{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,12,27]],"date-time":"2023-12-27T00:09:19Z","timestamp":1703635759037},"reference-count":67,"publisher":"Oxford University Press (OUP)","issue":"4","license":[{"start":{"date-parts":[[2023,11,24]],"date-time":"2023-11-24T00:00:00Z","timestamp":1700784000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/pages\/standard-publication-reuse-rights"}],"funder":[{"name":"Mathematical Investigation of Neuroscience Dynamics for Meditative Model Identification","award":["ANR-17-CE40-0005"],"award-info":[{"award-number":["ANR-17-CE40-0005"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2023,12,24]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>In this paper, we study several existing quantitative models of the hypothalamic\u2013pituitary\u2013adrenal (HPA) axis from a control systems theory viewpoint, that is, we suppose that we can act on the dynamics of the HPA axis throughout some parameters, which are the system inputs. In particular, we will focus on flatness and Liouvillian properties of the considered control systems of the HPA axis. We first study the minimal three-dimensional model of Bangsgaard and Ottesen (2017, Math. Biosci., 287:24\u201335) and the semi-mechanistic four-dimensional model of Gupta et al. (2007, Theor. Biol. Medical Model., 4(1):8) which are shown to be flat, and then, we consider the more involved and important model proposed in Rao &amp; Androulakis (2019, Sci. Rep., 9(1):11212; 2020, IFAC-PapersOnLine, 53(2):15858\u201315863), with seven states, for which we prove that for the nominal values of the parameters involved in the model, flatness no longer holds. The more involved model satisfies however a similar but weaker property than flatness: it is a Liouvillian system.<\/jats:p>","DOI":"10.1093\/imamci\/dnad030","type":"journal-article","created":{"date-parts":[[2023,11,29]],"date-time":"2023-11-29T11:17:19Z","timestamp":1701256639000},"page":"746-788","source":"Crossref","is-referenced-by-count":1,"title":["HPA axis differential flatness and Liouvillian study for higher resiliency investigations"],"prefix":"10.1093","volume":"40","author":[{"given":"Florentina","family":"Nicolau","sequence":"first","affiliation":[{"name":"Quartz EA7393 Laboratory, ENSEA , 95014 Cergy-Pontoise , France"}]},{"given":"Hugues","family":"Mounier","sequence":"additional","affiliation":[{"name":"Laboratoire des Signaux et Syst\u00e8mes, Universit\u00e9 Paris-Saclay, CentraleSup\u00e9lec, CNRS (UMR 8605) , 91192 Gif-sur-Yvette , France"}]},{"given":"Ioannis P","family":"Androulakis","sequence":"additional","affiliation":[{"name":"Department of Chemical and Biochemical Engineering and Department of Biomedical Engineering, Rutgers, the State University of New Jersey , 599 Taylor Road, Piscataway, NJ 08854 , USA"}]}],"member":"286","published-online":{"date-parts":[[2023,11,24]]},"reference":[{"key":"2023122615292598700_ref1","doi-asserted-by":"crossref","first-page":"3722","DOI":"10.1016\/j.jmb.2020.01.014","article-title":"Amplitude effects allow short jet lags and large seasonal phase shifts in minimal clock models","volume":"432","author":"Ananthasubramaniam","year":"2020","journal-title":"J. Mol. Biol."},{"key":"2023122615292598700_ref2","doi-asserted-by":"crossref","first-page":"e1518","DOI":"10.1002\/wsbm.1518","article-title":"Circadian rhythms and the HPA axis: a systems view","volume":"13","author":"Androulakis","year":"2021","journal-title":"WIREs Mechanisms of Disease"},{"key":"2023122615292598700_ref3","doi-asserted-by":"crossref","first-page":"19","DOI":"10.4064\/-32-1-19-33","article-title":"Infinitesimal Brunovsky form for nonlinear systems with applications to dynamic linearization","volume":"32","author":"Aranda-Bricaire","year":"1995","journal-title":"Banach Center Publ."},{"key":"2023122615292598700_ref4","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1137\/060663891","article-title":"Design of positive linear observers for positive linear systems via coordinate transformations and positive realizations","volume":"47","author":"Back","year":"2008","journal-title":"SIAM J. Control Optim."},{"key":"2023122615292598700_ref5","doi-asserted-by":"crossref","first-page":"24","DOI":"10.1016\/j.mbs.2016.10.007","article-title":"Patient specific modeling of the HPA axis related to clinical diagnosis of depression","volume":"287","author":"Bangsgaard","year":"2017","journal-title":"Math. Biosci."