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1Department of Pharmacology and Toxicology, Michigan State University, East Lansing, Michigan 48824; and 2Department of Internal Medicine, University of Texas Southwestern Medical Center, Dallas, Texas 75390
Submitted 25 June 2004 ; accepted in final form 30 July 2004
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ABSTRACT |
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 TOP ABSTRACT METHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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Allan and Fano factors; breath amplitude and frequency; dispersional analysis; Hurst exponent
First, except for some short-range correlations (4, 10), the fluctuations in respiration might be random, i.e., uncorrelated (6, 16, 32). That is, although influenced by events (breaths) in the recent past, the present value of the measured parameter would not be related to events in the distant past.
The second possibility is that long-range correlations also exist among the fluctuations in one or more of the respiratory parameters. If so, it would be important to define the duration of the memory in the system. Here, the term "memory" is used in the context of the time frame over which a series of events are correlated. If the memory extends across more than one time scale, the fluctuations would be best modeled as arising from a fractal (time-scale-invariant) process in which the present value of the measured property is related to events in the distant past (2, 12, 23, 34). The term "time scale" refers to the temporal resolution used to measure the parameters of interest. In this paper, each time scale is represented by a decade of window sizes (in seconds) plotted on a log scale. The results of the present study demonstrate time-scale-invariant behavior in the system responsible for breathing in awake humans under resting conditions.
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METHODS |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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12 h before the recording session. Their ages ranged from 22 to 58 yr. Mean blood pressure measured with an automated sphygmomanometer (Welch Allen) averaged 91 ± 2 mmHg in the 20 subjects, and mean heart rate derived from lead II of ECG was 1.1 ± 0.1 beats/s. Recordings
Recording sessions lasted 75–208 min, with the subject in a supine position, relaxed and breathing spontaneously in a quiet room. We also studied two subjects in whom breathing was paced by a metronome at a frequency close to their normal respiratory rate. Respiratory movements were recorded with a strain-gauge pneumograph placed in a stable position over the upper abdomen. Upward deflections in the records denote inspiratory movements and downward deflections denote expirations. This methodology is routinely used in human studies to monitor respiratory variability because it avoids having the subject breathe through a mouthpiece, which is known to alter both the duration and depth of breathing (18, 25).
Data Analysis
Data originally saved by use of a recording adaptor (model 4000 PCM, Vetter, Rebersberg, PA) and a videocassette recorder (model SLV-750 HF, Sony) were played back and acquired with software and an analog-to-digital converter board (Axon Instruments, Union City, CA) by using a sampling rate of 200 Hz. The analyses were performed using software developed by Gebber et al. (15) and Lewis et al. (22) at Michigan State University.
Parameters. After the peak and trough of each respiratory movement were detected, the following measurements were made: 1) the interval between the peaks of successive breaths (referred to as respiratory period or peak-to-peak breath interval) and 2) the trough-to-peak breath amplitude (normalized on a scale of 0–1.0). Fluctuations in the former parameter reflect the variability of respiratory rate, whereas fluctuations in the latter reflect the variability of the depth of breathing. From these cycle-by-cycle measurements, time series and frequency distributions were constructed. For a respiratory period of 4 s, the error of measurement is 1% with a bin resolution of 40 ms and 1.5% with a resolution of 60 ms. Values in the text are means ± SE.
Fractal analysis of respiratory fluctuations.
Two methods were used to determine whether fluctuations in the number of breaths occurring in time windows of specified length were fractal or random. The primary method involved calculation of the Allan factor, A(T), for window sizes of different lengths. Thurner et al. (30) and Turcott and Teich (33) define A(T) as the ratio of the event-number Allan variance to twice the mean number of events (i.e., respiratory cycles) in a window size of specified length (T):
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The Allan factor curve is constructed by plotting A(T) as a function of the window size on a log-log scale. For a data block of length Tmax, the window size, T, is progressively increased from a minimum of a single bin (40 or 60 ms) to a maximum of Tmax/6 so that 6 nonoverlapping windows are used for each measure of A(T). For a random process in which fluctuations in the number of events are uncorrelated, A(T) = 1 for all window sizes (29, 33). For a periodic process, the variance decreases and A(T) approaches zero as the window size is increased (29, 33). For a fractal process, A(T) increases as a power of the window size and may reach values 1.0 (29, 33). This reflects the greater variance in number of events with increasing window size. The increase in variance relative to the mean occurs because long window lengths are more apt to reveal rarer clusters of events (e.g., periods of relatively rapid or slowed breathing). The power law relationship between A(T) and window size appears as a straight line with a positive slope, 
, on the log-log scale. The , which is also known as the scaling exponent, is the power to which fluctuations in number of events measured on one time scale are proportional (i.e., statistically self-similar) to those measured on other time scales. The correlation coefficient (r) is used to test for linearity on the log-log scale, and linear regression is used to calculate .
