Abstract
We investigated whether breathtobreath fluctuations in tidal volume (Vt) and endtidal O_{2} and CO_{2}exhibit longrange correlations and whether parameters describing the correlations can be used as noninvasive descriptors of control of breathing. We measured Vt and endtidal O_{2} and CO_{2} over n = 352 ± 104 breaths in 26 term, healthy, unsedated infants (mean age ± SD: 36 ± 6 days) and calculated the detrended fluctuation function [F(n)]. The F(n) of the breathtobreath time series of Vt, O_{2}, and CO_{2} revealed a linear increase with a breath number on loglog plots with a slope that was significantly different from 0.5 (random) and thus consistent with scaleinvariant behavior. Longrange correlations were stronger for O_{2} than for Vt and CO_{2.} The F(n) of many individual signals exhibited a crossover behavior indicating that control mechanisms regulating fluctuations of Vt, O_{2}, and CO_{2} may be different on different time scales. Thus breathing has a memory up to at least 400 breaths that can be characterized by the simple indicator α.
 control of breathing
 lung
 airway
 longrange correlation
the breathing pattern of infants is highly irregular. Patterns of regular breathing interrupted by periods of insufficient breathing (hypopneas) or quiescence (apnea) are commonly observed even in healthy neonates. Although certain agerelated variability is a physiological feature of infancy (19), the exaggeration of this variability might be interpreted in terms of a delayed maturation of control of breathing (12) and higher risk for hypopneas and apneas in infants. This is particularly true for premature infants and infants with increased risk for sudden infant death syndrome.
The size, shape, and timing of each breath are controlled by a neural oscillator, which drives the respiratory muscles (28). A variety of feedback and feedforward mechanisms have been proposed to explain matching of the tidal volume (Vt), oxygen (O_{2}), and carbon dioxide (CO_{2}) outputs from the lungs with changes in airway mechanics and variations in total body O_{2} consumption and CO_{2} production (9,35).
It has been proposed that, in infants, breathing can be considered as the output of a regulatory system that is attracted to a steady state (12). In other words, regulation of breathing may be a homeostatically controlled process. However, recent works in neurorespiratory and other biological regulatory systems have emphasized the nonlinear and dynamic nature of such feedback controls (6, 11, 16, 31, 32). The interactions among the various complex feedback and feedforward mechanisms often result in weak but correlated fluctuations of the physiological quantity in question (6, 16). In general, correlations imply that successive values of the physiological variable are not independent of each other. Past values of variables describing the control system will have an influence on the future values of the variable. In other words, the system exhibits memory. When the correlations extend over at least one order of time decade, the variable is said to have longrange correlations. If the correlation function follows a powerlaw functional form, the correlations are also said to exhibit scaling behavior. Such scaling behavior has been found in heart rate variability (1, 25), in fluctuations in breath intervals (14), in firing rate of respiratory related neurons (20), in variations in lung volume (10), in transport of ions and molecules across biological membranes (21), and in the time intervals between crackle sounds (2).
The aim of this paper was to test whether correlations and, in particular, longrange correlations exist in the fluctuations of the breathtobreath Vt and endtidal O_{2} and CO_{2} in infant breathing and whether they can be described by simple mathematical parameters, potentially useful to characterize immature breathing in infants. Furthermore, if correlations existed, we also aimed to determine whether control mechanisms regulating fluctuations of breathing parameters are different on different time scales and how these correlations may change with age.
METHODS
Study Design
We have quantified the breathtobreath variability and ordering, or correlations, in Vt, O_{2}, and CO_{2} by measuring tidal breathing time series in 26 healthy, unsedated infants in quiet sleep.
First, robustness of the methodology was studied by using numerical simulations to investigate the effects of finite record length on data analysis. Second, we aimed to establish whether longrange correlations exist in tidal breathing in infants and to examine whether these correlations could be characterized by a simple parameter as a descriptor of control of breathing in infants. Analysis of breathtobreath fluctuations in Vt and endtidal O_{2} and CO_{2} was done by using detrended fluctuation analysis (DFA) (24). The existence of longrange correlations in individual time series was then established by detecting differences in ordering compared with the randomized surrogates of the original time series. Third, we investigated whether the ordering properties of the Vt and endtidal O_{2} and CO_{2} fluctuations were different from each other. Last, age dependence of longrange correlations in Vt, O_{2}, and CO_{2} was studied to explore the maturational differences in the control of breathing in infants of different age.
