| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
|
|
|
|||||||
|
|||||||||
University Laboratory of Physiology, Parks Road, Oxford OX1 3PT, United Kingdom
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
ACKNOWLEDGEMENTS
FOOTNOTES
REFERENCES
ABSTRACT 
Liang, Pei-Ji, Jaideep J. Pandit, and Peter A. Robbins. Statistical properties of breath-to-breath
variations in ventilation at constant
PETCO2 and
PETO2 in humans. J. Appl. Physiol. 81(5): 2274-2286, 1996.
The purpose of
this study was to provide a statistical description of the
breath-to-breath variations in ventilation during steady breathing in
both rest and during light exercise, with the end-tidal gases
controlled by using an end-tidal forcing system. Sixty data sets were
studied, only one of which was white (i.e., did not show
autocorrelation). Three simple autoregressive moving average (ARMA)
models, i.e., AR1, AR2, and
AR1MA1, and one simple state-space model were
fitted to the data and resulted in white residuals in 15, 31, 46, and
48 out of 60 occasions, respectively. Evolutionary spectral analysis
revealed that only 13 data sets had a constant power spectrum, although
50 were uniformly modulated. An autoregressive estimate of variance
could be used to "demodulate" the data in most cases, but the
results were not significantly affected by fitting the model to the
demodulated data. The results indicate that 1) both simple
ARMA models and a simple state-space model can describe the
autocorrelation present; 2) variations in spectral power were
present in the data that cannot be described by these models; and
3) these variations were often due to a uniform modulation
and did not significantly affect the coefficients for the models. For
these kinds of data, a heteroscedastic form of state-space model
provides an attractive theoretical structure for the noise processes.
time-series models; state-space model; constant power spectrum; uniform modulation; heteroscedastic form; end-tidal partial pressure of
oxygen; end-tidal partial pressure of carbon dioxide
INTRODUCTION IN THE EARLY 1960S, Priban (17) demonstrated
statistically that, in resting human subjects, successive breaths were
correlated. Since this study, a number of investigators have used a
variety of models to describe the breath-to-breath variation in
breathing (1, 3, 8, 9, 11, 13, 15). These studies cover a wide range of
different conditions, including studies of breathing in animals as well
as in humans, studies during sleep and anesthesia as well as during
consciousness, and studies in the presence of changing stimulation as
well as in the presence of steady stimulation. This makes the results
from these studies difficult to compare. Furthermore, many of these
studies are somewhat limited in scope. In particular 1) the
investigators often have not examined some of the basic statistical
properties of the sequences before applying their models to their data;
2) the investigators often have studied only one model rather
than undertaking a comparison across a range of possible models; and
3) the investigators often have not checked the adequacy of
the model for describing the correlation once the model had been
fitted.
The present study attempts to meet some of the criticisms that can be
leveled against these earlier studies. It focuses on steady breathing
in conscious humans during both rest and mild exercise under euoxic
conditions. The CO2 was slightly elevated compared with air
breathing so that the technique of end-tidal forcing could be used to
hold end-tidal PCO2 and
PO2 (PETCO2
and PETO2, respectively) as
constant as possible. The reason for this was that we wished to
concentrate on the statistical properties of the ventilatory controller
in the open-loop setting rather than on the properties of the entire
closed-loop system, which would include the dynamics of the gas stores
within the body. Thus the purpose of the study was to provide a
statistical description of the breath-by-breath ventilatory sequences
associated with the respiratory controller in an open-loop state in
conscious humans.
As well as the description's own intrinsic interest, it has a
particular application for studies of the respiratory control system
using the end-tidal forcing technique to generate dynamic stimuli in
the end-tidal gases. These studies often involve fitting one or more
models to the ventilatory sequences obtained, and to do this the model
should contain elements to describe both the deterministic and the
stochastic parts of the process. Therefore, a particular aim of the
present study was to develop a suitable structure for the stochastic
part of such models.
METHODS Data
Ventilatory data during steady-state breathing were used for model fitting and statistical analysis. Data collected during the first 50 breaths for protocol A and the first 300 breaths for protocol B were discarded to remove any initial transients in the response.
Model-Independent Statistical Analysis
The mean value for ventilation together with any linear trend term were removed from the data before any statistical analysis and model fitting was undertaken. The values for the mean ventilations and the trend terms are shown in Table 1.
|
|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
(
) over its real value h() approximately follows a
2 distribution such that
{()/h()} ~ a2v, with
both values of a and v determined by the window and the
length of the original data set (19). This enables the 95% confidence intervals for the spectral density to be calculated.
Truncation of each data set from the same subject and the same protocol
to the same length ensures that the estimated spectral density
() over its real value h() will follow a
2 distribution with the same degrees of freedom
v for each of the data sets (since this depends entirely on the
structure of the window and the length of the data set). Thus the
average value for the six estimated spectral densities (for the same
subject and the same protocol) over its real value will also follow a 2 distribution, with the degrees of freedom being
simply six times that of the original data sets (6v). This
enables the 95% confidence intervals for the mean spectral density to
be calculated in a similar way to that used for the spectral densities
from the individual data sets.
Evolutionary spectral analysis.
Evolutionary spectral analysis is a technique for assessing whether
there are changes occurring in the power spectrum of a series over time
(18, 20). The principle is to apply a double window in both time and
frequency domains to calculate values for evolutionary spectral density
corresponding to the selected times and frequencies. First, the
estimate (Ut) for the spectrum for a central
frequency of angular velocity 0 is calculated by
using
|
(1) |
|
(2) |
/h)/(2) = 1/14. So
the spacings between the selected central frequencies were chosen as
3/40 (giving central frequencies of 1/40, 4/40, 7/40, 10/40, 13/40,
16/40, and 19/40). From this, estimates of the spectrum
(Vt) for the selected central time points are
calculated by using
|
(3) |
,t (corresponding to a
Daniell window) is given by
|
(4) |
is a window length, and it was chosen
as 100 (see Eqs. 11.2.58 and 11.2.53 in Ref. 19).
Vt is an (approximately) unbiased estimate of the
(weighted) average value of the power spectrum at the frequency
0 in the neighborhood of the time instant t
(19). A two-way analysis of variance on the logarithmic values of the
spectral density using factors of time and frequency is then applied to
test whether there is significant variation in the spectral density
with time.