},{"key":"2023122615292598700_ref6","doi-asserted-by":"crossref","first-page":"3023","DOI":"10.1529\/biophysj.104.040824","article-title":"Modeling feedback loops of the mammalian circadian oscillator","volume":"87","author":"Becker-Weimann","year":"2004","journal-title":"Biophys. J."},{"key":"2023122615292598700_ref7","volume-title":"Flatness-based constrained control and model-free control applications to quadrotors and cloud computing","author":"Bekcheva","year":"2019"},{"key":"2023122615292598700_ref8","first-page":"173","article-title":"A classification of linear controllable systems","volume":"3","author":"Brunovsky","year":"1970","journal-title":"Kybernetika (Prague)"},{"key":"2023122615292598700_ref9","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1677\/joe.0.1770017","article-title":"The biological clock tunes the organs of the body: timing by hormones and the autonomic nervous system","volume":"177","author":"Buijs","year":"2003","journal-title":"J. Endocrinol."},{"key":"2023122615292598700_ref10","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1111\/j.1600-0447.2007.00967.x","article-title":"Pathophysiology of hypercortisolism in depression","volume":"115","author":"Carroll","year":"2007","journal-title":"Acta Psychiatr. Scand."},{"key":"2023122615292598700_ref11","doi-asserted-by":"crossref","first-page":"12","DOI":"10.24033\/bsmf.938","article-title":"Sur l\u2019\u00e9quivalence absolue de certains syst\u00e8mes d\u2019\u00e9quations diff\u00e9rentielles et Sur certaines familles de courbes","volume":"2","author":"Cartan","year":"1914","journal-title":"Bull. Soc. Math. France"},{"key":"2023122615292598700_ref12","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1016\/0167-6911(89)90031-5","article-title":"On dynamic feedback linearization","volume":"13","author":"Charlet","year":"1989","journal-title":"Systems Control Lett."},{"key":"2023122615292598700_ref13","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1137\/0329002","article-title":"Sufficient conditions for dynamic state feedback linearization","volume":"29","author":"Charlet","year":"1991","journal-title":"SIAM J. Control Optim."},{"key":"2023122615292598700_ref14","doi-asserted-by":"crossref","first-page":"4268","DOI":"10.1109\/CDC.1997.649507","article-title":"Extensions of differential flat fields and Liouvillian systems","volume-title":"Proceedings of the 36th IEEE Conference on Decision and Control","author":"Chelouah","year":"1997"},{"key":"2023122615292598700_ref15","article-title":"Diffieties and Liouvillian systems","author":"Chelouah","year":"2010","journal-title":"CoRR"},{"key":"2023122615292598700_ref16","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1007\/s10867-009-9134-3","article-title":"Modeling the hypothalamus\u2013pituitary\u2013adrenal system: homeostasis by interacting positive and negative feedback","volume":"35","author":"Conrad","year":"2009","journal-title":"J. Biol. Phys."},{"key":"2023122615292598700_ref17","first-page":"95","article-title":"Real Liouvillian extensions of partial differential fields","volume":"17","author":"Crespo","year":"2021","journal-title":"Symmetry Integr. Geom.: Methods Appl."},{"key":"2023122615292598700_ref18","volume-title":"Theoretical neuroscience: computational and mathematical modeling of neural systems","author":"Dayan","year":"2001"},{"key":"2023122615292598700_ref19","doi-asserted-by":"crossref","DOI":"10.1002\/9781118033029","volume-title":"Positive linear systems: theory and applications","author":"Farina","year":"2000"},{"key":"2023122615292598700_ref20","doi-asserted-by":"crossref","first-page":"1327","DOI":"10.1080\/00207179508921959","article-title":"Flatness and defect of non-linear systems: introductory theory and examples","volume":"61","author":"Fliess","year":"1995","journal-title":"Internat. J. Control"},{"key":"2023122615292598700_ref21","doi-asserted-by":"crossref","first-page":"922","DOI":"10.1109\/9.763209","article-title":"A lie-B\u00e4cklund approach equivalence and flatness of nonlinear systems","volume":"44","author":"Fliess","year":"1999","journal-title":"IEEE Trans. Automat. Control"},{"key":"2023122615292598700_ref22","doi-asserted-by":"crossref","first-page":"110652","DOI":"10.1016\/j.mce.2019.110652","article-title":"Rhythmicity matters: circadian and ultradian patterns of hpa axis activity","volume":"501","author":"Focke","year":"2020","journal-title":"Mol. Cell. Endocrinol."