Subjects
We measured breathtobreath tidal breathing parameters in 26 term, healthy, unsedated, quietsleeping infants with a mean age of 36 ± 6 (SD) days and gestational age of 40.1 ± 1.0 wk. None of the infants had respiratory infections. The study was approved by the ethics committee of the University Hospital of Berne, and parental consent was also obtained for each study. Parents were usually present at the time of measurement.
Measurements
Infants were measured in the supine position with the head in the midline position. Quiet sleep was defined as Prechtl state I (26), meaning closed eyes, regular respiration, and absence of eye and gross body movements. O_{2} saturation and heart rate were monitored throughout the measurements (model Biox 3700, DatexOhmeda, Helsinki, Finland). A total of 10 min of recordings [352 ± 104 (SD) breaths] of Vt, O_{2}, and CO_{2} was assessed by using a measurement set up (Exhalyser, EcoMedics, Switzerland), which is in accordance with the recent specifications for tidal breathing measurements in infants (15). The dead space of the flow, O_{2}, and CO_{2}measuring equipment was 3 ml. A compliant silicon infant face mask (infant mask, size 0; Homedica, Cham, Switzerland) was placed over the infant's nose and mouth. The dead space of the mask had a total volume of 15 ml (measured by water displacement). Hence, the effective dead space of the measurement head was 10.5 ml (50% contribution of the face mask dead space).
Flow, O_{2}, and CO_{2} were measured during tidal breathing by using commercially available infant lung function equipment (Exhalyser, EcoMedics). Flow measurements were assessed by using an ultrasonic flowmeter (Spiroson model M30.8001, EcoMedics) connected to a bias flow of 14 l/min. Flowvolume loops were inspected for a leak before starting measurement. Vt were calculated from the flow signal. Endexpiratory CO_{2} concentration was obtained by using an infrared technique: a mainstream sensor and an infrared analyzer with a resolution of 0.05% and a response time of 60 ms (Pyron).
Endexpiratory O_{2} concentration was measured by using a sidestream laser diode O_{2} sensor with visiblespectrum absorption spectroscopy analysis with a resolution of 0.02%, accuracy of ±0.2% in air mixtures and a response time of 100 ms (Oxygraf).
All tidal breathing signals were sampled at a rate of 200 Hz with the use of a 12bit analogtodigital converter. Flow, O_{2}, and CO_{2} signals were corrected for the time delay due to sampling. Analysis of the signals was carried out by using a customwritten analysis package (MATLAB, Mathworks).
Analysis
Theoretical background.
dfa.
First, the endexpiratory values of Vt, O_{2}, and CO_{2} were assessed from the original time series as a function of breath number (n). Next, breathtobreath time series of Vt, O_{2}, and CO_{2} were created by plotting the endexpiratory values of Vt, O_{2}, and CO_{2} as a function of n. Finally, the DFA introduced by Peng et al. (24) was applied to each time series as follows.
DFA is a technique suitable to quantify the correlation properties of nonstationary time series. Accordingly, this method can detect intrinsic correlation properties of a complex physiological signal and avoids the detection of false correlations due to the nonstationarity nature of the time series. According to Peng et al. (24), the DFA method estimates the fluctuation function of a time series as follows
If the F(n) shows a linear increase with increasing n on a loglog plot, then F(n) is said to follow a powerlaw functional form
For a random process, α takes the value of 0.5. For a positively correlated signal (large fluctuations are likely to be followed by large fluctuations), α is >0.5, and for an anticorrelated signal (large fluctuations are likely to be followed by small fluctuations), α is between 0 and 0.5 (25). If F(n) follows a power law over at least an order of magnitude time scale with an α different from 0.5, the corresponding variable is said to exhibit longrange correlations or scaleinvariant behavior. For example, α = 1 corresponds to 1/f noise and α = 1.5 corresponds to Brownian noise.
To ascertain that the correlations in breathtobreath fluctuations of Vt, O_{2}, and CO_{2} are real, we randomized (shuffled) the order of the original breathtobreath time series. Such a rearrangement of the data results in an uncorrelated time series. Thus, although this procedure does not alter the distribution of the amplitudes in the time series, the correlated ordering should disappear, and hence α of the shuffled time series should be 0.5. Thus the existence of longrange correlations in the individual original traces can be established if the value of α is significantly different from that after shuffling.
FINITE SIZE EFFECTS.
The expected value of α for an infinitely long random time series (white noise) is exactly 0.5. However, for a finite realization of a random sequence, the calculated value of α is usually different from 0.5 and depends on several factors including the length of the record, the signaltonoise ratio, and potentially the true value of α. This may lead to two important problems. The first is that the estimated α can be in error due to the finite record length, and the second is that recognizing weak correlations (i.e., when α is close to 0.5) from finite data sets can be ambiguous. For example, it is possible that the original time series had a weak correlation with an α of 0.58 and, after shuffling, α became 0.56 due to the short data record. In this case, it is not simple to identify the correlations from the original data set.
