomposing the predictive information about an assigned target system into amounts quantifying the information stored inside the system and the information transferred to it. While information storage and transfer are computed through the known self-entropy (SE) and transfer entropy (TE), an alternative decomposition evidences the so-called cross entropy (CE) and conditional SE (cSE), quantifying the cross information and internal information of the target system, respectively. This study presents a thorough evaluation of SE, TE, CE and cSE as quantities related to the causal statistical structure of coupled dynamic processes. First, we investigate the theoretical properties of these measures, providing the conditions for their existence and assessing the meaning of the information theoretic quantity that each of them reflects. Then, we present an approach for the exact computation of information dynamics based on the linear Gaussian approximation, and exploit this approach to characterize the behavior of SE, TE, CE and cSE in benchmark systems with known dynamics. Finally, we exploit these measures to study cardiorespiratory dynamics measured from healthy subjects during head-up tilt and paced breathing protocols. Our main result is that the combined evaluation of the measures of information dynamics allows to infer the causal effects associated with the observed dynamics and to interpret the alteration of these effects with changing experimental conditions.<\/jats:p>","DOI":"10.3390\/e17010277","type":"journal-article","created":{"date-parts":[[2015,1,12]],"date-time":"2015-01-12T13:30:46Z","timestamp":1421069446000},"page":"277-303","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":138,"title":["Information Decomposition in Bivariate Systems: Theory and Application to Cardiorespiratory Dynamics"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3271-5348","authenticated-orcid":false,"given":"Luca","family":"Faes","sequence":"first","affiliation":[{"name":"BIOtech, Department of Industrial Engineering, University of Trento, and IRCS PAT-FBK Trento, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6720-9824","authenticated-orcid":false,"given":"Alberto","family":"Porta","sequence":"additional","affiliation":[{"name":"Department of Biomedical Sciences for Health, University of Milan, Via R. Galeazzi 4, 20161 Milan, Italy"},{"name":"IRCCS Galeazzi Orthopedic Institute, Via R. Galeazzi 4, 20161 Milan, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9837-083X","authenticated-orcid":false,"given":"Giandomenico","family":"Nollo","sequence":"additional","affiliation":[{"name":"BIOtech, Department of Industrial Engineering, University of Trento, and IRCS PAT-FBK Trento, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2015,1,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"669","DOI":"10.1113\/jphysiol.2002.017483","article-title":"Short-term cardiovascular oscillations in man: measuring and modelling the physiologies","volume":"542","author":"Cohen","year":"2002","journal-title":"J. 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