},{"key":"2023122615292598700_ref23","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1098\/rspb.1995.0153","article-title":"A model for circadian oscillations in the drosophila period protein (per)","volume":"261","author":"Goldbeter","year":"1995","journal-title":"Proc. R. Soc. Lond. B Biol. Sci."},{"key":"2023122615292598700_ref24","doi-asserted-by":"crossref","first-page":"857","DOI":"10.1007\/s10441-020-09379-8","article-title":"The Goodwin oscillator and its legacy","volume":"69","author":"Gonze","year":"2021","journal-title":"Acta Biotheor."},{"key":"2023122615292598700_ref25","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1016\/0065-2571(65)90067-1","article-title":"Oscillatory behavior in enzymatic control processes","volume":"3","author":"Goodwin","year":"1965","journal-title":"Adv. Enzym Res."},{"key":"2023122615292598700_ref26","doi-asserted-by":"crossref","first-page":"110484","DOI":"10.1016\/j.automatica.2022.110484","article-title":"An approximate characterisation of the set of feasible trajectories for constrained flat systems","volume":"144","author":"Greco","year":"2022","journal-title":"Automatica J. IFAC"},{"key":"2023122615292598700_ref27","doi-asserted-by":"crossref","first-page":"827","DOI":"10.1140\/epjs\/s11734-021-00310-x","article-title":"Strengthen the circadian rhythms by the mathematical model of the SCN","volume":"231","author":"Gu","year":"2022","journal-title":"Eur. Phys. J. Spec. Top."},{"key":"2023122615292598700_ref28","doi-asserted-by":"crossref","DOI":"10.1186\/1742-4682-4-8","article-title":"Inclusion of the glucocorticoid receptor in a hypothalamic pituitary adrenal axis model reveals bistability","volume":"4","author":"Gupta","year":"2007","journal-title":"Theor. Biol. Medical Model."},{"key":"2023122615292598700_ref29","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1016\/j.tins.2007.11.002","article-title":"Melanopsin: an exciting photopigment","volume":"31","author":"Hankins","year":"2008","journal-title":"Trends Neurosci."},{"key":"2023122615292598700_ref30","doi-asserted-by":"crossref","first-page":"728","DOI":"10.1109\/TAC.1977.1101601","article-title":"Nonlinear controllability and observability","volume":"22","author":"Hermann","year":"1977","journal-title":"IEEE Trans. Automat. Control"},{"key":"2023122615292598700_ref31","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1007\/BF01456663","article-title":"\u00dcber den Begriff der Klasse von Differentialgleichungen","volume":"73","author":"Hilbert","year":"1912","journal-title":"Math. Ann."},{"key":"2023122615292598700_ref32","first-page":"119","article-title":"Linear equivalents of nonlinear time varying systems","volume-title":"Proc. MTNS. Santa Monica","author":"Hunt","year":"1981"},{"key":"2023122615292598700_ref33","doi-asserted-by":"crossref","first-page":"369","DOI":"10.1016\/j.ejcon.2013.05.014","article-title":"The zero dynamics of a nonlinear system: from the origin to the latest progresses of a long successful story","volume":"19","author":"Isidori","year":"2013","journal-title":"Eur. J. Control"},{"key":"2023122615292598700_ref34","first-page":"203","article-title":"A sufficient condition for full linearization via dynamic state feedback","volume-title":"Proc. IEEE CDC","author":"Isidori","year":"1986"},{"key":"2023122615292598700_ref35","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1137\/1035003","article-title":"Qualitative theory of compartmental systems","volume":"35","author":"Jacquez","year":"1993","journal-title":"SIAM Rev."},{"key":"2023122615292598700_ref36","first-page":"517","article-title":"On linearization of control systems","author":"Jakubczyk","year":"1980","journal-title":"Bull. Acad. Polonaise Sci. Ser. Sci. Math."},{"key":"2023122615292598700_ref37","doi-asserted-by":"crossref","first-page":"443","DOI":"10.1177\/027836402321261959","article-title":"On motion planning for robotic manipulation with permanent rolling contacts","volume":"21","author":"Kiss","year":"2002","journal-title":"Int. J. Rob. Res."},{"key":"2023122615292598700_ref38","doi-asserted-by":"crossref","DOI":"10.1137\/0323016","article-title":"Nonlinear observers with linearizable error dynamics","volume":"23","author":"Krener","year":"1985","journal-title":"SIAM J. Control Optim."},{"key":"2023122615292598700_ref39","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1016\/j.biopsycho.2004.11.009","article-title":"Sex differences in HPA axis responses to stress: a review","volume":"69","author":"Kudielka","year":"2005","journal-title":"Biol. Psychol."