To resolve the first issue, we investigated the effects of record length on the estimated α from simulated time series with a known α. Time series with a record length of 4,096 data points were created as follows. The signal was generated in the frequency domain by first prescribing the squared magnitude spectrum to follow a strain line with frequency on a loglog plot. The slope of the line is the negative exponent (β) of the power law spectrum, which was specified as β = 2α − 1 (24), where α is the desired exponent in the fluctuation function defined in Eq.3 . For example, α = 0.5 corresponds to a white noise with β = 0, and α = 1 corresponds to a 1/f noise with β = 1. The phases were randomly selected, and the time domain signal was obtained by using an inverse Fourier transform. Next, eight data segments with lengths of 128, 256, or 512 points were selected, and DFA was applied as described previously (see Theoretical background) to each segment before and after shuffling the segment. This provided estimates of the mean and SD of α as a function of the record length and the true value of α. Additionally, these simulations also tested whether the effectiveness of shuffling depends on the record length and the true value of α.
With regard to the second issue, we established the statistical properties of α from the shuffled Vt, O_{2}, and CO_{2} data. We shuffled the original time series of each individual data set. If a particular shuffling resulted in a value of α that was very different from 0.5 than the average, we repeated the shuffling of that time series 10 times and estimated α as the average obtained from the 10 shufflings. Next, we calculated the mean, SD, and 95% confidence interval of α for Vt, O_{2}, and CO_{2}. Finally, correlations in any time series (e.g., Vt) were established if the α from that record was outside the range of the group mean α ± 95% confidence interval for that type of variable (e.g., Vt).
CROSSOVER PHENOMENA.
In most cases, a single regression line was adequate to fit the F(n) function on a loglog graph providing a single exponent α. However, similar to the interbeat interval fluctuations in heart rate time series (24), F(n) of many individual signals exhibited a crossover behavior characterized by two separate regions of linear increase of F(n) on the loglog graph with two distinct slopes. In these cases, two separate exponents (α_{1} and α_{2}) could be determined from the data by using two regression lines. The time scale (which is the index) at which the two regions were separated is the crossover time (N _{x}). The two regions in F(n) were selected by maximizing the correlation coefficients (r) in the regression of both regions. In each case, r ≥ 0.95 was required.
Statistical analysis of physiological data.
In an attempt to compare the correlation properties of the different tidal breathing time series, we compared α (or α_{1}, α_{2}, and N _{x} where a crossover was observed) from the time series of Vt, O_{2}, and CO_{2} by using paired ttests for the whole group. For data that failed the normality test (e.g.,N _{x}), we used oneway ANOVA on ranks (KruskalWallis oneway ANOVA on ranks). In the case of a single slope, we assumed α_{1} to be the same as α_{2}. The estimation of α can be influenced by the number of points used in the linear regression. Thus, to exclude the possibility that the difference between the mean values of α of the groups (Vt, O_{2}, and CO_{2}) was simply a consequence of the different number of individual time series exhibiting crossover phenomena, we tested whether this number was not statistically different in the compared groups (χ^{2}test). To examine whether individual processes of breathtobreath fluctuations in Vt, O_{2}, or CO_{2} were governed by similar mechanisms, the exponents from Vt, O_{2}, and CO_{2} were also correlated with each other by using linear regression analysis.
Additionally, to detect maturational effects, we determined the age dependence of α (or α_{1} and α_{2}) as a function of gestation age (GA) and postnatal age (PNA). We also examined whether the crossover pattern was age related by examining the dependence of the ratio α_{1}/α_{2} andN _{x} on GA or PNA by using linear regression analysis. Whereas α_{1}, α_{2}, and all the ages were normally distributed, the distribution ofN _{x} of Vt, O_{2}, and CO_{2} were skewed. Thus, before the linear regression analysis was performed, a log transformation was applied to data, which transformed the distribution of these variables to a normal distribution.
RESULTS
Robustness of the Methodology (Finite Size Effects)
The results of the numerical simulations are summarized in Table1. The error in the estimated value of α decreases significantly from ∼4 to 0.5% when the record length increases from 128 to 512 data points. The error slightly increases when the true theoretical value of α increases from 0.6 to 1. Thus α can be estimated to within 1% error if the length of the record is at least 500 points independent of actual correlation properties of the signal. The errors in α from the shuffled time series are generally higher, reaching 16.6%, and even for the record length of 512 points the error is between 4 and 7%. This suggests that randomization of the ordering of the time series does not in general work for short time series. However, the error for the entire original 4,096 points record length is below 1% independent of the original α.
