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From conservative to dissipative non-linear differential systems. An application to the cardio-respiratory regulation
1. | Université Grenoble Alpes, AGEIS, Team Tools for e-Gnosis Medical, Faculté de Médecine, Domaine de la Merci 38706 La Tronche, France |
2. | University of Technology of Compiègne, UMR CNRS 7338 Biomechanics and Bioengineering, 60200 Compiègne, France |
3. | Université Pierre et Marie Curie, UMR 8256 - Adaptation Biologique et Vieillissement, 7 quai Saint Bernard, 75 252 PARIS CEDEX, France |
4. | Escuela de Ingeniería Civil en Informática, Universidad de Valparaíso, General Cruz 222, Valparaíso, Chile |
5. | Université libre de Bruxelles, Avenue Franklin Roosevelt 50, 1050 Bruxelles, Belgium |
We start by coupling negative 2-circuits, which are characteristic of the presence of a regulation loop in a dynamical system. This loop can be modelled with coupled differential equations represented in a first approach by a conservative differential system. Then, an example of regulation loop with a dissipative component will be given in human physiology by the vegetative system regulating the cardio-respiratory rhythms.
References:
[1] |
C. Antonopoulos, V. Basios, J. Demongeot, P. Nardone and R. Thomas, Linear and nonlinear arabesques: A study of closed chains of negative 2-element circuits, Int. J. Bifurcation and Chaos, 23 (2013).
doi: 10.1142/S0218127413300334. |
[2] |
D. M. Baekey, Y. I. Molkov, J. F. R. Paton, I. A. Rybak and T. E. Dick,
Effect of baroreceptor stimulation on the respiratory pattern: Insights into respiratory-sympathetic interactions, Respiratory Physiology & Neurobiology, 174 (2010), 135-145.
doi: 10.1016/j.resp.2010.09.006. |
[3] |
T. G. Bautista, Q. J. Sun and P. M. Pilowsky,
The generation of pharyngeal phase of swallow and its coordination with breathing: Interaction between the swallow and respiratory central pattern generators, Prog. Brain Res., 212 (2014), 253-275.
doi: 10.1016/B978-0-444-63488-7.00013-6. |
[4] |
T. Beauchaine,
Vagal tone, development, and Gray's motivational theory: Toward an integrated model of autonomic nervous system functioning in psychopathology, Development and Psychopathology, 13 (2001), 183-214.
doi: 10.1017/S0954579401002012. |
[5] |
E. Benoît, J. L. Callot, F. Diener and M. Diener,
Chasse au canard, Collect. Math., 31 (1981), 37-74.
|
[6] |
K. Bold, C. Edwards, J. Guckenheimer, S. Guharay, K. Hoffman, J. Hubbard, R. Oliva and W. Weckesser,
The forced van der Pol equation. Ⅱ: Canards in the reduced system, SIAM J. Appl. Dyn. Syst., 2 (2003), 570-608.
doi: 10.1137/S1111111102419130. |
[7] |
M. Brøns,
Bifurcations and instabilities in the Greitzer model for compressor system surge, Mathematical Engineering in Industry, 2 (1988), 51-63.
|
[8] |
J. Burke, M. Desroches, A. Granados, T. J. Kaper, M. Krupa and T. Vo,
From canards of folded singularities to torus canards in a forced van der Pol equation, J. Nonlinear Sci., 26 (2016), 405-451.
doi: 10.1007/s00332-015-9279-0. |
[9] |
M. Canalis-Durand, J. P. Ramis, R. Schafke and Y. Sibuya,
Gevrey solutions of singularly perturbed differential equations, J. Reine Angew. Math., 518 (2000), 95-129.
doi: 10.1515/crll.2000.008. |
[10] |
J. Demongeot, G. Virone, F. Duchêne, G. Benchetrit, T. Hervé, N. Noury and V. Rialle,
Multi-sensors acquisition, data fusion, knowledge mining and alarm triggering in health smart homes for elderly people, Comptes Rendus Biologies, 325 (2002), 673-682.
doi: 10.1016/S1631-0691(02)01480-4. |
[11] |
J. Demongeot and J. Waku,
Application of interval iterations to the entrainment problem in respiratory physiologye, Phil. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 367 (2009), 4717-4739.
doi: 10.1098/rsta.2009.0177. |
[12] |
J. Demongeot, M. Noual and S. Sené,
Combinatorics of Boolean automata circuits dynamics, Discrete Appl. Math., 160 (2012), 398-415.