An example of this form of analysis is shown in Table 2 for subject
797 at rest. The values for logarithmic spectral densities for
selected time and frequency points are listed in Table
2, whereas the results of variance analysis
are listed in Table 3. The first step is to
test the "interaction + residual" term
[2 = (sum of squares)/
2; in this
study,
2 = 4h/3T = (4 × 7)/(3 × 100) =
7/75 (see Ref. 20), where h and T , selected to be
7 and 100, respectively, are the parameters for the windows of the
frequency and time domains]. If the interaction is not significant, we
conclude that the sequence is a uniformly modulated process and proceed
to examine the "between times" term. The sequence is accepted as
having a constant power spectrum provided the between times term is not
significant.
|
|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|
||||||||||||||||||||||||||||
2
corresponding to P value of 0.0025 as significant for this test (see DISCUSSION). In the example in Table 3, the
value of 2 for interaction + residual results in a
P value of 0.0074, which implies that the sequence analyzed can
be accepted as uniformly modulated. However, the value of
2 for between times is high, which results in a
P value close to zero and means that the sequence cannot be
accepted as having a constant power spectrum.
All the statistical calculations were done by using S-plus statistical
software (Statistical Sciences UK, Oxford, UK).
Model-Dependent Statistical Analysis
Two simple linear time-independent approaches to modeling the sequences were employed. The first is the simple autoregressive moving average (ARMA) model. The second is a simple model in state-space form. Both of these models have a time-independent structure. Both of the models assume there is no exogenous input. These assumptions are explored further in both RESULTS and DISCUSSION. Simple ARMA model fitting. The general form for an ARMA model is
|
(5) |
|
|
(6) |
k is the estimated
autocorrelation value for the sequence of residuals with lag of
k; and K is an integer, the exact choice of which is
arbitrary (we used 10 + p + q in this study).
The coefficients of ARMA models were estimated by using S-plus
statistical software.
State-space model fitting.
As an alternative to specifying the noise processes in terms of an ARMA
model, the noise processes can also be modeled in state-space form
|
(7) |
|
(8) |
|
(9) |
|
(10) |
|
![+ W(<IT>t</IT> + 1)[ <IT>y</IT>(<IT>t</IT> + 1)− <IT><A><AC>x</AC><AC>ˆ</AC></A></IT>(<IT>t</IT> + 1‖<IT>t</IT>)]](/content/vol81/issue5/fulltext/2274/img015.gif)
|
(11) |
![<IT>P</IT>(<IT>t</IT> + 1‖<IT>t</IT> + 1) = [1 − W(<IT>t</IT> + 1)]P(<IT>t</IT> + 1‖<IT>t</IT>)[1 − W(<IT>t</IT> + 1)]](/content/vol81/issue5/fulltext/2274/img016.gif)
|
|
(12) |
![W(<IT>t</IT> + 1) = <IT>P</IT>(<IT>t</IT> + 1‖<IT>t</IT>)[<IT>P</IT>(<IT>t</IT> + 1‖<IT>t</IT>) + <IT>R</IT><SUB><IT>w</IT></SUB>]<SUP>−1</SUP>](/content/vol81/issue5/fulltext/2274/img018.gif)
|
(13) |
= P/Rw. The procedure for
prediction and update then becomes
|
(14) |
|
(15) |
|
![+ W(<IT>t</IT> + 1)[ <IT>y</IT>(<IT>t</IT> + 1)− <IT><A><AC>x</AC><AC>ˆ</AC></A></IT>(<IT>t</IT> + 1‖<IT>t</IT>)]](/content/vol81/issue5/fulltext/2274/img022.gif)
|
(16) |
![<IT>P</IT> ′(<IT>t</IT> + 1‖<IT>t</IT> + 1) = [1 − W(<IT>t</IT> + 1)]](/content/vol81/issue5/fulltext/2274/img023.gif)
|
![· <IT>P</IT> ′(<IT>t</IT> + 1‖<IT>t</IT>)[1 − W(<IT>t</IT> + 1)] + W(<IT>t</IT> + 1)W(<IT>t</IT> + 1)](/content/vol81/issue5/fulltext/2274/img024.gif)
|
(17) |
![W(<IT>t</IT> + 1) = <IT>P</IT> ′(<IT>t</IT> + 1‖<IT>t</IT>)[<IT>P</IT> ′(<IT>t</IT> + 1‖<IT>t</IT>) + 1]<SUP>−1</SUP>](/content/vol81/issue5/fulltext/2274/img025.gif)
|
(18) |
(1|0) were set arbitrarily at 0 and
Rv/Rw,
respectively. The initial transient lasts for only a few steps of
iteration.
By minimizing the prediction error
t [ y(t) 
(t|t 1)]2,
the system parameter f along with the ratio
Rv/Rw can be estimated. The sequences obtained using the Kalman filter for
v(t) and w(t)
[
(t) = (t + 1|t + 1)
f(t|t) and
(t) = y(t)
(t|t)] are such that
one is simply a scalar multiple of the other. It should be noted that
this procedure provides a maximum likelihood estimate for the
parameters and is different from the common procedure of using an
extended Kalman filter to estimate parameters on line (14).
The parameters of the state-space model were estimated using a least
squares method provided in the Numerical Algorithms Group Fortran
library (NAG; Oxford, UK), subroutine E04FDF.
RESULTS
Model-Independent Statistical Analysis
Specific periodicities. The smoothed periodicities for the six data sets for one subject and protocol (subject 811, exercise) are shown in Fig. 1. These show that the power is concentrated in the low frequencies, suggesting the presence of correlation within the data sets. No particular peaks are apparent that are consistent across the six data sets. Furthermore, in general, a smooth decline in power can be traced within the 95% confidence intervals. These findings were generally true for all the individual data sets, and following this observation the periodograms were averaged to produce a single periodogram for each subject and protocol; the results are shown in Fig. 2. Again, smooth lines for the power can be traced within the 95% confidence intervals, suggesting the absence of marked specific periodicities within the data.