The Allan factor curve for the original time series is routinely compared with those of 10 surrogate data sets in which the intervals between successive breaths have been shuffled. Specifically, we assigned random numbers to the peak-to-peak breath intervals in the original time series and then sorted the random numbers by size (7, 11, 22). This creates a randomized data set for which the mean, variance, and frequency distribution are identical to those of the original time series of peak-to-peak breath intervals, but with no correlations among events (7, 22, 29, 33). If shuffling of the data eliminates the power law in the Allan factor curve, it can be concluded that fluctuations in the number of breaths were fractal.
Alpha, which is bounded in a range of 0–3 (30, 33), is used to calculate the Hurst exponent (H, range 0–1). As described by Eke et al. (9) and Thurner et al. (30), H is calculated with the following formula when 0 < < 1
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< 3
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0.5 implies that the time series is fractal providing that the power law in the Allan factor curve extends over more than one time scale (decade on log scale). When H > 0.5, events are positively correlated [persistence; values larger (smaller) than the mean tend to be followed by values also larger (smaller) than the mean]. When H < 0.5, events are negatively correlated (antipersistence; values larger than the mean tend to be followed by values smaller than the mean and vice versa).
The second method used to test whether fluctuations in the number of breaths were fractal was Fano factor analysis. Teich (28) and Turcott and Teich (33) define the Fano factor, F(T), as the ratio of the variance of the number of events, var[Ni(T)], to the mean number of events, mean[Ni(T)], in a window size of specified length T:
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T1, thereby limiting the range of the slope () of the power law in the Fano factor curve to 0–1.0 (29). Thus the slope of the Fano factor curve should not be used to estimate H because this might lead to erroneous conclusions as to whether the long-range correlations among events were persistent or antipersistent. Dispersional analysis was used to test directly whether fluctuations in respiratory period and breath amplitude were fractal. The algorithm as originally described by Bassingthwaighte and Raymond (3) involves calculation of the standard deviation (SD) of the mean values of the measured parameter for groups of data points of a specified number (m). Specifically, the mean value for each group of m data points is obtained, and the SD of these values is calculated for the total number of groups. The process is repeated each time m is increased progressively from a minimum of one data point to a maximum of one-quarter of the total number of data points. In the present study, dispersional analysis was performed on first differences derived from the original time series. As described in our earlier studies (7, 11), a new time series of the absolute differences between successive respiratory periods or breath amplitudes is constructed and the SD of the mean first differences is calculated for all groups of size m. This modification of the basic method removes slow trends (i.e., nonstationarities) in the data such as progressive increases or decreases in the value of the measured parameter (17). Allan factor analysis, which is also based on first differences, similarly removes slow trends in the data (30, 33).
SD is plotted against m on a log-log scale, yielding a straight line with a negative slope. For a random process with no correlations among events, the slope of the dispersional analysis plot is –0.5 (3). For a fractal process, the slope is different from –0.5 over a range of m extending more than one decade (3). The decision on whether fluctuations are fractal is made by comparing the slope of the dispersional analysis plot for the original time series with those of the plots for 10 surrogates derived by shuffling the peak-to-peak breath intervals or breath amplitudes.
Although H can be calculated as the difference between the negative slope of the dispersional analysis curve and 1.0 (p. 67–70 in Ref. 2), Eke et al. (9) showed that the estimate provided may misidentify long-range correlations as persistent when, in fact, they are antipersistent. For this reason, values of H in this study are derived only from in the Allan factor curve and, thus, refer solely to fluctuations in number of breaths.