Measurements in infants.
The anthropometric data including weight, height, and age at the measurement, average number of breaths in time series, single breath mean Vt, and mean endtidal O_{2} and CO_{2} concentrations for the group of 26 infants are given in Table 2. Examples of the raw Vt, O_{2}, and CO_{2} signals from a representative infant are shown in Fig. 1as a function of time. The corresponding time series of the breathtobreath endtidal values of Vt, O_{2}, and CO_{2} from the same subject are shown in Fig.2. It can be seen that each time series displays considerable irregularities.
Evidence of LongRange Correlations in Breathing
Figure 3 shows the fluctuation functions corresponding to the data in Fig. 2 on loglog graphs before and after shuffling. The linear regression line fits are also shown in Fig. 3. For all three breathing parameters, F(n) follows a straight line over a time scale of about 1.5 decades. Exponents are above 0.8, and they decrease to 0.53 after shuffling. One example of the crossover behavior can be seen in Fig.4. The first region has a slope of nearly 1 over a range of somewhat less than a decade, whereas the second region in this case has a slope of 0.51 covering an entire decade. F(n) of all 26 individual traces revealed a similar behavior to those seen either in Figs. 3 or 4. The individual and group means and SD for α_{1}, α_{2}, andN _{x} for Vt, O_{2}, and CO_{2} (and median values for nonnormally distributed data, e.g., N _{x}) are summarized in Table3.
In the case of a single slope, the group means of the scaling exponent α were 0.75 ± 0.20, 0.95 ± 0.17, and 0.83 ± 0.15 for Vt, O_{2}, and CO_{2}, respectively. For the group with two scaling regions, the mean values of α_{1}were 1.00 ± 0.31, 1.19 ± 0.34, and 0.96 ± 0.27 for Vt, O_{2}, and CO_{2}, respectively, and the mean values of α_{2} were 0.73 ± 0.42 for Vt, 1.01 ± 0.29 for O_{2}, and 0.92 ± 0.33 for CO_{2}. The values of α from the shuffled time series are centered around 0.5 with a narrow 95% CI (Table4). Thus these data provide evidence for the presence of longrange correlations in the endtidal fluctuations of Vt, O_{2}, and CO_{2} in normal infants.
Considering the crossover phenomena, out of 26 time series, 11 records of Vt, 9 records of O_{2}, and 15 records of CO_{2} exhibited a crossover in scaling. There was no significant difference in the number of data points in the signals between one or twoslope pattern groups (ttest,P = 0.95). In most cases where F(n) showed a twoslope pattern, both exponents were different from 0.5, indicating that although the correlation properties of the variable were different for different time scales, longrange correlations still persisted throughout the whole trace. On the other hand, in three of Vt and one of CO_{2} time series, the second slope α_{2} approached 0.5, indicating that after a certain number of breaths, crossover point N _{x}, fluctuations in the signals became random. This phenomenon was not seen in the O_{2} traces. Although α_{2} was usually smaller than α_{1}, which was also true for the whole group of subjects with twoslopes pattern, 2 of 11, 3 of 9, and 7 of 15 time series for Vt, O_{2}, and CO_{2}, respectively, exhibited a reverse crossover (24) with a scaling exponent α_{2} bigger than α_{1}. The median of N _{x} defining the time scale at which the fluctuations in tidal breathing parameters (Vt, O_{2}, and CO_{2}) exhibited an obvious change in their scaling behavior was 20 for Vt, 10 for CO_{2}, and 14 for O_{2}.
Comparisons of the Correlations in Vt, O_{2}, and CO_{2}
There was no statistically significant difference in the distribution of crossover phenomena in the different parameter groups (χ^{2}test). Considering the group means of the exponents, α_{1} for O_{2} was significantly higher than α_{1} for Vt (paired ttest,P = 0.001) and CO_{2} (P = 0.02). The α_{1} of Vt and the α_{1}of CO_{2}, however, were not different from each other. The same was true for α_{2}; that is, α_{2} for O_{2} was higher than α_{2} for CO_{2}(P = 0.02) or Vt (P < 0.001). In the individuals with twoslope pattern, group comparisons ofN _{x} (oneway ANOVA on ranks) of different breathing parameters (Vt, O_{2}, and CO_{2}) revealed no statistically significant difference in the number of breaths at which scaling behavior of the fluctuations changed.
