Hopf bifurcations and the stability of the respiratory control system

Time delay in physiological systems: analyzing and modeling its impact

This article examines the functional and clinical impact of time delays that arise in human physiological systems, especially control systems. An overview of the mathematical and physiological contexts for considering time delays will be illustrated, from the system level to cell level, by examining models that incorporate time delays. This examination will highlight how such delays in combination with other system structures and parameters influence system dynamics. Model analysis that reveals the influence of delays can also reveal related physiological effects which may have medical consequences and clinical applications.

Model-Based Analysis of Mechanisms Responsible for Sleep-Induced Carbon Dioxide Differences

This work describes a comprehensive mathematical model of the human respiratory control system which incorporates the central mechanisms for predicting sleep-induced changes in chemical regulation of ventilation. The model integrates four individual compartments for gas storage and exchange, namely alveolar air, pulmonary blood, tissue capillary blood, body tissues, and gas transport between them. An essential mechanism in the carbon dioxide transport is its dissociation into bicarbonate and acid, where a buffering mechanism through hemoglobin is used to prevent harmfully low pH levels. In the current model, we assume high oxygen levels and consider intracellular hydrogen ion concentration as the principal respiratory control variable. The resulting system of delayed differential equations is solved numerically. With an appropriate choice of key parameters, such as velocity of blood flow and gain of a non-linear controller function, the model provides steady-state results consistent with our experimental observations measured in subjects across sleep onset. Dynamic predictions from the model give new insights into the behaviour of the system in subjects with different buffering capacities and suggest novel hypotheses for future experimental and clinical studies.

Cyclic model of respiration applied to asymmetrical ventilation and periodic breathing

A model taking into account the cyclic character of respiration in humans is developed using two classical simplifications: CO2 is the only respiratory gas involved; and respiration is regulated only by a CO2 linear controller. The model is used to investigate two important clinical aspects of respiratory disease: asymmetrical ventilation and periodic breathing. We show that asymmetry in ventilation significantly influences the time course of the CO2 partial pressure in the expired alveolar air at the mouth and the elimination of CO2 through the lungs. Furthermore, the CO2 controller delay plays a major role in periodic breathing.

A dynamic analysis of the ventilatory response to carbon dioxide inhalation in man

1. The dynamics of the ventilatory response to carbon dioxide inhalation were studied in ten healthy young men using four different inspired fractions of carbon dioxide (FI, CO2) in air (0.015, 0.030, 0.045 and 0.060) successively increasing and decreasing stepwise. 2. Seven such different progressions were performed for each subject and each of seven different durations of the steps (t) ranging between 0.1 (i.e. one ventilatory cycle) and 10 min ('steady-state' conditions). The overall duration of one test (T) was taken as the sum of the seven successive FI, CO2 steps (t) plus one step, t, of air breathing. Thus, the values of T ranged between 0.8 (i.e. eight ventilatory cycles) and 80 min. Three subjects were tested twice. 3. We measured, as a function of T, the magnitude of the loops formed by the curves PA, CO2-VE and the value of the highest ventilatory response (VE max) to each progression. For all ten subjects, both functions had two maxima, one for T values of 2.6 or 8.0 min and one for T values of 24 or 40 min, and one minimum at T equal to 12 min. 4. The same measurements were made on tidal volume-response curves (PA, CO2-VT) and ventilatory frequency-response curves (PA, CO2-f) and yielded the same results except for the ventilatory frequency-response curves, for which we only found a statistically insignificant single maximum for T values of 24 or 40 min. 5. The locations of the maxima in loop magnitude and VE max were similar in duplicate tests in three subjects, whereas the quantitative values of these variables showed wide differences. 6. We compared our results with what is expected from the current linear dynamic model of ventilatory control submitted to the same forcing function: the first maximum in the loop magnitude is predicted by the model, but the second is not. The model shows no peak in the evolution of VE max. 7. We conclude that controlled system dynamics, which are the only ones included in dynamic models of ventilatory control, cannot by themselves account for our observations, and that one should take into consideration the dynamics of the controlling neuronal network.