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RESULTS |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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Figure 1 illustrates the variability in respiratory period and in breath amplitude observed in a spontaneously breathing 41-yr-old man. The raw record of respiratory movements (90 s in length) in Fig. 1A shows a period of reasonably constant peak-to-peak breath intervals and breath amplitudes followed by a relatively long peak-to-peak breath interval (12.6 s), and then considerable variability of the two measured parameters. Cycle-by-cycle measurements of the peak-to-peak breath interval and normalized breath amplitude for the complete time series (8,940 s in length) are shown in Fig. 1, B and C, respectively. The variability depicted in these time series is similar to that observed in all of our subjects as well as that reported in previous studies of healthy humans (5, 8, 18, 31). The corresponding frequency distributions of 1,932 peak-to-peak breath intervals and 1,933 breath amplitudes recorded from this subject are shown in Fig. 3, A and B, respectively. The mean peak-to-peak breath interval was 4.62 s (13.0 breaths/min), and the maximum peak-to-peak breath interval was close to 15 s. As was most often the case, such frequency distributions were gammalike in shape (22, 26), with a longer tail extending to the right than to the left of the mode. The coefficient of variation (CV) of the distribution of peak-to-peak breath intervals was 0.24 and that of the distribution of breath amplitudes was 0.29. The distribution of peak-to-peak breath intervals was positively skewed (0.43), whereas the distribution of breath amplitudes was negatively skewed in this case (–0.12).
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Fractal Analysis
In nine spontaneously breathing subjects (group 1 in Table 1), fluctuations in the number of breaths, respiratory period, and breath amplitude were fractal with long-range correlations among events. The results from a 22-yr-old man are presented in Fig. 2. The distribution of 1,884 peak-to-peak breath intervals from a time series whose length was 8,580 s is shown in Fig. 2A. The CV of the distribution was 0.50 and skewness was 0.55 reflecting the long tail to the right of the modal interval. The mean peak-to-peak breath interval was 4.55 s (13.2 breaths/min).
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10 s in the Fano factor curve and 100 s in the Allan factor curve. The dips can be attributed to the strong periodic component of the respiratory signal that leads to a decrease in the variance with increasing window size (30, 33). The power laws in these curves extended over more than one time scale (up to the maximum allowable window size of 1,430 s, one-sixth of the data block length). The slopes () of the power laws in the Fano and Allan factor curves were 0.84 and 1.35, respectively. H derived from of the power law in the Allan factor curve was 0.18, thereby indicating that the long-range correlations among the fluctuations in number of breaths were antipersistent. That the power laws indeed reflected long-range correlations among events is indicated by the fact that the curves derived from the original time series fell outside of the range of the curves for 10 surrogates (gray region) at window sizes between 10 and 1,430 s (Fano factor) and 100 and 1,430 s (Allan factor). Note that after the initial dip in the Fano and Allan factors below 1.0, the slopes of the curves for the surrogates were essentially flat.
Dispersional analysis demonstrated that fluctuations in respiratory period and breath amplitude were fractal in the same subject. The frequency distribution of 1,885 breath amplitudes is shown in Fig. 2B. The CV of the distribution was 0.43, and skewness was 0.36. The slope of the dispersional analysis plot derived from the original time series of peak-to-peak breath intervals (dark trace) was –0.18 for m 5 (Fig. 2E), whereas the slope of the plot for the time series of breath amplitudes was –0.10 for m 23 (Fig. 2F). These slopes fell outside of the range of those (near –0.5) of the plots for 10 surrogates (gray regions).
For the nine spontaneously breathing subjects in whom fluctuations in breath number, respiratory period, and breath amplitude were fractal, H values derived from the slope of the power law in the Allan factor curve were antipersistent in three cases (H = 0.11 ± 0.04) and persistent in six cases (H = 0.81 ± 0.07).
In four spontaneously breathing subjects (group 2 in Table 1), fluctuations in the number of breaths and breath amplitude were fractal, but fluctuations in respiratory period were not. The results from a 41-yr-old man are presented in Fig. 3. The distributions of the 1,932 peak-to-peak breath intervals (Fig. 3A) and 1,933 breath amplitudes (Fig. 3B) were described earlier with reference to Fig. 1. The slopes of the power laws (extending over more than one time scale) in the Fano factor (Fig. 3C) and Allan factor (Fig. 3D) curves for the original time series were 0.54 and 0.41, respectively. H = 0.71 calculated from the slope of the power law in the Allan factor curve indicated the presence of long-range persistent correlations for window sizes between 41 and 1,490 s. As was typically the case, the slopes of the Allan and Fano factor curves for the surrogates were essentially flat in this range. The slope of the dispersional analysis plot for peak-to-peak breath interval fell within the range of those of the plots for 10 surrogates at m 5 (Fig. 3E). Thus fluctuations in peak-to-peak breath interval were considered random. In contrast, the slope (–0.29) of the dispersional analysis curve for breath amplitude (m > 11) fell outside the range of those of the surrogates (Fig. 3F). Thus fluctuations in breath amplitude were considered fractal.