doi: 10.1016/j.dam.2011.11.005. |
[13] |
J. Demongeot, H. Ben Amor, H. Hazgui and A. Lontos, La simplexité, dernier avatar de la complexit, OpenEdition, Marseille, 2014. Available from: http://books.openedition.org/cdf/3393. |
[14] |
J. Demongeot, J. Bezy-Wendling, J. Mattes, P. Haigron, N. Glade and J. L. Coatrieux,
Multiscale modeling and imaging: The challenges of biocomplexity, Proceedings of the IEEE Society, 91 (2003), 1723-1737.
doi: 10.1109/JPROC.2003.817878. |
[15] |
O. Dergacheva, K. J. Griffioen, R. A. Neff and D. Mendelowitz,
Respiratory modulation of premotor cardiac vagal neurons in the brainstem, Respiratory Physiology & Neurobiology, 174 (2010), 102-110.
doi: 10.1016/j.resp.2010.05.005. |
[16] |
M. Desroches, J. P. Francoise and L. Mgret,
Canard-induced loss of stability across a homoclinic bifurcation, ARIMA Rev. Afr. Rech. Inform. Math. Appl., 20 (2015), 47-62.
|
[17] |
F. Dumortier and R. Roussarie, Canard cycles and center manifolds, Mem. Amer. Math. Soc., 121 (1996).
doi: 10.1090/memo/0577. |
[18] |
W. Eckhaus, Relaxation oscillations including a standard chase on French ducks, in Asymptotic Analysis II, Lecture Notes in Math., 985, Springer, Berlin, 1983,449–494.
doi: 10.1007/BFb0062381. |
[19] |
D. G. S. Farmer, M. Dutschmann, J. F. R. Paton, A. E. Pickering and R. M. McAllen,
Brainstem sources of cardiac vagal tone and respiratory sinus arrhythmia, J. Physiology, 594 (2016), 7249-7265.
doi: 10.1113/JP273164. |
[20] |
M. Fliess and C. Join, Dynamic compensation and homeostasis: A feedback control perspective, preprint, arXiv: math/1801.04959. |
[21] |
L. Forest, N. Glade and J. Demongeot,
Liénard systemes and potential-Hamiltonian decomposition - Applications in biology, C. R. Biologies, 330 (2007), 97-106.
doi: 10.1016/j.crvi.2006.12.001. |
[22] |
J. Grasman, H. Nijmeijer and E. J. M. Veling,
Singular perturbations and a mapping on an interval for the forced van der Pol relaxation oscillator, Phys. D, 13 (1984), 195-210.
doi: 10.1016/0167-2789(84)90277-X. |
[23] |
R. Grave de Peralta, S. Gonzalez Andino and S. Perrig,
Patient machine interface for the control of mechanical ventilation devices, Brain Sci., 3 (2013), 1554-1568.
doi: 10.3390/brainsci3041554. |
[24] |
H. Khlaifi, D. Istrate, J. Demongeot, J. Boudy and D. Malouche,
Swallowing sound recognition at home using GMM, IRBM, 39 (2018), 407-412.
doi: 10.1016/j.irbm.2018.10.009. |
[25] |
É. Matzinger,
Étude des solutions sur-stables de l'équation de van der Pol, Ann. Fac. Sci. Toulouse Math. (6), 10 (2001), 713-744.
doi: 10.5802/afst.1010. |
[26] |
L. Mégret and J. Demongeot, Gevrey solutions of singularly perturbed differential equations, an extension to the non-autonomous case, Discrete Contin. Dyn. Syst., preprint. |
[27] |
D. J. A. Moraes, B. H. Machado and D. B. Zoccal,
Coupling of respiratory and sympathetic activities in rats submitted to chronic intermittent hypoxia, Prog. Brain Res., 212 (2014), 25-38.
doi: 10.1016/B978-0-444-63488-7.00002-1. |
[28] |
T. Pham Dinh, J. Demongeot, P. Baconnier and G. Benchetrit,
Simulation of a biological oscillator: The respiratory rhythm, J. Theor. Biol., 103 (1983), 113-132.
|
[29] |
B. van der Pol and J. van der Mark,
The heart beat considered as a relaxation oscillator and an electrical model of the heart, Philos. Mag., 6 (1928), 763-775.
doi: 10.1080/14786441108564652. |
[30] |
G. Virone, N. Noury and J. Demongeot,
A system for automatic measurement of circadian activity deviations in telemedicine, IEEE Trans. Biomed. Eng., 49 (2002), 1463-1469.
