Structured population on two patches: modeling dispersal and delay

We derive from the age-structured model a system of delay differential equations to describe the interaction of spatial dispersal (over two patches) and time delay (arising from the maturation period). Our model analysis shows that varying the immature death rate can alter the behavior of the homogeneous equilibria, leading to transient oscillations around an intermediate equilibrium and complicated dynamics (in the form of the coexistence of possibly stable synchronized periodic oscillations and unstable phase-locked oscillations) near the largest equilibrium.     

Relative stability of human respiration during progressive hypoxia

We have systematically studied the relationship between the relative stability (R) of respiration and the loop gain (LG) of the CO2 control system in 15 healthy awake adult males during progressive hypoxia. R was measured by the ventilatory oscillations after brief (less than 10 s) CO2 challenges. Control theory suggests that such oscillations are completely governed by LG. A significant positive correlation was found between R and LG (r = 0.74, P less than 0.01, n = 85). A minimal mathematical model of respiratory control was used to predict R as a function of LG. Serial correlation analysis (r = 0.09, P greater than 0.1) of the residuals indicated statistical agreement between predictions and observations. The mean residual (0.011) was not significantly different from zero (P greater than 0.1). Also, as the model predicted, sustained periodic breathing (PB) occurred whenever the estimated LG was greater than unity. The mean LG breathing room air was 0.51 and for the 13 epochs of PB was 1.17 (range 0.71-1.65). It is concluded that PB is a quantitative extension of the relative stability continuum and corresponds to unstable operation of the CO2 control system. Furthermore, relative stability can be quantitatively predicted for each subject by a minimal mathematical model.     

Transient oscillations induced by delayed growth response in the chemostat

In this paper, in order to try to account for the transient oscillations observed in chemostat experiments, we consider a model of single species growth in a chemostat that involves delayed growth response. The time delay models the lag involved in the nutrient conversion process. Both monotone response functions and nonmonotone response functions are considered. The nonmonotone response function models the inhibitory effects of growth response of certain nutrients when concentrations are too high. By applying local and global Hopf bifurcation theorems, we prove that the model has unstable periodic solutions that bifurcate from unstable nonnegative equilibria as the parameter measuring the delay passes through certain critical values and that these local periodic solutions can persist, even if the delay parameter moves far from the critical (local) bifurcation values.When there are two positive equilibria, then positive periodic solutions can exist. When there is a unique positive equilibrium, the model does not have positive periodic oscillations and the unique positive equilibrium is globally asymptotically stable. However, the model can have periodic solutions that change sign. Although these solutions are not biologically meaningful, provided the initial data starts close enough to the unstable manifold of one of these periodic solutions they may still help to account for the transient oscillations that have been frequently observed in chemostat experiments. Numerical simulations are provided to illustrate that the model has varying degrees of transient oscillatory behaviour that can be controlled by the choice of the initial data.Mathematics Subject Classification: 34D20, 34K20, 92D25Research was partially supported by NSERC of Canada.This work was partly done while this author was a postdoc at McMaster.     

A Model of Postural Control in Quiet Standing: Robust Compensation of Delay-Induced Instability Using Intermittent Activation of Feedback Control