Evolutionary spectral analysis. Evolutionary spectral analysis was used to determine whether there were changes occurring over time in the power spectrum. The results are shown in Table 4. When the values for the interaction + residual term are examined, they show that 24 of these 30 data sets for rest can be accepted as uniformly modulated but only 7 out of the 24 could be accepted as having a constant power spectrum over time. For exercise, 26 out of the 30 data sets could be treated as uniformly modulated, but only 6 of these 26 as having a constant power spectrum.
|
|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Model-Dependent Statistical Analysis
The results from the evolutionary spectral analysis suggest that simple linear time-invariant models are unlikely to describe the data fully. Nevertheless, we start with these models to investigate what aspects of the data the models fail to fit and also to determine whether there are aspects of the data that these models can describe well despite inadequacies elsewhere. Simple ARMA model fitting. Three kinds of low-order time-series models were applied to each individual data set. They were AR1, a single autoregressive term; AR2, two autoregressive terms; and AR1MA1, a single autoregressive term together with a moving average term. The result of each model fitting was tested by assessing the whiteness of the residuals using a portmanteau test. The numbers of data sets with white residuals for each subject and protocol are listed in Table 5. The AR1MA1 model is the most satisfactory model for both protocols. This model described 22 of the 30 resting data sets and 24 of 30 exercise data sets with overall white residuals.
|
||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1 coefficient model; AR2, 1 and 2 coefficient
model; AR1MA1, 1 and
c1 coefficient model. Top: only
AR1MA1 model is adequate. Bottom:
none of 3 models is adequate. Solid lines, autocorrelation functions;
dashed lines, 95% confidence intervals.
The mean and SD values for each coefficient of the three fitted models for each subject and protocol are listed in Table 6. In general, these coefficients are not too different between protocols and subjects, although the values for subject 807 at rest are unusual. The autocorrelation function for this subject at rest appeared much closer to white than in all other subjects and protocols.
|
|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|
|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|
(19) |
|
(20) |
After demodulating the original sequence by C(t), we obtain the demodulated sequence. Evolutionary spectral analysis on this demodulated sequence was then used to determine whether the variation in power spectrum had been removed by demodulation. The results are shown in Table 4. The number of data sets that had a constant power spectrum rose from 7 and 6 before demodulation to 24 and 26 after demodulation for resting and exercising data, respectively. It is possible that these figures could rise further with a different choice of demodulation function. Finally, it is worth noting that similar results are obtained if demodulation is applied to the residuals from the state-space model instead of the original data sequences (Table 4). Suitability of models. Both the ARMA models and the state-space model used above are linear time-invariant models. However, the results from evolutionary spectral analysis show that in most sequences the power spectrum is varying, and this may suggest that the process underlying the generation of the sequence is nonstationary and/or nonlinear. This variation is clearly not described by these models. The question thus arises as to whether these models are at all useful for describing the data sequences. One approach to answering this question is to apply the models to the data after demodulation, which removes, or at least attenuates, the variation in the power spectrum from the sequence. There was little difference between the number of residual data sets accepted as white before demodulation compared with after demodulation (Table 5). Also, the coefficients from the ARMA model fitting on the demodulated data (Table 6, bottom) are similar to the values obtained before demodulation (Table 6, top). Analysis of variance performed on the AR1MA1 model coefficients revealed no significant differences between the values obtained with the modulated and demodulated data. These findings suggest that the autocorrelation in the sequence can usefully be described by a simple model despite the presence of a slowly varying variance. The state-space model is equivalent to the AR1MA1 model with certain restrictions, and similar results for whiteness were obtained before and after demodulation (Table 5).
DISCUSSION
The major findings of this study are as follows. 1) Both a simple ARMA model and a simple linear time-invariant state-space model for the breath-to-breath correlations present in the ventilatory data gathered by using an end-tidal forcing system to hold end-tidal gas tensions constant can produce residuals that, in the majority of cases, are overall white. 2) Neither the simple ARMA models nor the simple linear time-invariant state-space model fully describe the data because the power spectra associated with the residuals sequences vary with time. 3) In a sizable proportion of cases, this variation in power spectrum can be explained by a process of uniform modulation of the ventilatory data or residuals. 4 ) One chosen modulating function was shown to be effective in reducing these data sequences to ones with constant power spectra in the majority of cases.
Comparison with Previous Studies
In general, it is quite difficult to draw direct comparisons between the results from our study and results from previous studies. First, by using data gathered with the end-tidal forcing technique, our study has concentrated on the properties of the respiratory controller, whereas most other studies have examined breath-by-breath variations when the chemical feedback loop is intact. Such a feedback loop might be expected to affect the correlational structure very significantly. Second, studies have varied in terms of the background level of stimulation employed (if any), the species employed, and whether the study was conducted with subjects awake, asleep, or under anesthesia. Specific periodicities. In our study, the results from spectral analysis failed to reveal any sustained specific periodicities in the data. In conscious humans, by displaying the spectral densities for sequential subsets of the original data series, Jensen (11) failed to find any peaks or troughs that were persistently present throughout the length of the series. This is also in keeping with a study of human subjects during stage 2 sleep (15). These results support the absence of specific periodic components in the data series. However, in the present study, the results of evolutionary spectral analysis generally show that the power spectrum is not constant, and thus none of the above studies can exclude the presence of brief periods of periodicity that are not present throughout the data sequence. Evolutionary spectral analysis. Most studies have not considered whether the data sequence retains a constant power spectrum throughout. In conscious humans, Jensen (11) calculated power spectra for sequential subsets of ventilatory data sequences but did not test whether they differed significantly from one another. Ackerson et al. (1) commented that data sequences were often "nonstationary" when assessed by using a "likelihood ratio test," but did not provide any results to compare with those in this study. Nevertheless, in their study, Ackerson et al. developed an adaptive time-series model for describing nonstationary ventilatory variations. This model has many similarities to the modulation model proposed in the present study. ARMA models. The results from our study suggest that, of AR1, AR2, and AR1MA1 model structures, the AR1MA1 structure was the most satisfactory for describing the breath-by-breath correlation in the data sequence. We have not found another study that has investigated the AR1MA1 structure for ventilatory sequences. Jensen (11) reports values in humans for coefficients from multivariate AR1 and AR2 models for tidal volume, inspired time, and expired time pooled from a range of different experimental conditions. All coefficients for the lagged effects of each variable on itself are