For the four spontaneously breathing subjects in whom fluctuations in breath number and breath amplitude (but not respiratory period) were fractal, H values derived from the slope of the power law in the Allan factor curve were persistent in three cases (H = 0.75 ± 0.12) and antipersistent in one case (H = 0.24).
In three spontaneously breathing women (group 3 in Table 1), fluctuations in the number of breaths were fractal, but fluctuations in respiratory period and breath amplitude were not. The results from a 46-yr-old subject appear in Fig. 4. The distribution of 1,470 peak-to-peak breath intervals (Fig. 4A) had a CV of 0.46 and was positively skewed (0.36). The mean peak-to-peak breath interval was 3.87 s (15.5 breaths/min). The distribution of 1,471 breath amplitudes (Fig. 4B) had a CV of 0.61 and was essentially normal (skewness, 0.01). The time series were 5,700 s long. The slopes of the power laws in the Fano factor (Fig. 4C) and Allan factor (Fig. 4D) curves for the original time series were 0.40 and 0.44, respectively. A persistent H = 0.72 was calculated from the slope of the power law in the Allan factor curve, which was measured over a range of window sizes between 60 and 600 s. Although this range encompassed only one decade on a log scale, the fluctuations in number of breaths were considered fractal because the power law of the Fano factor curve extended over more than one time scale. Note also that the slopes of the curves for the surrogates were flat. In contrast, the slopes of the dispersional analysis plots for peak-to-peak breath interval (–0.41 in Fig. 4E) and breath amplitude (–0.53 in Fig. 4F) fell within the range of the slopes for 10 surrogates. Thus the fluctuations in peak-to-peak breath interval and breath amplitude were considered random.
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In two spontaneously breathing subjects (group 4 in Table 1), fluctuations in breath number and respiratory period were fractal, but fluctuations in breath amplitude were not. The results from a 31-yr-old man are presented in Fig. 5. The distribution of 1,501 peak-to-peak breath intervals (Fig. 5A) had a CV of 0.15 and was negatively skewed (–0.24). The mean peak-to-peak breath interval was 3.99 s (15.0 breaths/min). The distribution of 1,502 breath amplitudes (Fig. 5B) had a CV of 0.51 and was positively skewed (0.98). The time series were 6,000 s long. In this case, the power law in the Allan factor curve (Fig. 5D) extended over less than one decade of window sizes. This likely was the consequence of the large window size (150 s) at which the power law began and the fact that the time series was below average in length, thereby limiting the maximum allowable window size to 1,000 s. Nonetheless, we consider the fluctuations in number of breaths to be fractal on the basis of the power law in the Fano factor curve (Fig. 5C), which extended over more than one time scale. The slope of the power law in the Allan factor curve for window sizes between 150 and 1,000 s was 0.94, yielding a persistent H = 0.97. The H value (0.6) obtained from the second subject in this group also was persistent. Dispersional analysis showed that fluctuations in the peak-to-peak breath interval were fractal because the slope of the plot (–0.24) for m 8 fell outside of the range of the slopes for the surrogates (Fig. 5E). However, this was not the case for the dispersional analysis plot for breath amplitude, which had a slope of –0.52 for m 5. Offsetting of the y-intercept of the plot for the original time series from the y-intercepts of the plots for the surrogates caused the former curve to fall outside of the gray region for m 
60, even though their slopes were essentially the same. Offsetting of the y-intercepts occurred in some cases because the surrogates were constructed by shuffling the peak-to-peak breath intervals or breath amplitudes rather than their first differences.
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DISCUSSION |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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Peng et al. (24) used detrended fluctuation analysis to demonstrate the fractal nature of interbreath interval fluctuations in healthy adult subjects. We have confirmed this finding using a different method of fractal analysis, and, in addition, demonstrated fractal fluctuations of two other respiratory parameters. Fractal analysis revealed long-range correlations among the fluctuations in the number of breaths and breath amplitude, as well as peak-to-peak breath interval. Such long-range correlations were reflected by the power law relationships in the Fano factor and Allan factor curves as well as the dispersional analysis plots. That the power law relationships indeed monitored the presence of fractal (i.e., time-scale-invariant) behavior was shown to be the case by comparing the curves for the original time series with those of surrogate data blocks. Time-scale-invariant fluctuations in respiration reflect a form of memory in that the present value of the measured parameter is related to those in the distant past. As such, and in contrast to a random process with no correlations among events, recognition of the fact that one is dealing with a fractal process should be of value in roughly predicting future behavior of the respiratory system.