The main purpose of this study is to compare two different feedback controllers for the stabilization of quiet standing in humans, taking into account that the intrinsic ankle stiffness is insufficient and that there is a large delay inducing instability in the feedback loop: 1) a standard linear, continuous-time PD controller and 2) an intermittent PD controller characterized by a switching function defined in the phase plane, with or without a dead zone around the nominal equilibrium state. The stability analysis of the first controller is carried out by using the standard tools of linear control systems, whereas the analysis of the intermittent controllers is based on the use of Poincaré maps defined in the phase plane. When the PD-control is off, the dynamics of the system is characterized by a saddle-like equilibrium, with a stable and an unstable manifold. The switching function of the intermittent controller is implemented in such a way that PD-control is ‘off’ when the state vector is near the stable manifold of the saddle and is ‘on’ otherwise. A theoretical analysis and a related simulation study show that the intermittent control model is much more robust than the standard model because the size of the region in the parameter space of the feedback control gains (P vs. D) that characterizes stable behavior is much larger in the latter case than in the former one. Moreover, the intermittent controller can use feedback parameters that are much smaller than the standard model. Typical sway patterns generated by the intermittent controller are the result of an alternation between slow motion along the stable manifold of the saddle, when the PD-control is off, and spiral motion away from the upright equilibrium determined by the activation of the PD-control with low feedback gains. Remarkably, overall dynamic stability can be achieved by combining in a smart way two unstable regimes: a saddle and an unstable spiral. The intermittent controller exploits the stabilizing effect of one part of the saddle, letting the system evolve by alone when it slides on or near the stable manifold; when the state vector enters the strongly unstable part of the saddle it switches on a mild feedback which is not supposed to impose a strict stable regime but rather to mitigate the impending fall. The presence of a dead zone in the intermittent controller does not alter the stability properties but improves the similarity with biological sway patterns. The two types of controllers are also compared in the frequency domain by considering the power spectral density (PSD) of the sway sequences generated by the models with additive noise. Different from the standard continuous model, whose PSD function is similar to an over-damped second order system without a resonance, the intermittent control model is capable to exhibit the two power law scaling regimes that are typical of physiological sway movements in humans.     

Slow oscillations in blood pressure via a nonlinear feedback model

Blood pressure is well established to contain a potential oscillation between 0.1 and 0.4 Hz, which is proposed to reflect resonant feedback in the baroreflex loop. A linear feedback model, comprising delay and lag terms for the vasculature, and a linear proportional derivative controller have been proposed to account for the 0.4-Hz oscillation in blood pressure in rats. However, although this model can produce oscillations at the required frequency, some strict relationships between the controller and vasculature parameters must be true for the oscillations to be stable. We developed a nonlinear model, containing an amplitude-limiting nonlinearity that allows for similar oscillations under a very mild set of assumptions. Models constructed from arterial pressure and sympathetic nerve activity recordings obtained from conscious rabbits under resting conditions suggest that the nonlinearity in the feedback loop is not contained within the vasculature, but rather is confined to the central nervous system. The advantage of the model is that it provides for sustained stable oscillations under a wide variety of situations even where gain at various points along the feedback loop may be altered, a situation that is not possible with a linear feedback model. Our model shows how variations in some of the nonlinearity characteristics can account for growth or decay in the oscillations and situations where the oscillations can disappear altogether. Such variations are shown to accord well with observed experimental data. Additionally, using a nonlinear feedback model, it is straightforward to show that the variation in frequency of the oscillations in blood pressure in rats (0.4 Hz), rabbits (0.3 Hz), and humans (0.1 Hz) is primarily due to scaling effects of conduction times between species.     

Respiratory system stability and abnormal carbon dioxide homeostasis.

We have tested the hypothesis that interactions among eight parameters of the respiratory and cardiovascular systems that determine the loop gain (LG) of the respiratory CO2 feedback control system might account for the degree of stability or instability of breathing patterns in healthy sleeping volunteers as well as in familial dysautonomia (FD) and congenital central hypoventilation syndrome (CCHS) patients. The predictability of cycle duration was tested as well. We measured the values of CO2 sensitivity, CO2 delivery capacity in the circulation, circulation delay, mean lung volume for CO2, and mixed venous PCO2 in 8 FD patients, 2 CCHS patients, and 19 healthy controls. The values of these parameters were used in a mathematical model to compute the LG of the respiratory control system during sleep for each epoch of respiration analyzed. The strength of the ventilatory oscillations (R) was quantified using power density spectra of the ventilation time series. All subjects were studied at inspiratory O2 concentrations (FIO2) of 0.21 and 0.15; CCHS patients and controls were also studied at 0.12 FIO2 to examine the effect of steady-state hypoxia on respiratory system stability. In 2 FD patients, LG was elevated at both levels of FIO2 and periodic breathing was observed; the values of R were elevated. Elevated mixed venous PCO2 and reduced CO2 delivery capacity were chiefly responsible for the abnormally high LG observed. In three healthy volunteers, high LG and unstable patterns were associated with high chemosensitivity. The CCHS patients, however, remained stable even at 0.12 FIO2 because LG remained equivalent to zero due to a lack of chemosensitivity.(ABSTRACT TRUNCATED AT 250 WORDS)     