positive. Modarreszadeh et al. (15) examined an AR2 model for describing ventilatory variability in sleeping humans. They found that the AR1 coefficient was significantly positive but that the AR2 coefficient did not differ significantly from zero. In the anesthetized cat, Benchetrit and Bertrand (3) used an "isolated respiratory centre" preparation to examine the statistical properties of the phrenic nerve output. This was done so as to circumvent the influence of the chemical feedback system, rather as we have attempted with the end-tidal forcing system in our present study. The study employed a multivariate AR1 structure and found a positive autocorrelation coefficient, suggesting that there was indeed breath-to-breath correlation arising within the respiratory controller. The first-order model structure was not always adequate. Khatib et al. (13) used an AR1 model structure to compare the correlation between tidal volumes in anesthetized spontaneously breathing rats, with the correlation between the integrated phrenic neurogram in anesthetized vagotomized artificially ventilated rats. In hyperoxia and hypercapnia, they found positive autoregressive coefficients for the tidal volume data in the spontaneously breathing rats but not for the integrated phrenic neurogram in the artificially ventilated rats. From these results, they conclude that the autocorrelation in tidal volume arises from the presence of noise within the chemical feedback system rather than as a function of the respiratory controller. They suggest that the difference between their results and those of Benchetrit and Bertrand (3) could either be due to species differences or to Benchetrit and Bertrand failing to eliminate experiments in which there was nonstationarity. The results from our study in conscious humans accord much more closely with those of Benchetrit and Bertrand than with those of Khatib et al. (13). State-space models. We are unaware of any studies in which a state-space formulation has been used for the noise processes in the absence of additional deterministic input stimuli. However, there have been studies of different noise models in experiments involving step changes in PETCO2 in both anesthetized cats and in conscious humans (8, 9). These studies do not report any of the parameter values associated with the stochastic part of the model, but the example autocorrelation functions suggest that both a "process" and a "parallel" noise model lead to a considerable whitening of the residuals when compared with the "measurement only" noise model. Effects of exercise. We are unaware of any previous studies that have provided a comparison of these models between the states of rest and exercise. In general, the results from the two states were surprisingly similar given the differences in the mean ventilations, the associated differences in respiratory cycle durations, and the somewhat different mental state that may exist between rest and exercise. Inspection of Tables 6 and 7 suggests that there may be some increase in the degree of autocorrelation observed in exercise compared with rest, but this did not reach significance when the values of the coefficients were compared for the AR1MA1 and state-space models.Evolutionary Spectral Analysis, Stationarity, and Time- and State-Dependent Models
Evolutionary spectral analysis. Evolutionary spectral analysis is a means of assessing whether the power spectrum of a data sequence is constant by estimating the power spectrum at certain specified points in time. This has been applied as a test of whether the underlying data sequence is stationary (18, 20). This section discusses the utility of evolutionary spectral analysis for this purpose, together with what may be inferred about the underlying model structure from this analysis. It also considers the reliability of the analysis for the particular form and length of data sequence that has been used in this study. A series is (weakly) stationary if the first and second moments are constant over time (i.e., the expected value of x, E(x) = µ and the variance of x Var (x) =2). If there are multiple
independent realizations of a time series, then stationarity may be
tested for by estimating the mean and variance at each point in the
sequence. However, in other cases, multiple realizations may not exist,
as is the case with our data. In this case, it is necessary to test for
stationarity by sampling the series at intervals that are sufficiently
far apart for the samples to be considered independent. Evolutionary
spectral analysis is one method based on this approach. This leaves the
question of how far apart do the samples have to be in order to be
sufficiently far apart? The problem is that very slow variations in the
structure of a time series may occur (especially with nonlinear series) despite the fact that the series is actually still stationary. Thus it
is safer to interpret results from evolutionary spectral analysis in
terms of whether there is long-term variation in a sequence than
whether the process is truly stationary. Indeed, the autoregressive
model used to estimate the slowly changing variance in this study is a
time-invariant model, despite the fact that it is being used to
describe "nonstationary" behavior detected by evolutionary
spectral analysis.
There is an interesting form of duality between the ideas of
nonstationarity and nonlinearity (20). A general stationary state-dependent model of order (k, l) may be written
as
|
|
(21) |
l 1, ... , et,
Xt k + 1, ... , Xt); 
u and
u are AR and MA coefficients, respectively; and µ is local mean.
This form is a stationary nonlinear model. However, for a single
realization, it is also possible to think of the state-dependent coefficients
u(xt), u(xt) as time-dependent coefficients
u(t), u(t) via the definitions u(t) 
u(xt) and
u(t) u(xt). In this formulation, the model may be thought of as a linear nonstationary model (19). This duality illustrates clearly the difficulty of
distinguishing between linear nonstationary models and stationary nonlinear models when there is only a single realization of the data
sequence.
The success, or otherwise, of using evolutionary spectral analysis to
classify time series must depend on the type and length of the actual
sequences used together with the particular forms of window chosen. To
explore this further, 60 AR1MA1 data sets were
generated, each set matching one of the real data sets for AR1 and MA1 coefficients and for record length.
Evolutionary spectral analysis was then undertaken on these sequences.
Using a significance level of P < 0.05 for the
2 test, over 20% of the sequences were incorrectly
classified as arising from a process generating sequences without a
constant power spectrum. However, modulation of each of the sequences
with a modulating function that was the same as that which had been determined for the matching experimental data sequence and then repetition of the evolutionary spectral analysis on the modulated sequences generally resulted in levels of significance for the 2 test that were very much higher than those for the
unmodulated sequences. When a value for 2
corresponding to P < 0.0025 was used, it resulted in only 4 of the 60 stationary data sets being misclassified as not having constant power, but only 7 of the 60 modulated data sets were classified as being of constant power. Furthermore, it must be remembered that in a number of cases the modulating function may have
been very "mild." The results from this simulation determined the
choice of significance level used for the 2 test in
this study.
Adequacy of simple linear time-invariant models.
A major result of this study is that the correlation between breaths
can be modeled by simple linear time-invariant ARMA or state-space
models but that there remains a longer-term variation in the variance
of the sequence. One question that naturally arises is to what extent
can we separate these two features of the sequence and, in particular,
to what degree may a longer term variation in ventilatory variance
affect the estimation of the coefficients describing the shorter term
correlational structure? A partial answer to this question is found in
RESULTS, where the coefficients of the
AR1MA1 model were estimated both before and
after demodulation. No significant differences were found. However, it
might be argued that this result arises because both the modulation and
the autoregressive coefficients were estimated from the same data
sequence.
