Hopf bifurcations and the stability of the respiratory control system

A simple model of a feedback loop controlling ventilation is analysed. This model is intended to describe the response of the system, initially at equilibrium, to a sudden fall in CO2 concentration in the lung, brought about by a deep sigh. A previous paper described the model in detail and the general method of analysis. Here we continue the discussion of stability, first in terms of local stability after a small displacement from equilibrium and then by computer simulation to illustrate the behaviour after large displacements. The local analysis is used to select representative sets of system parameters to illustrate the different types of trajectory obtained by computer simulation. When the equilibrium point is stable the response to a disturbance is overdamped, underdamped or critically damped. When the equilibrium point is unstable the system responds by going into a limit cycle. The transition between these two cases proceeds via a Hopf Bifurcation. The limit cycle type of ventilatory pattern, i.e. a periodic, underdamped waxing and waning of ventilation is commonly seen in premature infants and in term infants between 1 and 6 months of age.     

大熊猫生态系统动态模型稳定性及数值分析

本文主要建立大熊猫与两种竹子共生生态系统三种群的微分方程模型,得到Volterra系统,此生态系统是捕食与被捕食关系．分析了该系统在平衡状态处的稳定性,证明了其全局稳定性．最后,通过有关数据的调控确定大熊猫种群数量范围以及计算出该生态系统在平衡状态处受到标准扰动后的恢复时间．     

Brownian motion of the wormlike chain and segmental diffusion of DNA

The dynamics of the wormlike chain model for a polymer in solution is investigated in the case of free torsional and no longitudinal variations. A Langevin equation is derived and solved for circularly closed chains, neglecting hydrodynamic interactions. The local diffusion behavior of particular segments is described, and it is found that the mean-square displacements are proportional to t^3/4 at short times. Also, the equilibrium correlation functions for the closed chain are derived from the dynamic model in both the discrete and wormlike cases.     

Chaotic dynamics of a three species prey-predator competition model with bionomic harvesting due to delayed environmental noise as external driving force

We consider a biological economic model based on prey-predator interactions to study the dynamical behaviour of a fishery resource system consisting of one prey and two predators surviving on the same prey. The mathematical model is a set of first order non-linear differential equations in three variables with the population densities of one prey and the two predators. All the possible equilibrium points of the model are identified, where the local and global stabilities are investigated. Biological and bionomical equilibriums of the system are also derived. We have analysed the population intensities of fluctuations i.e., variances around the positive equilibrium due to noise with incorporation of a constant delay leading to chaos, and lastly have investigated the stability and chaotic phenomena with a computer simulation.     

Dynamics of the changes in homogenate respiratory activity after rat whole liver storage at 4 degrees C

Homogenate respiratory activity was studied after different storage terms of the whole rat liver at 4 degrees C in sucrose-based solution and following normothermic reperfusion. Preservation of homogenate respiratory activity in all metabolic states after normothermic reperfusion of the control liver (60 min, 4 degrees C) is shown. Further storage (6 and 24 hrs) of isolated liver under the mentioned above conditions strengthens the substrate respiration of homogenate both after storage and after normothermic reperfusion. At the same time oxidative phosphorylation does not practically change. No change was noted in respiratory activity in the states 3, 4ADP and 3DNP after 24 hrs of liver storage in respect of a previous term. Following normothermic liver reperfusion contributes to a statistically true reduction of mentioned parameters of respiration, that correlates with a decrease in the degree of respiration and phosphorylation coupled of the studied system.     

Stability of the human respiratory control system I. Analysis of a two-dimensional delay state-space model