Effect of reducing anatomic dead space on arterial PCO2 during CO2 inhalation

Carotid body-denervated (CBD) ponies have a less than normal increase in arterial PCO2 (PaCO2) when inspired CO2 (PICO2) is increased, even when pulmonary ventilation (VE) and breathing frequency (f) are normal. We studied six tracheostomized ponies to determine whether this change 1) might be due to increased alveolar ventilation (VA) secondary to a reduction in upper airway dead space (VD) or 2) is dependent on an upper airway sensory mechanism. Three normal and three chronic CBD ponies were studied while they were breathing room air and at 14, 28, and 42 Torr PICO2. While the ponies were breathing room air, physiological VD was 483 and 255 ml during nares breathing (NBr) and tracheostomy breathing (TBr), respectively. However, at elevated PICO2, mixed expired PCO2 often exceeded PaCO2; thus we were unable to calculate physiological VD using the Bohr equation. At all PICO2 in normal ponies, PaCO2 was approximately 0.3 Torr greater during NBr than during TBr (P less than 0.05). In CBD ponies this NBr-TBr difference was only evident while breathing room air and at 28 Torr PICO2. At each elevated PICO2 during both NBr and TBr, the increase in PaCO2 above control was always less in CBD ponies than in normal ponies (P less than 0.01). The VE-PaCO2, f-PaCO2, and tidal volume-PaCO2 relationships did not differ between NBr and TBr (P greater than 0.10) nor did they differ between normal and CBD ponies (P greater than 0.10). We conclude that the attenuated increase in PaCO2 during CO2 inhalation after CBD is not due to a relative increase in VA secondary to reducing upper airway VD.(ABSTRACT TRUNCATED AT 250 WORDS)     

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.     

Avian ventilatory responses to dynamic CO2 signals.

This study uses an awake unidirectionally ventilated avian preparation to examine the effects of dynamic CO2 signals on the respiratory drive. Results show that minute ventilation is affected by both 1) mean CO2 level and 2) amplitude of CO2 oscillations at the frequency of breathing. An increase in mean CO2 level increased minute ventilation. Comparisons of the effects of CO2 oscillations at the same mean CO2 level, however, showed minute ventilation to be less with the larger amplitudes of oscillations than with smaller ones. Graphs of minute ventilation (V) versus mean CO2 for families of oscillation sizes (0.5%, 1% and 2%) showed that the ventilatory sensitivity (slop) was least for the 2% oscillations and greatest for the 0.5% oscillations. Therefore, a static model for the respiratory regulator is not adequate. However, the apneic level of CO2 (V = O intercept) was independent of the size of the CO2 oscillations.     

Models of Cheyne-Stokes respiration with cardiovascular pathologies

Cheyne-Stokes respiration (CSR) is a periodic breathing pattern, characterized by short intervals of very little or no breathing (apnea), each followed by an interval of very heavy breathing (hyperpnea). This work presents a new compartmental model of the human cardio-respiratory system, simulating the factors that determine the concentrations of carbon dioxide in the compartments of the cardiovascular system and the lungs. The parameter set on which a Hopf bifurcation gives birth to stable CSR oscillations has been determined. The model predicts that the onset of CSR oscillations may result from an increase in any of: ventilation-perfusion ratio, feedback control gain, transport delay, left heart volume, lung congestion, or cardiovascular efficiency. The model is employed to investigate the relationship between CSR and serious cardiovascular pathologies, such as congestive heart failure and encephalitis, as well as the effects of acclimatization to higher altitudes. In all cases, the model is consistent with medical observations.     

