{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,15]],"date-time":"2026-03-15T03:08:06Z","timestamp":1773544086107,"version":"3.50.1"},"reference-count":56,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2019,2,14]],"date-time":"2019-02-14T00:00:00Z","timestamp":1550102400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["GRK1772"],"award-info":[{"award-number":["GRK1772"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The entropy production in stochastic dynamical systems is linked to the structure of their causal representation in terms of Bayesian networks. Such a connection was formalized for bipartite (or multipartite) systems with an integral fluctuation theorem in [Phys. Rev. Lett. 111, 180603 (2013)]. Here we introduce the information thermodynamics for time series, that are non-bipartite in general, and we show that the link between irreversibility and information can only result from an incomplete causal representation. In particular, we consider a backward transfer entropy lower bound to the conditional time series irreversibility that is induced by the absence of feedback in signal-response models. We study such a relation in a linear signal-response model providing analytical solutions, and in a nonlinear biological model of receptor-ligand systems where the time series irreversibility measures the signaling efficiency.<\/jats:p>","DOI":"10.3390\/e21020177","type":"journal-article","created":{"date-parts":[[2019,2,14]],"date-time":"2019-02-14T11:54:13Z","timestamp":1550145253000},"page":"177","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Information Thermodynamics for Time Series of Signal-Response Models"],"prefix":"10.3390","volume":"21","author":[{"given":"Andrea","family":"Auconi","sequence":"first","affiliation":[{"name":"Theoretische Biophysik, Humboldt-Universit\u00e4t zu Berlin, Invalidenstra\u00dfe 42, D-10115 Berlin, Germany"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8113-8188","authenticated-orcid":false,"given":"Andrea","family":"Giansanti","sequence":"additional","affiliation":[{"name":"Dipartimento di Fisica, Sapienza Universit\u00e0 di Roma, 00185 Rome, Italy"},{"name":"INFN, Sezione di Roma 1, 00185 Rome, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0567-7075","authenticated-orcid":false,"given":"Edda","family":"Klipp","sequence":"additional","affiliation":[{"name":"Theoretische Biophysik, Humboldt-Universit\u00e4t zu Berlin, Invalidenstra\u00dfe 42, D-10115 Berlin, Germany"}]}],"member":"1968","published-online":{"date-parts":[[2019,2,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1146\/annurev-conmatphys-062910-140506","article-title":"Equalities and inequalities: Irreversibility and the second law of thermodynamics at the nanoscale","volume":"2","author":"Jarzynski","year":"2011","journal-title":"Annu. Rev. Condens. Matter Phys."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"073008","DOI":"10.1088\/1367-2630\/11\/7\/073008","article-title":"Entropy production and the arrow of time","volume":"11","author":"Parrondo","year":"2009","journal-title":"New J. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"090602","DOI":"10.1103\/PhysRevLett.101.090602","article-title":"Length of time\u2019s arrow","volume":"101","author":"Feng","year":"2008","journal-title":"Phys. Revi. Lett."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2690","DOI":"10.1103\/PhysRevLett.78.2690","article-title":"Nonequilibrium equality for free energy differences","volume":"78","author":"Jarzynski","year":"1997","journal-title":"Phys. Rev. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2721","DOI":"10.1103\/PhysRevE.60.2721","article-title":"Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences","volume":"60","author":"Crooks","year":"1999","journal-title":"Phys. Rev. E"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1529","DOI":"10.1080\/00018730210155133","article-title":"The fluctuation theorem","volume":"51","author":"Evans","year":"2002","journal-title":"Adv. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"080602","DOI":"10.1103\/PhysRevLett.98.080602","article-title":"Dissipation: The phase-space perspective","volume":"98","author":"Kawai","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1023\/A:1018670721277","article-title":"Hamiltonian derivation of a detailed fluctuation theorem","volume":"98","author":"Jarzynski","year":"2000","journal-title":"J. Stat. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"P08001","DOI":"10.1088\/1742-5468\/2006\/08\/P08001","article-title":"Path-integral analysis of fluctuation theorems for general Langevin processes","volume":"2006","author":"Chernyak","year":"2006","journal-title":"J. Stat. Mech. Theory Exp."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"180603","DOI":"10.1103\/PhysRevLett.111.180603","article-title":"Information thermodynamics on causal networks","volume":"111","author":"Ito","year":"2013","journal-title":"Phys. Rev. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"021104","DOI":"10.1103\/PhysRevE.85.021104","article-title":"Nonequilibrium thermodynamics of feedback control","volume":"85","author":"Sagawa","year":"2012","journal-title":"Phys. Rev. E"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"090602","DOI":"10.1103\/PhysRevLett.104.090602","article-title":"Generalized Jarzynski equality under nonequilibrium feedback control","volume":"104","author":"Sagawa","year":"2010","journal-title":"Phys. Rev. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1002\/bs.3830090402","article-title":"On the decrease of entropy in a thermodynamic system by the intervention of intelligent beings","volume":"9","author":"Szilard","year":"1964","journal-title":"Syst. Res. Behav. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1038\/nphys3518","article-title":"Brownian carnot engine","volume":"12","author":"Dinis","year":"2016","journal-title":"Nat. Phys."},{"key":"ref_15","first-page":"021051","article-title":"Experiments in stochastic thermodynamics: Short history and perspectives","volume":"7","author":"Ciliberto","year":"2017","journal-title":"Phys. Rev. X"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"988","DOI":"10.1038\/nphys1821","article-title":"Experimental demonstration of information-to-energy conversion and validation of the generalized Jarzynski equality","volume":"6","author":"Toyabe","year":"2010","journal-title":"Nat. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"13786","DOI":"10.1073\/pnas.1406966111","article-title":"Experimental realization of a Szilard engine with a single electron","volume":"111","author":"Koski","year":"2014","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_18","first-page":"031015","article-title":"Thermodynamics with continuous information flow","volume":"4","author":"Horowitz","year":"2014","journal-title":"Phys. Rev. X"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"10007","DOI":"10.1209\/0295-5075\/116\/10007","article-title":"Continuous information flow fluctuations","volume":"116","author":"Rosinberg","year":"2016","journal-title":"EPL (Europhysics Letters)"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"P03006","DOI":"10.1088\/1742-5468\/2015\/03\/P03006","article-title":"Multipartite information flow for multiple Maxwell demons","volume":"2015","author":"Horowitz","year":"2015","journal-title":"J. Stat. Mech. Theory Exp."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Ito, S. (2016). Backward transfer entropy: Informational measure for detecting hidden Markov models and its interpretations in thermodynamics, gambling and causality. Sci. Rep., 6.","DOI":"10.1038\/srep36831"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"022135","DOI":"10.1103\/PhysRevE.94.022135","article-title":"Transfer entropy in physical systems and the arrow of time","volume":"94","author":"Spinney","year":"2016","journal-title":"Phys. Rev. E"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1038\/nphys3230","article-title":"Thermodynamics of information","volume":"11","author":"Parrondo","year":"2015","journal-title":"Nat. Phys."},{"key":"ref_24","unstructured":"Ito, S. (arXiv, 2018). Unified framework for the second law of thermodynamics and information thermodynamics based on information geometry, arXiv."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"030605","DOI":"10.1103\/PhysRevLett.121.030605","article-title":"Stochastic thermodynamic interpretation of information geometry","volume":"121","author":"Ito","year":"2018","journal-title":"Phys. Rev. Lett."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"031129","DOI":"10.1103\/PhysRevE.85.031129","article-title":"Entropy production and Kullback-Leibler divergence between stationary trajectories of discrete systems","volume":"85","author":"Parrondo","year":"2012","journal-title":"Phys. Rev. E"},{"key":"ref_27","unstructured":"Cover, T.M., and Thomas, J.A. (2012). Elements of Information Theory, John Wiley & Sons."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1143\/PTPS.130.17","article-title":"Langevin equation and thermodynamics","volume":"130","author":"Sekimoto","year":"1998","journal-title":"Prog. Theor. Phys. Supp."},{"key":"ref_29","unstructured":"Crooks, G.E., and Still, S.E. (arXiv, 2016). Marginal and conditional second laws of thermodynamics, arXiv."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"042315","DOI":"10.1103\/PhysRevE.95.042315","article-title":"Causal influence in linear Langevin networks without feedback","volume":"95","author":"Auconi","year":"2017","journal-title":"Phys. Rev. E"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"126001","DOI":"10.1088\/0034-4885\/75\/12\/126001","article-title":"Stochastic thermodynamics, fluctuation theorems and molecular machines","volume":"75","author":"Seifert","year":"2012","journal-title":"Rep. Prog. Phys."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Shreve, S.E. (2004). Stochastic Calculus for Finance II: Continuous-Time Models, Springer Science & Business Media.","DOI":"10.1007\/978-1-4757-4296-1"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10955-006-9252-2","article-title":"Onsager-Machlup theory for nonequilibrium steady states and fluctuation theorems","volume":"126","author":"Taniguchi","year":"2007","journal-title":"J. Stat. Phys."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"040602","DOI":"10.1103\/PhysRevLett.95.040602","article-title":"Entropy production along a stochastic trajectory and an integral fluctuation theorem","volume":"95","author":"Seifert","year":"2005","journal-title":"Phys. Rev. Lett."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Risken, H. (1996). Fokker-planck equation. The Fokker-Planck Equation, Springer.","DOI":"10.1007\/978-3-642-61544-3"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"125007","DOI":"10.1088\/1367-2630\/16\/12\/125007","article-title":"Second-law-like inequalities with information and their interpretations","volume":"16","author":"Horowitz","year":"2014","journal-title":"New J. Phys."