A number of mathematical models of the human respiratory control system have been developed since 1940 to study a wide range of features of this complex system. Among them, periodic breathing (including Cheyne-Stokes respiration and apneustic breathing) is a collection of regular but involuntary breathing patterns that have important medical implications. The hypothesis that periodic breathing is the result of delay in the feedback signals to the respiratory control system has been studied since the work of Grodins et al. in the early 1950's [12]. The purpose of this paper is to study the stability characteristics of a feedback control system of five differential equations with delays in both the state and control variables presented by Khoo et al. [17] in 1991 for modeling human respiration. The paper is divided in two parts. Part I studies a simplified mathematical model of two nonlinear state equations modeling arterial partial pressures of O2 and CO2 and a peripheral controller. Analysis was done on this model to illuminate the effect of delay on the stability. It shows that delay dependent stability is affected by the controller gain, compartmental volumes and the manner in which changes in the ventilation rate is produced (i.e., by deeper breathing or faster breathing). In addition, numerical simulations were performed to validate analytical results. Part II extends the model in Part I to include both peripheral and central controllers. This, however, necessitates the introduction of a third state equation modeling CO2 levels in the brain. In addition to analytical studies on delay dependent stability, it shows that the decreased cardiac output (and hence increased delay) resulting from the congestive heart condition can induce instability at certain control gain levels. These analytical results were also confirmed by numerical simulations.     

A Retrospective Respiratory Gating System Based on Epipolar Consistency Conditions

Motion artifacts of in vivo imaging, due to rapid respiration rate and respiration displacements of the mice while free-breathing, is a major challenge in micro computed tomography(micro-CT). The respiratory gating is often served for either projective images acquisition or per projection qualification, so as to eliminate the artifacts brought by in vivo motion. In this paper, we propose a novel respiratory gating method, which firstly divides one rotation cycle into a number of segments, and extracts the respiratory signal from the projective image series of current segment by the value of the epipolar consistency conditions (ECC), then in terms of the measured average respiratory period, sets next segment’s start-up time and rotation speed of the gantry for respiratory phase synchronization, and so on so forth. The gating procedure is through the whole projections of three cycles, only one among three projections at each angle is qualified by their phase value and is retained for future use for tomographic image reconstruction. In practical experiment, the ECC based gating method and the conventional hardware gating method are employed on micro CT imaging of C57BL/6 mice respectively. The result shows that, compared with the hardware based one, the proposed method not only achieve much better consistency in the projection images, but also suppresses the streak artifacts more effectively on the different parts like the breast, abdomen and head of in vivo mice.     

Stability of the human respiratory control system II. Analysis of a three-dimensional delay state-space model

A number of mathematical models of the human respiratory control system have been developed since 1940 to study a wide range of features of this complex system. Among them, periodic breathing (including Cheyne-Stokes respiration and apneustic breathing) is a collection of regular but involuntary breathing patterns that have important medical implications. The hypothesis that periodic breathing is the result of delay in the feedback signals to the respiratory control system has been studied since the work of Grodins et al. in the early 1950's [1]. The purpose of this paper is to study the stability characteristics of a feedback control system of five differential equations with delays in both the state and control variables presented by Khoo et al. [4] in 1991 for modeling human respiration. The paper is divided in two parts. Part I studies a simplified mathematical model of two nonlinear state equations modeling arterial partial pressures of O2 and CO2 and a peripheral controller. Analysis was done on this model to illuminate the effect of delay on the stability. It shows that delay dependent stability is affected by the controller gain, compartmental volumes and the manner in which changes in the ventilation rate is produced (i.e., by deeper breathing or faster breathing). In addition, numerical simulations were performed to validate analytical results. Part II extends the model in Part I to include both peripheral and central controllers. This, however, necessitates the introduction of a third state equation modeling CO2 levels in the brain. In addition to analytical studies on delay dependent stability, it shows that the decreased cardiac output (and hence increased delay) resulting from the congestive heart condition can induce instability at certain control gain levels. These analytical results were also confirmed by numerical simulations.     

渗透胁迫对小麦幼苗根系呼吸的影响

用PEG—6000调节培养液的渗透势,研究了渗透胁迫对小麦幼苗根系呼吸作用的影响。在-0.5 MPa的溶液中根总呼吸强度显著降低,不同苗龄根的反应差异明显;随胁迫加强呼吸强度随之降低;根系ATP含量减少。在胁迫初期呼吸废物对呼吸强度的降低无补偿作用,而在后期(72 h后)则可提高呼吸强度。 中度水分胁迫下,HMP支路活性上升,EMP-TCAO途径活性降低;抗氰呼吸活性增大,而对氰敏感的系统活性减低;细胞色素氧化酶活性显著低于对照。     

Human respiratory sensory system, its physiological role

Modern data on genesis and functional role of sensations associated with respiration, were analysed. Regularities of human perception of the respiratory cycle's parameters have been cited. Respiratory discomfort occurring in an increased loading of the respiratory system, as well as an imperative respiration stimulus developing in a prolonged apnoe or obvious hypoventilation, are regarded as defensive responses to a threat to normal ventilation of the lungs. The central link in mechanisms of the respiratory sensory system seems to involve pulses from receptors of the respiratory muscles.     