To overcome this objection, a simulation study was performed. A set of
10 data sequences was constructed with a known underlying AR1MA1 structure, with the coefficient values
equal to the mean values for each subject and protocol. First, the
coefficients were estimated from these simulated data by using the same
technique as used for the real data sequences. Second, the simulated
data sequences were modulated by using a randomly selected modulating function, one from each subject and protocol. Then the coefficients were reestimated by using these modulated sequences. Finally, the
values estimated from both the original and modulated sequences were
compared. Comparison of the parameter values between the modulated and
unmodulated data sets revealed an average deviation for the
a1 coefficient of 0.8 ± 8.35 (SD) % and for the
c1 coefficient of 5.9 ± 20.08%. Analysis of
variance revealed no significant differences for either coefficient.
These results support the notion that the shorter term correlations in
the data sequence can be estimated in the presence of the slower
variations in the variance of the sequence.
Influence of CO2 on Data Sequences
A major intent of this study was to examine the open-loop characteristics of the respiratory controller with chemical levels of stimulation held constant. However, the experimental technique of dynamic end-tidal forcing is imperfect, and small fluctuations in PETCO2 and PETO2 remain. Because all experiments were conducted at a mean PETO2 of 100 Torr, it is most unlikely that the variations in PETO2 will have any effect on the ventilatory sequence because the respiratory controller's sensitivity to changes in PETO2 around this mean level will be extremely low (6). The same may not necessarily be true for PCO2, since the ventilatory sensitivity to changes in PCO2 is much higher and, indeed, cross-correlation analysis reveals a significant effect of PETCO2 on minute ventilation (
E) in 30 out of 60 data sets
(portmanteau test for cross correlation). Thus one question that arises
is whether our results reflect the combined properties of the human
respiratory system coupled to our dynamic end-tidal forcing apparatus
rather than simply the open-loop properties of the respiratory
controller.
To try to answer this question, a multivariate time-series approach has been adopted. This was necessary because not only may the PETCO2 affect the ventilation but the breath-by-breath fluctuations in ventilation will affect PETCO2. Many different model structures are possible when using multivariate analysis, and to make progress we decided to focus on one simple model that has some structural basis in terms of the known properties of both the physiological and the dynamic end-tidal forcing systems.
The PETCO2 of a breath is determined (assuming that CO2 delivery to the lungs via the blood remains constant) by 1) the PETCO2 of the previous breath; 2) the inspired PCO2 of the current breath; and 3) the ventilation of the current breath. Thus we may write
![P<SC>et</SC><SUB>CO<SUB>2</SUB></SUB>(<IT>n</IT>) = <IT>h</IT>[P<SC>et</SC><SUB>CO<SUB>2</SUB></SUB>(<IT>n</IT> − 1), P<SC>i</SC><SUB>CO<SUB>2</SUB></SUB>(<IT>n</IT>), <A><AC>V</AC><AC>˙</AC></A><SC>e</SC>(<IT>n</IT>)]](/content/vol81/issue5/fulltext/2274/img033.gif)
|
(22) |
where h is a general function and PICO2 is inspiratory PCO2. The end-tidal forcing system uses a prediction-correction scheme, with the correction based on an integral-proportional controller structure. We may write for this
![P<SC>i</SC><SUB>CO<SUB>2</SUB></SUB>(<IT>n</IT>) = P<SC>i</SC><SUB>CO<SUB>2</SUB> p</SUB> + g<SUB>p</SUB>[P<SC>et</SC><SUB>CO<SUB>2</SUB> d</SUB> − P<SC>et</SC><SUB>CO<SUB>2</SUB> m</SUB>(<IT>n</IT> − 1)]](/content/vol81/issue5/fulltext/2274/img034.gif)
|
![+ g<SUB>i</SUB> <LIM><OP>∑</OP><LL><IT>j</IT> = 1</LL><UL><IT>n</IT> − 1</UL></LIM> [P<SC>et</SC><SUB>CO<SUB>2</SUB> d</SUB> − P<SC>et</SC><SUB>CO<SUB>2</SUB> m</SUB>( <IT>j</IT>)]](/content/vol81/issue5/fulltext/2274/img035.gif)
|
(23) |
where subscripts p, d, and m are predicted, desired, and measured gas tensions, respectively, and gp and gi are the proportional and integral feedback gains, respectively.
In our experiments,
PETCO2 d is
constant, and after a period of time the integral term should be
relatively constant. Thus PICO2(n) is really just a
linear function of
PETCO2 m(n 1). Hence, we may write Eq. 22 as
![P<SC>et</SC><SUB>CO<SUB>2</SUB></SUB>(<IT>n</IT>) = <IT>h</IT>′[P<SC>et</SC><SUB>CO<SUB>2</SUB></SUB>(<IT>n</IT> − 1), <A><AC>V</AC><AC>˙</AC></A><SC>e</SC>(<IT>n</IT>)]](/content/vol81/issue5/fulltext/2274/img036.gif)
|
(24) |
where h is a general function. For small
fluctuations in PETCO2, we
linearize around the means and write
|
|
|
(25) |
This now gives a simple expression for the way ventilation may affect CO2 in our system.
The PETCO2 may affect ventilation either through the peripheral chemoreceptors or through the central chemoreceptors or both. A reasonable estimate for the median lag to the peripheral chemoreceptors is two breaths (5). Estimates for the lag for the central chemoreceptors vary, but a value of three to four breaths might be reasonable (7). To keep the model simple, we chose to use a lag of two breaths, which should model a peripheral response and still be capable of modeling at least some of any central response that is present. When this is combined with our AR1MA1 model for ventilation and the model derived above for PETCO2, we get an entire multivariate model of the form
|
|
(26) |
|
|
(27) |
This model was fitted to our data by using the NAG library routine G13DCF with and without the coefficient a12 being fixed at zero, to examine the effects of either incorporating or not incorporating the effects of PETCO2 on ventilation. The exclusion of the effects of PETCO2 on ventilation reduced the average value of coefficient a11 by 7% and reduced the average value of coefficient c11 by 14%. Both these changes were significant when using analysis of variance. Thus we conclude that when the variations in PETCO2 are ignored, it causes a modest underestimation of the degree of correlation associated with the respiratory controller.
After fitting the multivariate model, the cross correlations between
the residuals for ventilation and the
PETCO2 were calculated. There was
no significant effect of PCO2 on
E in 51 out of 60 data sets, confirming
that the effect of fluctuations in
PETCO2 on
E had been modeled satisfactorily for
most data sequences.