By plotting α_{1} (or α_{2}) of O_{2}(or CO_{2}) as a function of the corresponding exponent for Vt, the α_{1} for O_{2}(P = 0.002; linear regression; Fig.5 B) and CO_{2}(P = 0.04; linear regression; Fig. 5 A) was significantly correlated with the corresponding exponent for Vt. The same was true for the correlation between α_{1} of CO_{2} and O_{2}(P = 0.02) as well as α_{2} of CO_{2} and O_{2} (P < 0.001; Fig.5 C).
Age Dependence of LongRange Correlations
Neither α_{1}, α_{2}, nor the α_{1}/α_{2} ratio was significantly correlated with GA or PNA for Vt, CO_{2}, and O_{2.}On the other hand, log(N _{x}) significantly increased with GA for O_{2} (P = 0.01, linear regression) but not with PNA. The relationship between log(N _{x}) for O_{2} and GA remained statistically significant also after adjusting for a study age (P = 0.06, linear regression). The linear decrease of log(N _{x}) vs. GA became nearly significant for Vt (P = 0.058), but no statistically significant correlation was found between CO_{2} and GA or PNA.
DISCUSSION
The control of breathing in infants undergoes developmental changes and can be altered in disease. Searching for a noninvasive descriptor of the control of breathing seems to be crucial in an attempt to describe and understand the physiological mechanisms involved in neurorespiratory control in infants and developmental process in the regulation of breathing during postnatal life. Furthermore, such a descriptor may be useful for the early detection and monitoring of disease, and the assessment of the therapeutic interventions.
In this study, we demonstrated that breathtobreath time series of tidal breathing parameters in infants (Vt, O_{2}, and CO_{2}) exhibit nontrivial, scaleinvariant behavior. We studied the variability of the outputs of the complex breathing control system by using DFA. We found that F(n) of all individual tidal breathing time series (Vt, O_{2}, and CO_{2}) as a function of breath lag (n) plotted on a loglog plot revealed linear behavior with a slope α, which was significantly different from α = 0.5, the value that signifies randomness. This implies that there are significant longrange correlations in time series of tidal breathing parameters. Thus the values of singlebreath Vt and endtidal O_{2} or CO_{2} levels are not independent of those in previous breaths. In other words, breathing has a memory. Because the longrange correlations followed a power law, this behavior is consistent with fractal properties of the respiratory control system in infants.
We identified two patterns in the scaling behavior of individual Vt, O_{2}, and CO_{2} traces. Whereas some of the traces exhibited a single slope α of the linear regression fit of F(n) vs. n on a loglog plot, denoting that the characteristics of correlations did not change on different time scales through 10min traces, others exhibited a crossover pattern (see Crossover Phenomena). However, there was no systematic difference in the number of data points (breaths) between one or twoslope pattern traces. In the case of a single slope, the group mean exponent α for O_{2}approached 1, which is consistent with 1/f behavior in O_{2}breathtobreath fluctuations. On the other hand, fluctuations of CO_{2} and Vt were rougher, with values of α of 0.83 and 0.75, respectively. This is consistent with persistent longrange, powerlaw correlations, such that a large difference in Vt (O_{2} or CO_{2} concentrations) between breaths separated by a certain breath lag was more likely to be followed by a large lag and vice versa. In other words, big fluctuations (in comparison to the average) are more likely to be close in time to bigger fluctuations for a certain time interval. Because the process is stochastic, the pattern can suddenly change so that a big fluctuation is followed first by a smaller one, which in turn is likely to be followed by even smaller fluctuations afterward. This effect is not present on a breathtobreath basis but on a wide range of time scales.
Other complex regulatory systems, such as heart rate control (24,25), have shown scaleinvariant properties persistent with longrange correlations. Heart rate variability in healthy adults has been shown to follow a 1/f behavior (α∼1), which is very similar to the behavior of oneslope pattern of O_{2} breathtobreath time series in our infants.