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.     

Increased propensity for apnea in response to acute elevations in left atrial pressure during sleep in the dog.

Periodic breathing is commonly observed in chronic heart failure (CHF) when pulmonary capillary wedge pressure is abnormally high and there is usually concomitant tachypneic hyperventilation. We hypothesized that acute pulmonary hypertension at pressures encountered in CHF and involving all of the lungs and pulmonary vessels would predispose to apnea/unstable breathing during sleep. We tested this in a chronically instrumented, unanesthetized dog model during non-rapid eye movement (NREM) sleep. Pulmonary hypertension was created by partial occlusion of the left atrium by means of an implanted balloon catheter in the atrial lumen. Raising mean left atrial pressure by 5.7 +/- 1.1 Torr resulted immediately in tachypneic hyperventilation [breathing frequency increased significantly from 13.8 to 19.9 breaths/min; end-tidal P(CO2) (P(ET(CO2))) fell significantly from 38.5 to 35.9 Torr]. This tachypneic hyperventilation was present during wakefulness, NREM sleep, and rapid eye movement sleep. In NREM sleep, this increase in left atrial pressure increased the gain of the ventilatory response to CO2 below eupnea (1.3 to 2.2 l.min(-1).Torr(-1)) and thereby narrowed the CO2 reserve [P(ET(CO2)) (apneic threshold) - P(ET(CO2)) (eupnea)], despite the decreased plant gain resulting from the hyperventilation. We conclude that acute pulmonary hypertension during sleep results in a narrowed CO2 reserve and thus predisposes toward apnea/unstable breathing and may, therefore, contribute to the breathing instability observed in CHF.     

A dynamical model for reflex activated head movements in the horizontal plane

 We present a controls systems model of horizontal-plane head movements during perturbations of the trunk, which for the first time interfaces a model of the human head with neural feedback controllers representing the vestibulocollic (VCR) and the cervicocollic (CCR) reflexes. This model is homeomorphic such that model structure and parameters are drawn directly from anthropomorphic, biomechanical and physiological studies. Using control theory we analyzed the system model in the time and frequency domains, simulating neck movement responses to input perturbations of the trunk. Without reflex control, the head and neck system produced a second-order underdamped response with a 5.2 dB resonant peak at 2.1 Hz. Adding the CCR component to the system dampened the response by approximately 7%. Adding the VCR component dampened head oscillations by 75%. The VCR also improved low-frequency compensation by increasing the gain and phase lag, creating a phase minimum at 0.1 Hz and a phase peak at 1.1 Hz. Combining all three components (mechanics, VCR and CCR) linearly in the head and neck system reduced the amplitude of the resonant peak to 1.1 dB and increased the resonant frequency to 2.9 Hz. The closed loop results closely fit human data, and explain quantitatively the characteristic phase peak often observed. Received: 15 April 1996 / Accepted in revised form: 1 July 1996     

Induction of oscillatory ventilation pattern using dynamic modulation of heart rate through a pacemaker