Regulation of respiration during muscular exertion

A brief critical review of literature shows that many authors still follow a classical theory that the respiration control is performed by feedback (by deviation of PCO2, PO2 and pH in blood). This point of view does not account for the exercise hyperpnea. The present paper contains the various data and considerations which show that respiration during muscular exercise is controlled by a combined self-learning system. The system is based on both disturbance (open-loop) control and feedback control. The signals of disturbance (of central origin and from receptors of exercising muscles) cause the increase of respiration during exercise. The signals of deviations (from peripheral and central chemoreceptors) correct the response of respiratory centre to disturbance signals. The self-learning takes place by the formation of conditioned reflexes that ensures the control of respiration (the stability of gaseous composition of blood during exercise).     

Variation of flux control coefficient of cytochrome c oxidase and of the other respiratory chain complexes at different values of protonmotive force occurs by a threshold mechanism

The metabolic control analysis was applied to digitonin-permeabilized HepG2 cell line to assess the flux control exerted by cytochrome c oxidase on the mitochondrial respiration. Experimental conditions eliciting different energy/respiratory states in mitochondria were settled. The results obtained show that the mitochondrial electrochemical potential accompanies a depressing effect on the control coefficient exhibited by the cytochrome c oxidase. Both the components of the protonmotive force, i.e. the voltage (ΔΨ(m)) and the proton (ΔpH(m)) gradient, displayed a similar effect. Quantitative estimation of the ΔΨ(m) unveiled that the voltage-dependent effect on the control coefficient of cytochrome c oxidase takes place sharply in a narrow range of membrane potential from 170-180 to 200-210mV consistent with the physiologic transition from state 3 to state 4 of respiration. Extension of the metabolic flux control analysis to the NADH dehydrogenase and bc(1) complexes of the mitochondrial respiratory chain resulted in a similar effect. A mechanistic model is put forward whereby the respiratory chain complexes are proposed to exist in a voltage-mediated threshold-controlled dynamic equilibrium between supercomplexed and isolated states.     

随机配对种群的两性模型的动力学性态

基于Kendall-Goodman模型，提出了一个两性具有不同生理性态的随机配对的两性模型．如果不考虑密度制约因素，那么模型存在一个全局渐近稳定的指数解；如果考虑密度制约因素，对于给定的一个出生函数，得到了唯一正平衡态存在及全局稳定的充要条件．结论表明，无论是否考虑密度制约因素，种群的性比总是稳定的．     

The respiratory system of Halarachne halichoeri (Halarachnidae: Gamasida: Anactinotrichida)

Halarachne halichoeri Allman (Halarachnidae: Anactinotrichida), a parasitic mite occuring in the nasal cavities of the grey seal, Halichoerus grypus , has an unusual apneustic respiratory system, which is described in detail. The main functions of this system are to facilitate gaseous (tracheal) respiration and prevent foreign material being forced into the tracheae under increased pressure experience when the host seal dives. Possible responses of the system to the rapidly changing pressure regimes are described.     

Mitochondrial respiratory control in the myocardium

The heart muscle has proved to be a practical model for studying respiratory control in intact tissues. It also demonstrates that control at the level of the respiratory chain is augmented by metabolic control at the substrate level as exemplified by the very narrow range of changes in the redox state of the mitochondrial NADH/NAD couple even during extensive changes in ATP and oxygen consumption. The behaviour of mitochondria when isolated can largely be duplicated in the intact myocardium. Moreover, the high intracellular concentrations of enzymes, coenzymes and adenine nucleotides create conditions of high reaction rates, enabling the formation of a near equilibrium network of certain main pathways. This equilibrium network in connection with metabolic regulation of the hydrogen pressure upon the matrix NADH/NAD pool is a prerequisite for the regulation of cellular respiration at a high efficiency of energy transfer. Experimentation on the intact myocardium also seems to be capable of resolving some of the uncertainties about prevailing mechanisms for the regulation of cellular respiration.     

