doi: 10.3934/dcdss.2020181

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

* Corresponding author: Jacques Demongeot

Received  February 2019 Published  November 2019

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.

Citation: Jacques Demongeot, Dan Istrate, Hajer Khlaifi, Lucile Mégret, Carla Taramasco, René Thomas. From conservative to dissipative non-linear differential systems. An application to the cardio-respiratory regulation. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020181
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.  Google Scholar

[2]

D. M. BaekeyY. I. MolkovJ. F. R. PatonI. 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.  Google Scholar

[3]

T. G. BautistaQ. 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.  Google Scholar

[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.  Google Scholar

[5]

E. BenoîtJ. L. CallotF. Diener and M. Diener, Chasse au canard, Collect. Math., 31 (1981), 37-74.   Google Scholar

[6]

K. BoldC. EdwardsJ. GuckenheimerS. GuharayK. HoffmanJ. HubbardR. 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.  Google Scholar

[7]

M. Brøns, Bifurcations and instabilities in the Greitzer model for compressor system surge, Mathematical Engineering in Industry, 2 (1988), 51-63.   Google Scholar

[8]

J. BurkeM. DesrochesA. GranadosT. J. KaperM. 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.  Google Scholar

[9]

M. Canalis-DurandJ. P. RamisR. 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.  Google Scholar

[10]

J. DemongeotG. VironeF. DuchêneG. BenchetritT. HervéN. 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.  Google Scholar

[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.  Google Scholar

[12]

J. DemongeotM. 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.  Google Scholar

[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. Google Scholar

[14]

J. DemongeotJ. Bezy-WendlingJ. MattesP. HaigronN. 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.  Google Scholar

[15]

O. DergachevaK. J. GriffioenR. 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.  Google Scholar

[16]

M. DesrochesJ. 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.   Google Scholar

[17]

F. Dumortier and R. Roussarie, Canard cycles and center manifolds, Mem. Amer. Math. Soc., 121 (1996). doi: 10.1090/memo/0577.  Google Scholar

[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.  Google Scholar

[19]

D. G. S. FarmerM. DutschmannJ. F. R. PatonA. 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.  Google Scholar

[20]

M. Fliess and C. Join, Dynamic compensation and homeostasis: A feedback control perspective, preprint, arXiv: math/1801.04959. Google Scholar

[21]

L. ForestN. 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.  Google Scholar

[22]

J. GrasmanH. 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.  Google Scholar

[23]

R. Grave de PeraltaS. 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.  Google Scholar

[24]

H. KhlaifiD. IstrateJ. DemongeotJ. Boudy and D. Malouche, Swallowing sound recognition at home using GMM, IRBM, 39 (2018), 407-412.  doi: 10.1016/j.irbm.2018.10.009.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[27]

D. J. A. MoraesB. 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.  Google Scholar

[28]

T. Pham DinhJ. DemongeotP. Baconnier and G. Benchetrit, Simulation of a biological oscillator: The respiratory rhythm, J. Theor. Biol., 103 (1983), 113-132.   Google Scholar

[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.  Google Scholar

[30]

G. VironeN. 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.  Google Scholar

[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.  Google Scholar

[32] M. Winter-ArboledaW. 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.  Google Scholar

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.  Google Scholar

[2]

D. M. BaekeyY. I. MolkovJ. F. R. PatonI. 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.  Google Scholar

[3]

T. G. BautistaQ. 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.  Google Scholar

[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.  Google Scholar

[5]

E. BenoîtJ. L. CallotF. Diener and M. Diener, Chasse au canard, Collect. Math., 31 (1981), 37-74.   Google Scholar

[6]

K. BoldC. EdwardsJ. GuckenheimerS. GuharayK. HoffmanJ. HubbardR. 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.  Google Scholar

[7]

M. Brøns, Bifurcations and instabilities in the Greitzer model for compressor system surge, Mathematical Engineering in Industry, 2 (1988), 51-63.   Google Scholar

[8]

J. BurkeM. DesrochesA. GranadosT. J. KaperM. 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.  Google Scholar

[9]

M. Canalis-DurandJ. P. RamisR. 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.  Google Scholar

[10]

J. DemongeotG. VironeF. DuchêneG. BenchetritT. HervéN. 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.  Google Scholar