Generalization of State-Space Model to Incorporate a Time- and State-Dependent Form
The results have shown that the degree of variation in ventilatory variance observed with a constant level of stimulation is such that it does not affect the fitting of the correlational structure unduly. However, it is not clear whether this would still be the case if the level of stimulation were varying because, in this case, the variation in ventilatory variance is likely to be substantially greater (cf. variances for rest and exercise in Fig. 4). In this sense, the results of the current study might be seen as rather limited when applying models to situations in which the level of stimulation is varying (i.e., there is a deterministic input present, rather than the purely stochastic processes considered in the present study).As an alternative to the linear time-invariant state-space model, a more general state-space model can be written as follows
|
(28) |
|
(29) |
where 
(t) and 
(t) are noise
processes with state- and time-dependent variances
![<IT>R</IT><SUB>&xgr;</SUB> = <IT>g</IT>[<IT>x</IT>(<IT>t</IT>), <IT>t</IT>]<IT>R</IT><SUB><IT>v</IT></SUB>](/content/vol81/issue5/fulltext/2274/img046.gif)
|
(30) |
![<IT>R</IT><SUB>&eegr;</SUB> = <IT>g</IT>[<IT>x</IT>(<IT>t</IT>), <IT>t</IT>]<IT>R</IT><SUB><IT>w</IT></SUB>](/content/vol81/issue5/fulltext/2274/img047.gif)
|
(31) |
This model allows the variances to vary with amplitude of ventilation
and with time through the function
g[x(t), t]. This is known as
a heteroscedastic form (10). The advantage is that provided
g[x(t), t] is fully
determined at time t, then the same Kalman filter can be used
to estimate the coefficients much as before, since the term in
g drops out while the ratio of the variances is estimated. The
only differences are that the prediction and update equations for the
variance P should be defined by using R
and R
instead of Rv
and Rw
|
(32) |
![<IT>P</IT>(<IT>t</IT> + 1‖<IT>t</IT> + 1) = [1 − W(<IT>t</IT> + 1)]](/content/vol81/issue5/fulltext/2274/img049.gif)
|
![· <IT>P</IT>(<IT>t</IT> + 1‖<IT>t</IT>)[1 − W(<IT>t</IT> + 1)] + W(<IT>t</IT> + 1)<IT>R</IT><SUB>&eegr;</SUB><IT>W</IT>(<IT>t</IT> + 1)](/content/vol81/issue5/fulltext/2274/img050.gif)
|
(33) |
![W(<IT>t</IT> + 1) = <IT>P</IT>(<IT>t</IT> + 1‖<IT>t</IT>)[<IT>P</IT>(<IT>t</IT> + 1‖<IT>t</IT>) + <IT>R</IT><SUB>&eegr;</SUB>]<SUP>−1</SUP>](/content/vol81/issue5/fulltext/2274/img051.gif)
|
is the Kalman gain at time t + 1, and that
P is defined as P = P/R for the estimating procedure. The
ratio R/R equals the
ratio Rv/Rw
and may be estimated in the estimation procedure.
The question now remains, is this heteroscedastic form likely to be an
adequate model for ventilatory variability? Two possible limitations
are 1) the same function g is used to modulate
variances of both noise sequences, and 2) the function
g has to be fully determined (i.e., nonstochastic) at time
t. For the scalar output of ventilation, the first of these
limitations is unimportant as (t) and (t) are
not separately identifiable and the sequences 
(t) and 
(t) are scalar
multiples of each other. Consequently, the same function
(t) may be constructed to modulate both
v(t) and w(t) to produce
(t) and (t), respectively.
The importance of the second of these limitations is less clear theoretically. However, if g(t) is varying sufficiently slowly with respect to the observations, then a good estimation of g(t) may be obtained from prior observations. The observation in this study that many sequences were uniformly modulated, combined with the practical construction of the demodulation function C(t) for the residual sequences, suggests that such a function does exist (or at least a good approximation exists).
For use in parameter estimation, the heteroscedastic noise model is no more complicated than the ordinary form, since the algorithms associated with each are the same. Thus, for a parallel noise structure, the heteroscedastic noise model can be combined with deterministic models in the same way as an ordinary noise model. The deterministic and stochastic components are simply added together. Because the algorithms are the same for both the ordinary and heteroscedastic models, parameter estimates obtained by using these models are the same in each case. The important point is that the heteroscedastic model provides the necessary theoretical extension to deal with noise processes that are not of constant power but that are, nevertheless, uniformly modulated.
In summary, the heteroscedastic structure allows the variation in ventilatory variance to be introduced into the model in a general way, without specifying the precise form that it takes. Although this does not contribute further to understanding the origins of the variation in ventilatory variance, it does provide a useful means to incorporate this feature into overall models of ventilatory control and still provide a structure that is amenable to parameter estimation.
ACKNOWLEDGEMENTS
P. A. Robbins thanks many colleagues, including Drs. Berkenbosch, DeGoede, and Marriott, and Profs. Swanson, and Ward for all the discussion that has helped to give rise to this paper.
FOOTNOTES
Received 29 March 1995; accepted in final form 12 July, 1996.
APPENDIX
Relationship between State-Space Form and AR1MA1 Form
The first section of the Appendix derives the appropriate ARMA representation of the state-space model employed in this study. The second section derives the relations between the parameters of the state-space model and those of the ARMA model.ARMA Representation of State-Space Model
The state-space model with noise processes and zero input employed in this study may be written as
|
(A1) |
|
(A2) |
From Eq. A2, we may write
|
(A3) |
|
![= f<IT>x</IT>(<IT>t</IT> − 1) + <IT>v</IT>(<IT>t</IT> − 1) + <IT>w</IT>(<IT>t</IT>) − <IT>a</IT><SUB>1</SUB>[<IT>x</IT>(<IT>t</IT> − 1) + <IT>w</IT>(<IT>t</IT> − 1)]](/content/vol81/issue5/fulltext/2274/img058.gif)
|
|
(A4) |
1) drops out and the equation may
be written as
|
(A5) |
k = 0 for all
k > 1, the combination of these two processes builds up a
MA1 process (12). So Eq. A5 can be rewritten as
|
(A6) |
Relationship between Parameters of AR1MA1 and State-Space Model
Define z(t) as
|
(A7) |
|
(A8) |
![E[<IT>z</IT>(<IT>t</IT>)]<SUP>2</SUP> = E[<IT>v</IT>(<IT>t</IT> − 1) + <IT>w</IT>(<IT>t</IT>) − f<IT>w</IT>(<IT>t</IT> − 1)]<SUP>2</SUP>](/content/vol81/issue5/fulltext/2274/img064.gif)
|
(A9) |
|
(A10) |
|
(A11) |
![E[<IT>z</IT>(<IT>t</IT>)]<SUP>2</SUP> = E[&egr;(<IT>t</IT>) − <IT>c</IT><SUB>1</SUB>&egr;(<IT>t</IT> − 1)]<SUP>2</SUP>](/content/vol81/issue5/fulltext/2274/img067.gif)
|
(A12) |
(t) is white
|
(A13) |
is the variance of sequence
.