For disease states characterized by oscillatory ventilation, an ideal dynamic therapy would apply a counteracting oscillation in ventilation. Modulating respiratory gas transport through the circulation might allow this. We explore the ability of repetitive alternations in heart rate, using a cardiac pacemaker, to elicit oscillations in respiratory variables and discuss the potential for therapeutic exploitation. By incorporating acute cardiac output manipulations into an integrated mathematical model, we observed that a rise in cardiac output should yield a gradual rise in end-tidal CO2 and, subsequently, ventilation. An alternating pattern of cardiac output might, therefore, create oscillations in CO2 and ventilation. We studied the effect of repeated alternations in heart rate of 30 beats/min with periodicity of 60 s, on cardiac output, respiratory gases, and ventilation in 22 subjects with implanted cardiac pacemakers and stable breathing patterns. End-tidal CO2 and ventilation developed consistent oscillations with a period of 60 s during the heart rate alternations, with mean peak-to-trough relative excursions of 8.4 +/- 5.0% (P < 0.0001) and 24.4 +/- 18.8% (P < 0.0001), respectively. Furthermore, we verified the mathematical prediction that the amplitude of these oscillations would depend on those in cardiac output (r = 0.59, P = 0.001). Repetitive alternations in heart rate can elicit reproducible oscillations in end-tidal CO2 and ventilation. The size of this effect depends on the magnitude of the cardiac output response. Harnessed and timed appropriately, this cardiorespiratory mechanism might be exploited to create an active dynamic responsive pacing algorithm to counteract spontaneous respiratory oscillations, such as those causing apneic breathing disorders.     

Sleep-induced periodic breathing and apnea: a theoretical study

To elucidate the mechanisms that lead to sleep-disordered breathing, we have developed a mathematical model that allows for dynamic interactions among the chemical control of respiration, changes in sleep-waking state, and changes in upper airway patency. The increase in steady-state arterial PCO2 accompanying sleep is shown to be inversely related to the ventilatory response to CO2. Chemical control of respiration becomes less stable during the light stage of sleep, despite a reduction in chemoresponsiveness, due to a concomitant increase in "plant gain" (i.e., responsiveness of blood gases to ventilatory changes). The withdrawal of the "wakefulness drive" during sleep onset represents a strong perturbation to respiratory control: higher magnitudes and rates of withdrawal of this drive favor instability. These results may account for the higher incidence of periodic breathing observed during light sleep and sleep onset. Periodic ventilation can also result from repetitive alternations between sleep onset and arousal. The potential for instability is further compounded if the possibility of upper airway occlusion is also included. In systems with high controller gains, instability is mediated primarily through chemoreflex overcompensation. However, in systems with depressed chemoresponsiveness, rapid sleep onset and large blood gas fluctuations trigger repetitive episodes of arousal and hyperpnea alternating with apneas that may or may not be obstructive. Between these extremes, more complex patterns can arise from the interaction between chemoreflex-mediated oscillations of shorter-cycle-duration (approximately 36 s) and longer-wavelength (approximately 60-80 s) state-driven oscillations.     

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.     

Inferior olive mirrors joint dynamics to implement an inverse controller

To produce smooth and coordinated motion, our nervous systems need to generate precisely timed muscle activation patterns that, due to axonal conduction delay, must be generated in a predictive and feedforward manner. Kawato proposed that the cerebellum accomplishes this by acting as an inverse controller that modulates descending motor commands to predictively drive the spinal cord such that the musculoskeletal dynamics are canceled out. This and other cerebellar theories do not, however, account for the rich biophysical properties expressed by the olivocerebellar complex’s various cell types, making these theories difficult to verify experimentally. Here we propose that a multizonal microcomplex’s (MZMC) inferior olivary neurons use their subthreshold oscillations to mirror a musculoskeletal joint’s underdamped dynamics, thereby achieving inverse control. We used control theory to map a joint’s inverse model onto an MZMC’s biophysics, and we used biophysical modeling to confirm that inferior olivary neurons can express the dynamics required to mirror biomechanical joints. We then combined both techniques to predict how experimentally injecting current into the inferior olive would affect overall motor output performance. We found that this experimental manipulation unmasked a joint’s natural dynamics, as observed by motor output ringing at the joint’s natural frequency, with amplitude proportional to the amount of current. These results support the proposal that the cerebellum—in particular an MZMC—is an inverse controller; the results also provide a biophysical implementation for this controller and allow one to make an experimentally testable prediction.     