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"011107","DOI":"10.1103\/PhysRevE.78.011107","article-title":"Lower bounds on dissipation upon coarse graining","volume":"78","author":"Parrondo","year":"2008","journal-title":"Phys. Rev. E"},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Ito, S., and Sagawa, T. (2016). Information flow and entropy production on Bayesian networks. Mathematical Foundations and Applications of Graph Entropy, Wiley-VCH Verlag GmbH & Co. KGaA.","DOI":"10.1002\/9783527693245.ch3"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"823","DOI":"10.1103\/PhysRev.36.823","article-title":"On the theory of the Brownian motion","volume":"36","author":"Uhlenbeck","year":"1930","journal-title":"Phys. Rev."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"2084","DOI":"10.1103\/PhysRevE.54.2084","article-title":"Exact numerical simulation of the Ornstein-Uhlenbeck process and its integral","volume":"54","author":"Gillespie","year":"1996","journal-title":"Phys. Rev. E"},{"key":"ref_41","unstructured":"R Core Team (2014). R: A Language and Environment for Statistical Computing, Foundation for Statistical Computing. Available online: www.r-project.org."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"052802","DOI":"10.1103\/PhysRevE.91.052802","article-title":"Exploration of synergistic and redundant information sharing in static and dynamical Gaussian systems","volume":"91","author":"Barrett","year":"2015","journal-title":"Phys. Rev. E"},{"key":"ref_43","unstructured":"Klipp, E., Liebermeister, W., Wierling, C., Kowald, A., and Herwig, R. (2016). Systems Biology: A Textbook, Wiley-VCH Verlag GmbH & Co. KGaA."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1038\/nrm1838","article-title":"Cell-signalling dynamics in time and space","volume":"7","author":"Kholodenko","year":"2006","journal-title":"Nat. Rev. Mol. Cell Biol."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"10040","DOI":"10.1073\/pnas.0504321102","article-title":"Physical limits to biochemical signaling","volume":"102","author":"Bialek","year":"2005","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"12265","DOI":"10.1073\/pnas.0806077105","article-title":"Information flow and optimization in transcriptional regulation","volume":"105","author":"Callan","year":"2008","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"022407","DOI":"10.1103\/PhysRevE.97.022407","article-title":"Statistics of optimal information flow in ensembles of regulatory motifs","volume":"97","author":"Crisanti","year":"2018","journal-title":"Phys. Rev. E"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"924","DOI":"10.1016\/j.bbagen.2011.07.009","article-title":"Information theory based approaches to cellular signaling","volume":"1810","author":"Waltermann","year":"2011","journal-title":"Biochimica et Biophysica Acta (BBA)-General Subjects"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"763","DOI":"10.1016\/j.devcel.2012.01.019","article-title":"Short-term integration of Cdc25 dynamics controls mitotic entry during Drosophila gastrulation","volume":"22","author":"Wieschaus","year":"2012","journal-title":"Dev. Cell"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"026003","DOI":"10.1088\/1478-3975\/9\/2\/026003","article-title":"Gain control in molecular information processing: Lessons from neuroscience","volume":"9","author":"Nemenman","year":"2012","journal-title":"Phys. Biol."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"L1","DOI":"10.1016\/j.bpj.2014.06.018","article-title":"Simple biochemical pathways far from steady state can provide switchlike and integrated responses","volume":"107","author":"Wieschaus","year":"2014","journal-title":"Biophys. J."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"031920","DOI":"10.1103\/PhysRevE.80.031920","article-title":"Optimizing information flow in small genetic networks","volume":"80","author":"Walczak","year":"2009","journal-title":"Phys. Rev. E"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"046105","DOI":"10.1103\/PhysRevE.73.046105","article-title":"Rare events and the convergence of exponentially averaged work values","volume":"73","author":"Jarzynski","year":"2006","journal-title":"Phys. Rev. E"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"7498","DOI":"10.1038\/ncomms8498","article-title":"Maxwell\u2019s demon in biochemical signal transduction with feedback loop","volume":"6","author":"Ito","year":"2015","journal-title":"Nat. Commun."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"238701","DOI":"10.1103\/PhysRevLett.103.238701","article-title":"Granger causality and transfer entropy are equivalent for Gaussian variables","volume":"103","author":"Barnett","year":"2009","journal-title":"Phys. Rev. Lett."},{"key":"ref_56","unstructured":"Duchi, J. (2019, February 11). Derivations for Linear Algebra and Optimization. Available online: https:\/\/docplayer.net\/30887339-Derivations-for-linear-algebra-and-optimization.html."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/2\/177\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:32:02Z","timestamp":1760185922000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/2\/177"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,2,14]]},"references-count":56,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2019,2]]}},"alternative-id":["e21020177"],"URL":"https:\/\/doi.org\/10.3390\/e21020177","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,2,14]]}}}