Crossover Phenomena
Peng et al. (24) found two slope patterns in some of their heart rate time series in healthy subjects as well as in diseased patients. They could distinguish between the healthy and the pathological data sets on the basis of this crossover phenomena. Similarly, we found crossover phenomena in approximately half of our infant breathing time series. Although the oneslope pattern of Vt, O_{2}, and CO_{2} time traces showed longrange correlations through the whole recorded trace with the same exponent, the twoslope pattern traces implied that the correlated structure of the breath to breath fluctuations in Vt, O_{2}, and CO_{2} changed at time scales corresponding to N _{x}, which was ∼18 breaths. On short time scales (n < N _{x}), the intrinsic dynamics of Vt, O_{2}, and CO_{2} fluctuations approached that of an ideal 1/f behavior for all parameters, i.e., α_{1} was close to 1. This behavior was similar for all parameters observed (Vt, O_{2}, and CO_{2}). The correlated structure of the time series then changed for longer time scales (n >N _{x}), which was characterized by a change in the correlation exponent (i.e., α_{2} was different from α_{1}). In most of the individual time series with a twoslope pattern, α_{2} was smaller than α_{1}but still different from 0.5. Nevertheless, the group means of α_{2} for Vt, O_{2}, and CO_{2} were not statistically significantly different from α_{1} for any of the parameters observed.
In four individual traces, α_{2} approached 0.5, indicating that, after a certain number of breaths (N _{x}), the fluctuations of the variable in question were not correlated any more. This behavior was found in Vt and CO_{2}time series but not in O_{2}. Thus, in these infants, the memory existed only over a short time scale, gradually became weaker, and eventually at large time scales (n >N _{x}) fluctuations in Vt and endtidal CO_{2} became independent of the previous ones. Thus, apart from four infant time series, scaling and hence memory effects existed through several hundred breaths. This behavior suggests that in a healthy infant who takes a breath in the absence of a strong external stimuli, the properties of that single breath can be influenced by those of many breaths ranging up to 400 previous breaths.
The physiological origin of the longrange correlations is not entirely clear. Theoretically, longrange correlations may serve as an organizing principle for the feedback mechanisms generating fluctuations on a wide range of scales. Alternatively, the longrange correlations may be a consequence of the combined effects of multiple nonlinear feedback loops in the control of breathing and fluctuating external stimuli. However, in a dynamic system operating far from equilibrium, significant fluctuations in the system variables can occur, even in the absence of external stimuli. The potential advantage of inherent longrange correlations in such systems may be that they allow for an improved functional responsiveness and adaptability to external perturbations (25). The lack of a characteristic scale may help prevent the system from being locked into a particular phase that would restrict the functional responsiveness of the organism (16). These arguments are supported by observations from severe diseased states where the breakdown of multiscale, longrange order is accompanied by the emergence of a dominant frequency mode characterized by a highly periodic behavior (trivial longrange correlations). The output of the system becomes nearly sinusoidal such as the lowfrequency oscillations seen in heart rate pattern of infants with fetal distress syndrome (18). In other cases, the breakdown of longrange correlations may be accompanied by the emergence of uncorrelated randomness as seen in certain cardiac arrhythmias, such as ventricular fibrillation (17).
The control of breathing is a nonlinear feedback control system with various input stimuli, which themselves can fluctuate and have several feedback loops. Numerical simulations, as described in the , using the triphasic model of the respiratory rhythm generator (5, 28), demonstrated that longrange correlations are already present in the phrenic output fluctuations of the respiratory oscillator. A representative simulation using a model of the respiratory oscillator (14) provided power law fluctuations with values for α between 0.58 and 0.65, which is consistent with longrange correlated behavior. Although this does not prove that the observed longrange correlations originate from the respiratory oscillator, it is consistent with this idea and opens the possibility for future research in this field.
Comparison of Vt, CO_{2}, and O_{2}LongRange Correlation
The comparison of the slopes α (α_{1} and α_{2}) of different tidal breathing parameters provided statistically significantly higher values for O_{2} than for Vt and CO_{2}, indicating that breathtobreath endtidal O_{2} concentration was more strongly correlated than breathtobreath Vt or endtidal CO_{2}concentration. Stronger correlation implies a more deterministic system and hence possibly a stronger regulatory mechanism controlling the output of the system of O_{2} compared with Vt and CO_{2}.
Furthermore, we tested whether the values of α for Vt, O_{2}, and CO_{2} were correlated. Such correlations are expected because, within an individual breath, Vt and endtidal O_{2} and CO_{2} must be related to each other on the basis of lung clearance mechanism and feedback regulation. We found that α for O_{2} and CO_{2} were correlated both for short (α_{1}) and long (α_{2}) time scales, but they were correlated to Vt only over short time scales. For short time scales, the correlation between α_{1} (O_{2}) and α_{1} (Vt) was significantly stronger than that between α_{1} (CO_{2}) and α_{1}(Vt). We cannot conclude from these data whether O_{2} or CO_{2} is dominating Vtregulation; we can only say that if the breathtobreath fluctuations in O_{2} are strongly correlated (high α), the same will be true for CO_{2}, independent of the time scale observed. For short time scales (α_{1}), the same is true for O_{2} and CO_{2} compared with Vt. However, for long time scales, we found a dissociation or uncoupling of α_{2} (Vt) and the corresponding α_{2} (O_{2}) and α_{2}(CO_{2}).