[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.  Google Scholar

[12]

J. DemongeotM. 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.  Google Scholar

[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. Google Scholar

[14]

J. DemongeotJ. Bezy-WendlingJ. MattesP. HaigronN. 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.  Google Scholar

[15]

O. DergachevaK. J. GriffioenR. 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.  Google Scholar

[16]

M. DesrochesJ. 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.   Google Scholar

[17]

F. Dumortier and R. Roussarie, Canard cycles and center manifolds, Mem. Amer. Math. Soc., 121 (1996). doi: 10.1090/memo/0577.  Google Scholar

[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.  Google Scholar

[19]

D. G. S. FarmerM. DutschmannJ. F. R. PatonA. 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.  Google Scholar

[20]

M. Fliess and C. Join, Dynamic compensation and homeostasis: A feedback control perspective, preprint, arXiv: math/1801.04959. Google Scholar

[21]

L. ForestN. 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.  Google Scholar

[22]

J. GrasmanH. 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.  Google Scholar

[23]

R. Grave de PeraltaS. 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.  Google Scholar

[24]

H. KhlaifiD. IstrateJ. DemongeotJ. Boudy and D. Malouche, Swallowing sound recognition at home using GMM, IRBM, 39 (2018), 407-412.  doi: 10.1016/j.irbm.2018.10.009.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[27]

D. J. A. MoraesB. 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.  Google Scholar

[28]

T. Pham DinhJ. DemongeotP. Baconnier and G. Benchetrit, Simulation of a biological oscillator: The respiratory rhythm, J. Theor. Biol., 103 (1983), 113-132.   Google Scholar

[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.  Google Scholar

[30]

G. VironeN. 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.  Google Scholar

[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.  Google Scholar

[32] M. Winter-ArboledaW. 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.  Google Scholar
Figure 1.  Jacobian graph of the system (1), the red corresponds to positive edges and the blue to negative ones
Figure 2.  Steady states of system 2 in phase space ($ xOy $), for $ a = 0 $ (left), $ a = 0.15 $ (middle) and $ a = 0.17 $ (right)
Figure 3.  Left: trajectory starting from the complex solution for $ a = 0.17 $ in the phase plane ($ Re(x)ORe(y) $). Right: trajectory starting from the complex solution for $ a = 0.17 $ in the phase plane ($ Re(x)ORe(y) $) for an initial state modified by simply adding $ 0.000001 $ to the initial value $ y(0) $ of the left trajectory
Figure 4.  Time evolution of $ Re(x(t)) $ for $ a = 0.30 $, showing the successive phases (left) quasi-constant, erratic (middle) and periodic (right)
Figure 5.  Cheyne-Stokes respiration
Figure 6.  The central vegetative system made of the bulbar respiratory centre with inspiratory (I) (composed of early eI and post pI inspiratory neurons) and expiratory (E) neurons, and the cardio-regulator centre, ruling the main peripheral actuators, like the diaphragm and the heart controlled by the sinus node (S), and the peripheral sensors represented by the baroreceptors (B). The variables x, y, w and z represent respectively the activity of the four sets of excitatory cells, namely E, I, B and S. The squared scheme (of same type as in Figure 1) shows the relationships between these variables, with inhibitions in blue and activations in red
Figure 7.  Evolution of the instantaneous cardiac period $ T $, which is anti-correlated with the duration $ t $ of the inspiration in which occurs the cardiac cycle (after [2])
Figure 8.  Experimental data (recorded on a voluntary healthy adult man) at the transition waking/sleep states indicated on the Beta wave logarithmic power of the EEG (top), showing in sleep state a decrease of the RR amplitude (middle) and the influence of the swallowing on the respiratory signal (bottom)
Figure 10.  Simulations of system (8) showing for the sleep state a cardiac rhythm (top) with a period slightly less and an amplitude slightly larger than the corresponding values calculated for the awake state (bottom) (obtained with the online simulation tool https://www.zweigmedia.com/RealWorld/deSystemGrapher/func.html)
Figure 9.  Experimental data showing that during sleep the amplitude of the cardiac signal increases and its period (RR interval) decreases, with an augmentation of the correlation between the lengths of successive cardiac cycles
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