},{"key":"2023122615292598700_ref40","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-00839-9","volume-title":"Analysis and Control of Nonlinear Systems: A Flatness-Based Approach","author":"L\u00e9vine","year":"2009"},{"key":"2023122615292598700_ref41","doi-asserted-by":"crossref","first-page":"4398","DOI":"10.1073\/pnas.070039597","article-title":"Navigation-related structural change in the hippocampi of taxi drivers","volume":"97","author":"Maguire","year":"2000","journal-title":"Proc. Natl. Acad. Sci. U.S.A."},{"key":"2023122615292598700_ref42","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1016\/0167-6911(86)90130-1","article-title":"On the largest feedback linearizable subsystem","volume":"6","author":"Marino","year":"1986","journal-title":"Systems Control Lett."},{"key":"2023122615292598700_ref43","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1016\/0005-1098(95)00099-2","article-title":"A different look at output tracking: control of a VTOL aircraft","volume":"32","author":"Martin","year":"1996","journal-title":"Automatica J. IFAC"},{"key":"2023122615292598700_ref44","article-title":"Flat systems","volume-title":"Mathematical control theory, Part 1, 2 (Trieste, 2001)","author":"Martin","year":"2002"},{"key":"2023122615292598700_ref45","doi-asserted-by":"crossref","first-page":"766","DOI":"10.1152\/physiolgenomics.00026.2014","article-title":"Mathematical modeling of light-mediated HPA axis activity and downstream implications on the entrainment of peripheral clock genes","volume":"46","author":"Mavroudis","year":"2014","journal-title":"Physiol. Genomics"},{"key":"2023122615292598700_ref46","volume-title":"Sur certaines familles de s\u00e9ries formelles","author":"Mi. Fliess.","year":"1972"},{"key":"2023122615292598700_ref47","doi-asserted-by":"crossref","DOI":"10.1016\/j.ifacol.2023.01.014","article-title":"Differential flatness and Liouvillian character of two HPA axis models","volume-title":"Proc. of the 9th International Conference on the Foundations of Systems Biology in Engineering (FOSBE 2022)","author":"Nicolau","year":"2022"},{"key":"2023122615292598700_ref48","first-page":"200","article-title":"Flatness of networks of two synaptically coupled excitatory-inhibitory neural modules","volume-title":"Proc. of the 25th MTNS","author":"Nicolau","year":"2022"},{"key":"2023122615292598700_ref49","doi-asserted-by":"crossref","DOI":"10.23919\/ECC57647.2023.10178333","article-title":"Flatness of networks of two synaptically coupled excitatory-inhibitory neural modules with maximal symmetry","volume-title":"Proc. of the 21st ECC","author":"Nicolau","year":"2023"},{"key":"2023122615292598700_ref50","doi-asserted-by":"crossref","first-page":"464","DOI":"10.1016\/j.tins.2008.06.006","article-title":"The HPA axis in major depression: classical theories and new developments","volume":"31","author":"Pariante","year":"2008","journal-title":"Trends Neurosci."},{"key":"2023122615292598700_ref51","doi-asserted-by":"crossref","first-page":"319","DOI":"10.4064\/-32-1-319-339","article-title":"A differential geometric setting for dynamic equivalence and dynamic linearization","volume":"32","author":"Pomet","year":"1995","journal-title":"Banach Center Publ."},{"key":"2023122615292598700_ref52","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1051\/cocv:1997107","article-title":"On dynamic feedback linearization of four-dimensional affine control systems with two inputs","volume":"2","author":"Pomet","year":"1997","journal-title":"ESAIM Control Optim. Calc. Var."},{"key":"2023122615292598700_ref53","doi-asserted-by":"crossref","first-page":"11212","DOI":"10.1038\/s41598-019-47605-7","article-title":"Allostatic adaptation and personalized physiological trade-offs in the circadian regulation of the HPA axis: a mathematical modeling approach","volume":"9","author":"Rao","year":"2019","journal-title":"Sci. Rep."},{"key":"2023122615292598700_ref54","doi-asserted-by":"crossref","first-page":"15858","DOI":"10.1016\/j.ifacol.2020.12.241","article-title":"The circadian rhythms of cortisol: modelling their role in regulating homeostasis and personalized resilience and adaptation","volume":"53","author":"Rao","year":"2020","journal-title":"IFAC-PapersOnLine"},{"key":"2023122615292598700_ref55","first-page":"133","article-title":"Right and left invertibility of nonlinear control systems","volume-title":"Nonlinear Controllability and Optimal Control","author":"Respondek","year":"1990"},{"key":"2023122615292598700_ref56","volume-title":"Advanced Models of Neural Networks. Nonlinear Dynamics and Stochasticity in Biological Neurons","author":"Rigatos","year":"2013"},{"key":"2023122615292598700_ref57","doi-asserted-by":"crossref","first-page":"R826","DOI":"10.1016\/j.cub.2008.07.017","article-title":"Modelling biological rhythms","volume":"18","author":"Roenneberg","year":"2008","journal-title":"Curr. Biol."