# American Institute of Mathematical Sciences

August  2020, 13(8): 2121-2134. 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  August 2020 Early access  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, 2020, 13 (8) : 2121-2134. doi: 10.3934/dcdss.2020181
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##### References:
Jacobian graph of the system (1), the red corresponds to positive edges and the blue to negative ones
Steady states of system 2 in phase space ($xOy$), for $a = 0$ (left), $a = 0.15$ (middle) and $a = 0.17$ (right)
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
Time evolution of $Re(x(t))$ for $a = 0.30$, showing the successive phases (left) quasi-constant, erratic (middle) and periodic (right)
Cheyne-Stokes respiration
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])
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)
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)
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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