After equating Eqs. A10 and A13 and eliminating Rz, we get
|
(A14) |
In addition to the above result, we may also calculate
E[z(t)z(t 1)]. From
Eq. A8 we
obtain
![E[<IT>z</IT>(<IT>t</IT>)<IT>z</IT>(<IT>t</IT> − 1)] = E{[<IT>v</IT>(<IT>t</IT> − 1) − f<IT>w</IT>(<IT>t</IT>) + <IT>w</IT>(<IT>t</IT> − 1)]](/content/vol81/issue5/fulltext/2274/img070.gif)
|
![&z.ccirf; [<IT>v</IT>(<IT>t</IT> − 2) − f<IT>w</IT>(<IT>t</IT> − 1) + <IT>w</IT>(<IT>t</IT> − 2)]}](/content/vol81/issue5/fulltext/2274/img071.gif)
|
(A15) |
![E[<IT>z</IT>(<IT>t</IT>)<IT>z</IT>(<IT>t</IT> − 1)]= −f<IT>R</IT><SUB><IT>w</IT></SUB>](/content/vol81/issue5/fulltext/2274/img072.gif)
|
(A16) |
![E[<IT>z</IT>(<IT>t</IT>)<IT>z</IT>(<IT>t</IT> − 1)]](/content/vol81/issue5/fulltext/2274/img073.gif)
|
![= E{[&egr;(<IT>t</IT>) − <IT>c</IT><SUB>1</SUB>&egr;(<IT>t</IT> − 1)][&egr;(<IT>t</IT> − 1) − <IT>c</IT><SUB>1</SUB>&egr;(<IT>t</IT> − 2)]}](/content/vol81/issue5/fulltext/2274/img074.gif)
|
(A17) |
is white, then
![E[<IT>z</IT>(<IT>t</IT>)<IT>z</IT>(<IT>t</IT> − 1)] = −<IT>c</IT><SUB>1</SUB><IT>R</IT><SUB>&egr;</SUB>](/content/vol81/issue5/fulltext/2274/img075.gif)
|
(A18) |
1)] yields
|
(A19) |
, we obtain
|
(A20) |
|
![<FR><NU><IT>R</IT><SUB><IT>v</IT></SUB> + (1 + f<SUP>2</SUP>)<IT>R</IT><SUB><IT>w</IT></SUB> ± <RAD><RCD>[<IT>R</IT><SUB><IT>v</IT></SUB> + (1 + f<SUP>2</SUP>)<IT>R</IT><SUB><IT>w</IT></SUB>]<SUP>2</SUP> − 4f<SUP>2</SUP><IT>R</IT> <SUP>2</SUP><SUB><IT>w</IT></SUB></RCD></RAD></NU><DE>2f<IT>R</IT><SUB><IT>w</IT></SUB></DE></FR>](/content/vol81/issue5/fulltext/2274/img079.gif)
|
(A21) |
|
(A22) |
![<IT>c</IT><SUB>1</SUB> = <FR><NU><IT>R</IT><SUB><IT>v</IT></SUB> + (1 + f<SUP>2</SUP>)<IT>R</IT><SUB><IT>w</IT></SUB> − <RAD><RCD>[<IT>R</IT><SUB><IT>v</IT></SUB> + (1 + f<SUP>2</SUP>)<IT>R</IT><SUB><IT>w</IT></SUB>]<SUP>2</SUP> − 4f<SUP>2</SUP><IT>R</IT> <SUP>2</SUP><SUB><IT>w</IT></SUB></RCD></RAD></NU><DE>2f<IT>R</IT><SUB><IT>w</IT></SUB></DE></FR>](/content/vol81/issue5/fulltext/2274/img081.gif)
|
(A23) |
REFERENCES
| 1. |
Ackerson, L. M.,
R. H. Jones,
and
E. N. Bruce.
Adaptive multivariate autoregressive modeling of respiratory cycle variables.
In: Respiratory ControlA Modeling Perspective. New York: Plenum, 1989, p. 309-316.
|
| 2. | Anderson, B. D. O., and J. B. Moore. Optimal Filtering. Englewood Cliffs, NJ: Prentice-Hall, 1979. |
| 3. | Benchetrit, G., and F. Bertrand. A short-term memory in the respiratory centres: statistical analysis. Respir. Physiol. 23: 147-158, 1975. |
| 4. |
Box, G. E. P.,
and
G. M. Jenkins.
Time Series AnalysisForecasting and Control (rev. ed.). San Francisco, CA: Holden-Day, 1976.
|
| 5. | Clement, I. D., and P. A. Robbins. Latency of the ventilatory chemoreflex response to hypoxia in humans. Respir. Physiol. 92: 277-287, 1993. |
| 6. | Cunningham, D. J. C., P. A. Robbins, and C. B. Wolff. Interaction of respiratory responses to changes in alveolar partial pressures of CO2 and O2 and in arterial pH. In: Handbook of Physiology. The Respiratory System. Control of Breathing. Bethesda, MD: Am. Physiol. Soc., 1986, vol. II. pt. 2, chapt. 15, p. 475-528. |
| 7. | Dahan, A., J. DeGoede, A. Berkenbosch, and I. C. W. Olievier. The influence of oxygen on the ventilatory response to carbon dioxide in man. J. Physiol. Lond. 428: 485-499, 1990. |
| 8. |
Dahan, A.,
I. C. W. Olievier,
A. Berkenbosch,
and
J. DeGoede.
Modeling the dynamic ventilatory response to carbon dioxide in healthy human subjects during normoxia.
