From conservative to dissipative non-linear differential systems. An application to the cardio-respiratory regulation

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