doi: 10.1109/TBME.2002.805452. |
[31] |
G. Virone, B. Lefebvre, N. Noury and J. Demongeot, Modeling and computer simulation of physiological rhythms and behaviors at home for data fusion programs in a telecare system, in IEEE Healthcom, Piscataway, 2003,111–117.
doi: 10.1109/HEALTH.2003.1218727. |
[32] |
M. Winter-Arboleda, W. S. Gray and L. A. D. Espinosa, On global convergence of fractional Fliess operators with applications to bilinear systems, in 51st Annual Conference on Information Sciences and Systems (CISS 1), IEEE Press, Piscataway, 2017.
doi: 10.1109/CISS.2017.7926119.![]() ![]() |
show all references
References:
[1] |
C. Antonopoulos, V. Basios, J. Demongeot, P. Nardone and R. Thomas, Linear and nonlinear arabesques: A study of closed chains of negative 2-element circuits, Int. J. Bifurcation and Chaos, 23 (2013).
doi: 10.1142/S0218127413300334. |
[2] |
D. M. Baekey, Y. I. Molkov, J. F. R. Paton, I. A. Rybak and T. E. Dick,
Effect of baroreceptor stimulation on the respiratory pattern: Insights into respiratory-sympathetic interactions, Respiratory Physiology & Neurobiology, 174 (2010), 135-145.
doi: 10.1016/j.resp.2010.09.006. |
[3] |
T. G. Bautista, Q. J. Sun and P. M. Pilowsky,
The generation of pharyngeal phase of swallow and its coordination with breathing: Interaction between the swallow and respiratory central pattern generators, Prog. Brain Res., 212 (2014), 253-275.
doi: 10.1016/B978-0-444-63488-7.00013-6. |
[4] |
T. Beauchaine,
Vagal tone, development, and Gray's motivational theory: Toward an integrated model of autonomic nervous system functioning in psychopathology, Development and Psychopathology, 13 (2001), 183-214.
doi: 10.1017/S0954579401002012. |
[5] |
E. Benoît, J. L. Callot, F. Diener and M. Diener,
Chasse au canard, Collect. Math., 31 (1981), 37-74.
|
[6] |
K. Bold, C. Edwards, J. Guckenheimer, S. Guharay, K. Hoffman, J. Hubbard, R. Oliva and W. Weckesser,
The forced van der Pol equation. Ⅱ: Canards in the reduced system, SIAM J. Appl. Dyn. Syst., 2 (2003), 570-608.
doi: 10.1137/S1111111102419130. |
[7] |
M. Brøns,
Bifurcations and instabilities in the Greitzer model for compressor system surge, Mathematical Engineering in Industry, 2 (1988), 51-63.
|
[8] |
J. Burke, M. Desroches, A. Granados, T. J. Kaper, M. Krupa and T. Vo,
From canards of folded singularities to torus canards in a forced van der Pol equation, J. Nonlinear Sci., 26 (2016), 405-451.
doi: 10.1007/s00332-015-9279-0. |
[9] |
M. Canalis-Durand, J. P. Ramis, R. Schafke and Y. Sibuya,
Gevrey solutions of singularly perturbed differential equations, J. Reine Angew. Math., 518 (2000), 95-129.
doi: 10.1515/crll.2000.008. |
[10] |
J. Demongeot, G. Virone, F. Duchêne, G. Benchetrit, T. Hervé, N. Noury and V. Rialle,
Multi-sensors acquisition, data fusion, knowledge mining and alarm triggering in health smart homes for elderly people, Comptes Rendus Biologies, 325 (2002), 673-682.
doi: 10.1016/S1631-0691(02)01480-4. |
[11] |
J. Demongeot and J. Waku,
Application of interval iterations to the entrainment problem in respiratory physiologye, Phil. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 367 (2009), 4717-4739.
doi: 10.1098/rsta.2009.0177. |
[12] |
J. Demongeot, M. Noual and S. Sené,
Combinatorics of Boolean automata circuits dynamics, Discrete Appl. Math., 160 (2012), 398-415.
doi: 10.1016/j.dam.2011.11.005. |
[13] |
J. Demongeot, H. Ben Amor, H. Hazgui and A. Lontos, La simplexité, dernier avatar de la complexit, OpenEdition, Marseille, 2014. Available from: http://books.openedition.org/cdf/3393. |
[14] |
J. Demongeot, J. Bezy-Wendling, J. Mattes, P. Haigron, N. Glade and J. L. Coatrieux,
Multiscale modeling and imaging: The challenges of biocomplexity, Proceedings of the IEEE Society, 91 (2003), 1723-1737.