},{"key":"2023122615292598700_ref58","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1152\/physiolgenomics.00128.2011","article-title":"Transcriptional implications of ultradian glucocorticoid secretion in homeostasis and in the acute stress response","volume":"44","author":"Scheff","year":"2012","journal-title":"Physiol. Genomics"},{"key":"2023122615292598700_ref59","article-title":"Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems: an introduction to the theory of competitive and cooperative systems","author":"Smith","year":"1995","journal-title":"American Mathematical Soc"},{"key":"2023122615292598700_ref60","doi-asserted-by":"crossref","first-page":"358","DOI":"10.1016\/j.jalgebra.2019.12.023","article-title":"Differential subfields of Liouvillian extensions","volume":"550","author":"Srinivasan","year":"2020","journal-title":"J. Algebra"},{"key":"2023122615292598700_ref61","doi-asserted-by":"crossref","first-page":"e1002379","DOI":"10.1371\/journal.pcbi.1002379","article-title":"Modeling cortisol dynamics in the neuro-endocrine axis distinguishes normal, depression, and post-traumatic stress disorder (ptsd) in humans","volume":"8","author":"Sriram","year":"2012","journal-title":"PLoS Comput. Biol."},{"key":"2023122615292598700_ref62","doi-asserted-by":"crossref","first-page":"84","DOI":"10.1016\/j.coche.2018.04.003","article-title":"Advances in mathematical modelling of the hypothalamic\u2013pituitary\u2013adrenal (hpa) axis dynamics and the neuroendocrine response to stress","volume":"21","author":"Stanojevi\u0107","year":"2018","journal-title":"Curr. Opin. Chem. Eng."},{"key":"2023122615292598700_ref63","doi-asserted-by":"crossref","first-page":"216","DOI":"10.18699\/VJ21.025","article-title":"The role of the corticotropin-releasing hormone and its receptors in the regulation of stress response","volume":"25","author":"Sukhareva","year":"2021","journal-title":"Vavilov Journal of Genetics and Breeding"},{"key":"2023122615292598700_ref64","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1016\/0022-0396(72)90007-1","article-title":"Controllability of nonlinear systems","volume":"12","author":"Sussmann","year":"1972","journal-title":"J. Differ. Equ."},{"key":"2023122615292598700_ref65","doi-asserted-by":"crossref","first-page":"663","DOI":"10.1007\/s00285-010-0384-2","article-title":"The minimal model of the hypothalamic\u2013pituitary\u2013adrenal axis","volume":"63","author":"Vinther","year":"2011","journal-title":"J. Math. Biol."},{"key":"2023122615292598700_ref66","doi-asserted-by":"crossref","first-page":"1627","DOI":"10.1098\/rspb.2009.2148","article-title":"Origin of ultradian pulsatility in the hypothalamic\u2013pituitary\u2013adrenal axis","volume":"277","author":"Walker","year":"2010","journal-title":"Proc. R. Soc. B: Biol. Sci."},{"key":"2023122615292598700_ref67","doi-asserted-by":"crossref","first-page":"1081","DOI":"10.1080\/00207178808906076","article-title":"Non-linear observer design by observer canonical forms","volume":"47","author":"Xia","year":"1988","journal-title":"Int. J. Control"}],"container-title":["IMA Journal of Mathematical Control and Information"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/imamci\/article-pdf\/40\/4\/746\/54804640\/dnad030.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/imamci\/article-pdf\/40\/4\/746\/54804640\/dnad030.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,12,26]],"date-time":"2023-12-26T15:30:23Z","timestamp":1703604623000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/imamci\/article\/40\/4\/746\/7450345"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,11,24]]},"references-count":67,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2023,11,24]]},"published-print":{"date-parts":[[2023,12,24]]}},"URL":"https:\/\/doi.org\/10.1093\/imamci\/dnad030","relation":{},"ISSN":["0265-0754","1471-6887"],"issn-type":[{"value":"0265-0754","type":"print"},{"value":"1471-6887","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2023,12]]},"published":{"date-parts":[[2023,11,24]]}}}