In: Respiratory ControlA Modeling Perspective. New York: Plenum, 1989, p. 265-273.
|
| 9. | DeGoede, J., and A. Berkenbosch. Dynamic end-tidal forcing technique: modeling the ventilatory response to carbon dioxide. In: Modeling and Parameter Estimation in Respiratory Control. New York: Plenum, 1989, p. 59-69. |
| 10. | Harvey, A. C. Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge, UK: Cambridge Univ. Press, 1989. |
| 11. | Jensen, J. I. An Analysis of Breath-to-Breath Variability in Steady-States of Breathing in Man (PhD thesis). Arahus Universitet, Denmark, 1987. |
| 12. | Kendall, M. G., and A. Stuart. The Advanced Theory of Statistics (3rd ed.). London: Griffin, 1969. |
| 13. | Khatib, M. F., Y. Oku, and E. N. Bruce. Contribution of chemical feedback loops to breath-to-breath variability of tidal volume. Respir. Physiol. 83: 115-128, 1991. |
| 14. |
Ljung, L.
System IdentificationTheory for the User. Englewood Cliffs, NJ: Prentice-Hall, 1987.
|
| 15. | Modarreszadeh, M., E. N. Bruce, and B. Gothe. Nonrandom variability in respiratory cycle parameters of humans during stage 2 sleep. J. Appl. Physiol. 69: 630-639, 1990. |
| 16. | Pandit, J. J., and P. A. Robbins. Ventilation and gas exchange during sustained exercise at normal and raised CO2 in man. Respir. Physiol. 88: 101-112, 1992. |
| 17. | Priban, I. P. An analysis of some short-term patterns of breathing in man at rest. J. Physiol. Lond. 166: 425-434, 1963. |
| 18. | Priestley, M. B. Spectral Analysis and Time Series. London: Academic, 1981. |
| 19. | Priestley, M. B. Non-linear and Non-Stationary Time Series Analysis. London: Academic, 1991. |
| 20. | Priestley, M. B., and S. S. Rao. A testing of non-stationarity of time-series. J. R. Stat. Soc. Ser. B 31: 140-149, 1969. |
This article has been cited by other articles:
|
|
 |
|
  C. Liu, G. M. Balanos, M. Fatemian, T. G. Smith, K. L. Dorrington, and P. A. Robbins Effects of hydralazine on the pulmonary vasculature and respiratory control in humans Exp Physiol, January 1, 2008; 93(1): 104 - 114. [Abstract] [Full Text] [PDF]
|
|
|
|
 |
|
 C. Liu, T. G. Smith, G. M. Balanos, J. Brooks, A. Crosby, M. Herigstad, K. L. Dorrington, and P. A. Robbins Lack of involvement of the autonomic nervous system in early ventilatory and pulmonary vascular acclimatization to hypoxia in humans J. Physiol., February 15, 2007; 579(1): 215 - 225. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
H. E. Wood, M. Fatemian, and P. A. Robbins Respiratory: Prior sustained hypoxia attenuates interaction between hypoxia and exercise as ventilatory stimuli in humans Exp Physiol, January 1, 2007; 92(1): 273 - 286. [Abstract] [Full Text] [PDF]
|
|
|
|
 |
|
 S. Donoghue, M. Fatemian, G. M. Balanos, A. Crosby, C. Liu, D. O'Connor, N. P. Talbot, and P. A. Robbins Ventilatory acclimatization in response to very small changes in PO2 in humans J Appl Physiol, May 1, 2005; 98(5): 1587 - 1591. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
A. Crosby, N. P. Talbot, G. M. Balanos, S. Donoghue, M. Fatemian, and P. A. Robbins Respiratory effects in humans of a 5-day elevation of end-tidal PCO2 by 8 Torr J Appl Physiol, November 1, 2003; 95(5): 1947 - 1954. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
M. Fatemian, D. J F Nieuwenhuijs, L. J Teppema, S. Meinesz, A. G L van der Mey, A. Dahan, and P. A Robbins The respiratory response to carbon dioxide in humans with unilateral and bilateral resections of the carotid bodies J. Physiol., June 15, 2003; 549(3): 965 - 973. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
F. Leon-Velarde, A. Gamboa, M. Rivera-Ch, J.-A. Palacios, and P. A. Robbins Plasticity in Respiratory Motor Control: Selected Contribution: Peripheral chemoreflex function in high-altitude natives and patients with chronic mountain sickness J Appl Physiol, March 1, 2003; 94(3): 1269 - 1278. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
M. Fatemian, A. Gamboa, F. Leon-Velarde, M. Rivera-Ch, J.-A. Palacios, and P. A. Robbins Plasticity in Respiratory Motor Control: Selected Contribution: Ventilatory response to CO2 in high-altitude natives and patients with chronic mountain sickness J Appl Physiol, March 1, 2003; 94(3): 1279 - 1287. [Abstract] [Full Text] [PDF]
|
|
|
|
 |
|
 E. N. Bruce Assessing Respiratory Control during Spontaneous Breathing . Practice May Be More Difficult than Theory Am. J. Respir. Crit. Care Med., April 15, 2002; 165(8): 1033 - 1034. [Full Text] [PDF]
|
|
|
|
|
|
M. Fatemian and P. A. Robbins Physiological and Genomic Consequences of Intermittent Hypoxia: Selected Contribution: Chemoreflex responses to CO2 before and after an 8-h exposure to hypoxia in humans J Appl Physiol, April 1, 2001; 90(4): 1607 - 1614. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
X. Ren, M. Fatemian, and P. A. Robbins Changes in respiratory control in humans induced by 8 h of hyperoxia J Appl Physiol, August 1, 2000; 89(2): 655 - 662. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
M. E F Pedersen, M. Fatemian, and P. A Robbins Identification of fast and slow ventilatory responses to carbon dioxide under hypoxic and hyperoxic conditions in humans J. Physiol., November 15, 1999; 521(1): 273 - 287. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
M. E F Pedersen, K. L Dorrington, and P. A Robbins Effects of somatostatin on the control of breathing in humans J. Physiol., November 15, 1999; 521(1): 289 - 297. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
P.-J. Liang, D. A. Bascom, and P. A. Robbins Extended models of the ventilatory response to sustained isocapnic hypoxia in humans J Appl Physiol, February 1, 1997; 82(2): 667 - 677. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
T. Busso, P.-J. Liang, and P. A. Robbins Breath-to-breath relationships between respiratory cycle variables in humans at fixed end-tidal PCO2 and PO2 J Appl Physiol, November 1, 1996; 81(5): 2287 - 2296. [Abstract] [Full Text] [PDF]
|
|
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
| Visit Other APS Journals Online |