One possible explanation for the uncoupling of O_{2} and CO_{2} from Vt at large time scales could be as follows. Within an individual breath, there must be strong correlations between the parameters because of lung clearance effects. However, different internal or external inputs to the individual feedback loops of these variables can result in small differences in the fluctuations. These differences could become amplified by system nonlinearities for long time scales when the central regulatory feedback loops become more dominant, leading to an uncoupling of α_{2} for Vt compared with O_{2} and CO_{2}. Another possibility could be that O_{2} and CO_{2}are simultaneously influenced by certain additional weak but longlasting memory effects, such as those due to bloodmediated slower feed back loops. These additional factors could introduce similarities in the long timescale fluctuations of O_{2} and CO_{2} but not in the fluctuations of Vt. Theoretical investigations of how these timing effects of the various control loops have been described by Khoo (22).
Chemoregulation undergoes maturational effects in infants and is distinctively different from adults (4, 10, 30). Although we were not able to assess longitudinal data sets, in our crosssectional data, we found that slope α did not change with GA or PNA in this small age range between 3.6 and 6.7 wk. However, the logarithm of N _{x} in the twoslope fluctuation function statistically significantly decreased with GA (38.0–42.3 wk) for O_{2} and nearly statistically significant increased with Vt. Thus, in most infants, the transition of stronger shortrange correlation for O_{2} to weaker longrange correlation occurs after a shorter time period in the older infants. However, in some infants, α_{1} was smaller than α_{2}.
Because the crossover phenomenon is sensitive to gestational age and because stronger correlation implies a more deterministic system and hence possibly a stronger regulatory mechanism, DFA could potentially be a marker of changes in the control of breathing in premature infants. Nevertheless, this should be tested in longitudinal studies.
Potential Limits of the Method
Sleep stage.
Although it has been known that sleep stage influences the control of breathing in infants (29), all the measurements in this study were performed in spontaneously quiet sleeping infants. Thus the length of time for collecting tidal breathing data was significantly influenced by their natural sleep. This resulted in relatively short breathing traces of 10 min, for which stable conditions with no change in sleep states were maintained.
Nonstationary data.
The nonstationary behavior of most physiological systems, including the neurorespiratory control system, could potentially influence the identification of intrinsic correlation properties of the system. In other words, correlations due to nonstationary trends have to be distinguished from more subtle systemrelated fluctuations. This problem was avoided by analyzing the data by using the DFA (24). The longrange correlation properties of Vt, O_{2}, and CO_{2} time series were further confirmed by using the statistical properties of simulated data as well as analyzing the shuffled time series. As a result, even more subtle correlation structures of the time series, such as a crossover characterized by the twoslope pattern, could reliably be detected.
Oscillations in the data sets.
For higher time window length, the DFA exhibited oscillations in some of the traces. Although the oscillations may possibly emerge from nonlinearities of the respiratory system, they are not necessarily due to them. We tested this by calculating the DFA of random noise sequences of different lengths. The DFA of these time series also showed oscillations. These oscillations were different after shuffling and were significantly reduced in the longer time series. This indicates that the oscillations were related to insufficient averaging of the fluctuations for large window lengths. The oscillations in the respiratory data may contain physiological information. However, although the DFA is sensitive to correlations, it is not suitable to study system nonlinearities.
Influence of a face mask.
It has to be noted that the control of breathing and, consequently, the pattern of breathing tend to become somewhat more regular after placement of a mask on the infants' faces. Thus the possible influence of a face mask on a breathing pattern cannot be excluded (13, 33). Nevertheless, when the mask was put on the infants' faces it was not removed until the recording was completed, thus minimizing the effect of an alteration of sleep and possibly the breathtobreath variability of Vt, O_{2}, and CO_{2}.