The coordination and interaction between respiration and deglutition in young pigs

The anatomical pathways for inspired air and ingested food cross in the pharynx of mammals, implying that breathing and swallowing must be separated either in space or in time. In this study we investigated the time relationship between swallowing and respiration in young pigs, as a model for suckling mammals. Despite the high morphological position of the larynx in young mammals, allowing liquid to pass in food channels lateral to the larynx, respiration and swallowing are not wholly independent events. Although, when suckling on a veterinary teat, the swallows occurred at various points in the respiratory cycle, there was always a period of apnea associated with the swallow. Finally, an increase in the viscosity of the milk altered this coordination, changing respiratory cycle length and also restricting the relative rate at which swallows occurred in some parts of the respiratory cycle. These results suggest that the subsequent changes in respiratory activity at weaning, associated with passage of a solid bolus over the larynx, is preceded by the ability of the animal to alter coordination between respiration and swallowing for a liquid bolus. Accepted: 29 September 1997     

A new explicit stability criterion for human periodic breathing<--RID="*"--><--ID="*" This research was supported by grants from DRED (96-920).-->

The aim of this paper is to carry out a stability analysis for periodic breathing in humans that incorporates the dynamic characteristics of ventilation control. A simple CO₂ model that takes into account the main elements of the respiratory system, i.e. the lungs and the ventilatory controller with its dynamic properties, is presented. This model results in a three-dimensional non-linear delay differential system for which there exists a unique equilibrium point. Our stability analysis of this equilibrium point leads to the definition of a new explicit stability criterion and to the demonstration of the existence of a Hopf bifurcation. Numerical simulations illustrate the influence of physiological parameters on the stability of ventilation, and particularly the major role of the dynamic characteristics of the respiratory controller. Received: 2 February 1999 / Revised version: 18 June 1999 / Published online: 23 October 2000     

Respiration and reproductive effort in Xanthium canadense

* BACKGROUND AND AIMS: The proportion of resources devoted to reproduction in the plant is called the reproductive effort (RE), which is most commonly expressed as the proportion of reproductive biomass to total plant biomass production (RE(W)). Reproductive yield is the outcome of photosynthates allocated to reproductive structures minus subsequent respiratory consumption for construction and maintenance of reproductive structures. Thus, RE(W) can differ from RE in terms of photosynthates allocated to reproductive structures (RE(P)). * METHODS: Dry mass growth and respiration of vegetative and reproductive organs were measured in Xanthium canadense and the amount of photosynthates and its partitioning to dry mass growth and respiratory consumption were determined. Differences between RE(W) and RE(P) were analysed in terms of growth and maintenance respiration. * KEY RESULTS: The fraction of allocated photosynthates that was consumed by respiration was smaller in the reproductive organ than in the vegetative organs. Consequently, RE(P) was smaller than RE(W). The smaller respiratory consumption in the reproductive organ resulted from its shorter period of existence and a seasonal decline in temperature, as well as a slower rate of maintenance respiration, although the fraction of photosynthates consumed by growth respiration was larger than in the vegetative organs. * CONCLUSIONS: Reproductive effort in terms of photosynthates (RE(P)) was smaller than that in terms of biomass (RE(W)). This difference resulted from respiratory consumption for maintenance, which was far smaller in the reproductive organ than in vegetative organs.     

Regulation of respiration: visceral and behavioural components

The review segregates two aspects of respiration regulation: autonomous respiration regulation as a visceral function ensuring metabolic needs of a body by maintaining stability of own respiratory environment, on the one hand, and behavioural regulation of respiration under control of the volitional sphere, on the other hand. The authors focus on respiratory rythmogenesis, the problem that has not yet been resolved, and on the mechanism of precise correlation of lung ventillation with the metabolic level, in case of muscular exercise, in particular. The authors discuss interaction of visceral and behavioural mechanisms of respiratory regulation. The substance of the phenomenon of respiratory embarrassment is considered in this connection as a visceral signal addressed to the behavioural sphere. Reasonableness of introduction of a new breathing system in a healthy person is doubted. The article justifies the pracice of bioregulation of the respiratory function.