doi: 10.1109/JPROC.2003.817878. |
[15] |
O. Dergacheva, K. J. Griffioen, R. A. Neff and D. Mendelowitz,
Respiratory modulation of premotor cardiac vagal neurons in the brainstem, Respiratory Physiology & Neurobiology, 174 (2010), 102-110.
doi: 10.1016/j.resp.2010.05.005. |
[16] |
M. Desroches, J. P. Francoise and L. Mgret,
Canard-induced loss of stability across a homoclinic bifurcation, ARIMA Rev. Afr. Rech. Inform. Math. Appl., 20 (2015), 47-62.
|
[17] |
F. Dumortier and R. Roussarie, Canard cycles and center manifolds, Mem. Amer. Math. Soc., 121 (1996).
doi: 10.1090/memo/0577. |
[18] |
W. Eckhaus, Relaxation oscillations including a standard chase on French ducks, in Asymptotic Analysis II, Lecture Notes in Math., 985, Springer, Berlin, 1983,449–494.
doi: 10.1007/BFb0062381. |
[19] |
D. G. S. Farmer, M. Dutschmann, J. F. R. Paton, A. E. Pickering and R. M. McAllen,
Brainstem sources of cardiac vagal tone and respiratory sinus arrhythmia, J. Physiology, 594 (2016), 7249-7265.
doi: 10.1113/JP273164. |
[20] |
M. Fliess and C. Join, Dynamic compensation and homeostasis: A feedback control perspective, preprint, arXiv: math/1801.04959. |
[21] |
L. Forest, N. Glade and J. Demongeot,
Liénard systemes and potential-Hamiltonian decomposition - Applications in biology, C. R. Biologies, 330 (2007), 97-106.
doi: 10.1016/j.crvi.2006.12.001. |
[22] |
J. Grasman, H. Nijmeijer and E. J. M. Veling,
Singular perturbations and a mapping on an interval for the forced van der Pol relaxation oscillator, Phys. D, 13 (1984), 195-210.
doi: 10.1016/0167-2789(84)90277-X. |
[23] |
R. Grave de Peralta, S. Gonzalez Andino and S. Perrig,
Patient machine interface for the control of mechanical ventilation devices, Brain Sci., 3 (2013), 1554-1568.
doi: 10.3390/brainsci3041554. |
[24] |
H. Khlaifi, D. Istrate, J. Demongeot, J. Boudy and D. Malouche,
Swallowing sound recognition at home using GMM, IRBM, 39 (2018), 407-412.
doi: 10.1016/j.irbm.2018.10.009. |
[25] |
É. Matzinger,
Étude des solutions sur-stables de l'équation de van der Pol, Ann. Fac. Sci. Toulouse Math. (6), 10 (2001), 713-744.
doi: 10.5802/afst.1010. |
[26] |
L. Mégret and J. Demongeot, Gevrey solutions of singularly perturbed differential equations, an extension to the non-autonomous case, Discrete Contin. Dyn. Syst., preprint. |
[27] |
D. J. A. Moraes, B. H. Machado and D. B. Zoccal,
Coupling of respiratory and sympathetic activities in rats submitted to chronic intermittent hypoxia, Prog. Brain Res., 212 (2014), 25-38.
doi: 10.1016/B978-0-444-63488-7.00002-1. |
[28] |
T. Pham Dinh, J. Demongeot, P. Baconnier and G. Benchetrit,
Simulation of a biological oscillator: The respiratory rhythm, J. Theor. Biol., 103 (1983), 113-132.
|
[29] |
B. van der Pol and J. van der Mark,
The heart beat considered as a relaxation oscillator and an electrical model of the heart, Philos. Mag., 6 (1928), 763-775.
doi: 10.1080/14786441108564652. |
[30] |
G. Virone, N. Noury and J. Demongeot,
A system for automatic measurement of circadian activity deviations in telemedicine, IEEE Trans. Biomed. Eng., 49 (2002), 1463-1469.
doi: 10.1109/TBME.2002.805452. |
[31] |
G. Virone, B. Lefebvre, N. Noury and J. Demongeot, Modeling and computer simulation of physiological rhythms and behaviors at home for data fusion programs in a telecare system, in IEEE Healthcom, Piscataway, 2003,111–117.
doi: 10.1109/HEALTH.2003.1218727. |
[32] |
M. Winter-Arboleda, W. S. Gray and L. A. D. Espinosa, On global convergence of fractional Fliess operators with applications to bilinear systems, in 51st Annual Conference on Information Sciences and Systems (CISS 1), IEEE Press, Piscataway, 2017.
doi: 10.1109/CISS.2017.7926119.![]() ![]() |









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