Summary and Hypothesis for Future Research
In this study, we found that breathtobreath time series of tidal breathing parameters (Vt, O_{2}, and CO_{2}) in infants exhibit powerlaw, longrange correlations, consistent with scaleinvariant behavior. We quantitatively characterized this memory of the respiratory control system by using the F(n). F(n) of the tidal breathing parameters as a function of breath lag (n) plotted on loglog graphs revealed a linear behavior with the slope α significantly different from 0.5 (i.e., uncorrelated behavior). Thus the time correlations present in the breathtobreath time series of tidal breathing parameters contain information on the control of breathing in infants, and α may serve as a simple noninvasive descriptor of the control of breathing. We have shown that the longrange correlations for O_{2} are stronger than those for Vt and CO_{2} in healthy infants. We found an uncoupling of Vt, and we also found crossover phenomena as described by Peng et al. (24) and reported by Alencar et al. (2). The crossover behavior is particularly interesting because it was sensitive to gestational age and hence could be used to assess the degree of immature breathing in premature infants. For clinical applications, the effects of sleep state, intermittent hypoxia, disease (infants with chronic lung disease and neurological diseases, sudden infant death syndrome), or toxic influences (e.g., maternal smoking) on changes of the control of breathing should be investigated in future studies. DFA could potentially also be used to monitor therapeutic effects of drugs (e.g., caffeine, theophilline). The new parameters are particularly interesting because they are related to the feedbackcontrol system properties, which is a novel approach of studying control of breathing in infants.
The longrange correlation analysis technique offers distinct advantages to probe the physiological mechanisms involved in developing the neurorespiratory control in healthy infants and infants with disease.
Acknowledgments
The authors thank the staff of the Respiratory Medicine Department for help in obtaining the data presented in this study and, in particular, Heidi Staub.
Footnotes

Address for reprint requests and other correspondence: U. Frey, Pediatric Respiratory Medicine, Dept. of Pediatrics, Univ. Hospital Inselspital, Bern, CH3010 Switzerland (Email:urs.frey{at}insel.ch).

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked “advertisement” in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

First published December 21, 2001;10.1152/japplphysiol.00675.2001
 Copyright © 2002 the American Physiological Society
Appendix
We propose a possible mechanism to explain how longrange correlations in breathtobreath fluctuations of Vt, O_{2}, and CO_{2} could originate from the neural respiratory network. There is evidence from animal models that a threephasic model of the respiratory oscillator is similarly appropriate to describe the breathing cycle in newborns as in adults (5). There is also evidence of noise in the respiratory rhythm generator. The firing of individual neurons has been found to be a probabilistic process with intrinsic noise (3). Recently, Hoop et al. (20) demonstrated the presence of noise in respiratoryrelated neural activity in the brain stem of neonatal rats. In a previous study (14), our laboratory introduced noise in the neural oscillator model proposed by Botros and Bruce (5). This model was able to quantitatively mimic the large variabilities and irregularities as well as the scaling behavior of interbreath time intervals observed in healthy and premature babies (14). In that study, scaling behavior was observed in the probabilitydensity function of interbreath intervals, which followed a powerlaw form and described the likelihood of extreme values (7, 14). However, this method does not examine the ordering of amplitudes of the simulated fluctuations in the phrenic output time series.
Here, we analyzed the longrange correlations in the fluctuations of the phrenic output of the Botros and Bruce model (5) similarly as described in methods and calculated α from the DFA. Briefly, to reproduce the observed irregularities, we modified the neural oscillator model proposed by Botros and Bruce (5), which transforms tonic neural inputs (TNI) into a regular rhythm and hence breathing (28). The model consisted of five coupled nonlinear differential equations corresponding to the activities of five neuron groups in the respiratory center. The rampinspiratory neuron group provided periodic outputs to the phrenic nerve similar to the measured data. We solved the network in the time domain by using MATLAB (Mathwork, Natick, MA) and examined the amplitudes of the peaks of the output of the rampinspiratory neuron group. However, after a short transient period, the solution of the network was a periodic waveform without any irregularities. Thus, to mimic irregularities in phrenic output amplitudes, we added a varying amplitude noise to the TNI of the first or rampinspiratory neuron group (TNI_{1}) based on considerations of Hoop et al. (20), suggesting that neural noise is not constant but does vary within the respiratory cycle, most likely because of varying chemoreceptor responses. Hoop et al. (20) found correlations in the neural noise itself. However, to test whether the respiratory oscillator alone is able to generate longrange correlations, we added random noise to the input of the oscillator. The model parameters are summarized in Table5. The mean value of TNI_{1} was 5 with a uniformly distributed noise (SD = 4), which changed on average four times within the respiratory cycle. We simulated 300 breaths by using this model, similar to the number of breaths in our infant measurements, and obtained large variations in phrenic amplitude similar to those observed in the measured infant Vt data. When we calculated F(n) from these simulated phrenic output series, we found a linear relationship in the loglog representation with α = 0.58 (r = 0.976; Fig.6). With the use of only the most linear range from n = 1–200, α was 0.61 (r = 0.991). After reshuffling this series, we found α to be 0.46 (r = 0.974; Fig. 6).