Simulation of gas transport due to cardiogenic oscillations

We simulated gas transport due to cardiogenic oscillations (CO) using a model developed to quantify the gas mixing due to high-frequency ventilation (16). The basic components of the model are 1) gas mixing by augmented transport, 2) symmetrical lung morphometry, and 3) a Lagrangian (moving) reference frame. The theoretical predictions of the model are in general agreement with published experimental studies that have examined the effect of CO on the nitrogen concentration obtained by intrapulmonary gas sampling and the effect of CO on regional and total anatomical dead space. Further, the model predicts that augmentation of gas transport due to CO is less, nearer to the alveolar regions of the lung, and that the effect of CO during normal tidal breathing is negligible, but that CO may contribute up to approximately 10% of the alveolar ventilation in patients with severe hypoventilation. The agreement between experimental and theoretical results suggests that it may not be necessary to invoke gas transport mechanisms specific to an asymmetrical bronchial tree to explain the major proportion of gas transport due to CO.     

Nonlinear control of bioreactors with input multiplicities—an experimental work

The phrase input multiplicities means that an input variable with more than one value produces the same output value as if there were a single input–single output process. With input multiplicities, the value of the process gain changes as the manipulated variable changes, and beyond a certain input value, the sign of the gain also changes. A conventional PI controller for processes with input multiplicities may give unstable, less economical, or oscillatory responses. In the present work, control problems of a continuous bioreactor exhibiting two input multiplicities in the dilution rate on productivity were experimentally analyzed. A regulatory problem for the evaluation of controllers was taken up, i.e. a step change was made in the feed substrate concentration from 20 to 25 g/l at steady state conduction at lower (0.09386 h⁻¹) and higher (0.2278 h⁻¹) dilution rates for the same productivity of 2.9 g/l h. The nonlinear PI controller gave a more stable and fast response at both input dilution rates. The linear PI controller designed for a lower input dilution rate was stable, with some oscillations at the lower dilution rate, but the response was unstable at a higher dilution rate due to the input multiplicity behaviour of the process. Thus, nonlinear PI controller performance was found to be superior to that of the linear controller, and earlier reported theoretical results have been validated by the present experimental work.     

V(O2) recovery kinetics in the horse following moderate, heavy, and severe exercise.

At the onset of exercise, horses exhibit O2 uptake (VO2) kinetics that are qualitatively similar to those of humans. In humans, there is a marked dissymmetry between on- and off-kinetics for VO2. This investigation sought to formally characterize the off-transient (recovery) VO2 kinetics in the horse within the moderate (M), heavy (H), and severe (S) exercise domains. Six horses were run on a high-speed treadmill at M, H, and S exercise intensities (i.e., that speed which yielded approximately 50, 85, 100% peak VO2, respectively, on the maximal incremental test). The time courses for the recovery were modeled by using a three-phase model with a single-exponential (fast component) or double-exponential (fast and slow component) phase 2. The single-exponential phase 2 model provided an excellent fit to the off-transient data, with the exception of one horse in the H domain which was best modeled by a double exponential. The time delay elicited no domain dependency (M, 18.0 +/- 1.0; H, 17.6 +/- 1.1; S, 17.8 +/- 2.0 s; P > 0.05), as was the case for the fast-component time constants (M, 16.3 +/- 2.0 s; H, 13.5 +/- 1.0 s; S, 14.6 +/- 0.3 s; P > 0.05). In the H and S (but not M) domains, the VO2 following resolution of the fast component was elevated above the preexercise baseline (H, 3.0 +/- 1.0 l/min; S, 5.7 +/- 1.1 l/min). This additional postexercise VO2 was correlated to the end-exercise increase in lactate (r = 0.94, P < 0.001) but not the end-exercise pulmonary arterial blood temperature (r = 0.45, P > 0.05). These data indicate that the time delay and subsequent kinetic response of the primary (fast-component) phase of exercise VO2 recovery in the horse is independent of the preceding exercise-intensity domain. However, in the H and S domains, the fast component resolves to an elevated baseline.