To our knowledge, the present study is the first to determine whether the fluctuations in one or more of three simultaneously measured respiratory parameters were fractal in the same subject. Previous to this report, Frey et al. (14) demonstrated that the probability density distribution of interbreath intervals in sleeping infants followed a power law. In another study using Fano factor analysis, Larsen et al. (21) concluded that the fluctuations in interbreath interval in sleeping infants were statistically self-similar (i.e., fractal). However, neither of these studies dealt with fluctuations in breath amplitude. This was also the case in the study of Szeto et al. (27), which demonstrated that the probability density distribution of interbreath intervals in fetal lambs followed a power law. Whereas Hoop et al. (19) used rescaled range analysis to demonstrate fractal fluctuations in breath volume in adult rats, their study did not deal with fluctuations in the breath number and respiratory period.
Our data suggest that the processes responsible for the fractal fluctuations in number of breaths, respiratory period, and breath amplitude can function independently of each other. Whereas there was a fractal component of the fluctuations in all three of these parameters in 9 of 20 spontaneously breathing subjects, there were four subjects in whom only the fluctuations in number of breaths and breath amplitude were fractal, three subjects in whom only the fluctuations in number of breaths were fractal, and two subjects in whom fluctuations in number of breaths and respiratory period but not breath amplitude were fractal. None of the parameters exhibited time-scale-invariant behavior in the two remaining spontaneously breathing subjects. Not surprisingly, there were no fractal fluctuations during breathing paced by a metronome, which essentially eliminated the variability in respiratory period. Barring major differences in the sensitivities of the methods used to test for time-scale-invariant behavior or differential degrees of randomness for the three measured parameters, these results point to the existence of potentially independent fractal mechanisms controlling the number of breaths, respiratory period, and breath amplitude. These processes may function at different levels of the central networks controlling respiration. For example, fractal fluctuations in number of breaths and respiratory period presumably would involve the rhythm generator in some way, whereas the fluctuations in breath amplitude might occur in a pattern generator made up of follower circuits (13).
The data are suggestive of two types of encoding of time-scale-invariant fluctuations in breathing frequency. We refer to the first as a fractal form of rate coding (20). A fractal rate code is suggested by seven cases in which time-scale-invariant fluctuations in the number of breaths occurred in the absence of fractal ordering of the peak-to-peak breath intervals. In such cases, the Fano and Allan factor curves contained a power law extending over more than one time scale, but the dispersional analysis plot for peak-to-peak breath intervals had a slope in the range of those of the plots for the surrogate data sets. For these cases, it is assumed that random shuffling of the peak-to-peak breath intervals to construct the surrogates indirectly destroyed fractal ordering of the number of breaths occurring over time, thereby eliminating the power laws in the Fano factor and Allan factor curves.
There were 11 subjects in whom the slope of the dispersional analysis plot for peak-to-peak breath interval fell outside of the range of the slopes of the plots for the surrogates. This always occurred in combination with the appearance of a power law in the corresponding Fano factor and Allan factor curves. Such cases point to fractal ordering of the peak-to-peak breath intervals, and we consider this to be a form of temporal coding (20). The physiological implications of rate vs. temporal coding of time-scale-invariant fluctuations in respiratory frequency remain to be determined.
For reasons given in METHODS, we derived values of H only from the slope () of the Allan factor curve. In the 18 subjects in whom fluctuations in the number of breaths were fractal, the long-range correlations were persistent in 14 subjects and antipersistent in 4 subjects. Thus the data could be either positively or negatively correlated. H measures the smoothness of a fractal time series with antipersistent values (negatively correlated data) reflecting a high degree of roughness and persistent values (positively correlated data) reflecting relative smoothness in the time series (2, 12, 23). The functional implications attached to positive vs. negative long-range correlations among the number of breaths remain to be investigated.
The mechanisms responsible for the fractal fluctuations in the number of breaths, respiratory period, and breath amplitude remain to be determined. One or more of the following factors might be involved in producing fractal fluctuations in each of these parameters. First, fractal fluctuations might reflect the inherent properties of the respiratory oscillator. Second, peripheral feedback from the lungs and/or thoracic mechanoreceptors might lead to fractal fluctuations in respiratory pattern. Third, suprapontine inputs to the respiratory network might be a source of the observed fractal fluctuations. Fourth, a feedback loop involving fractal fluctuations in blood-gas concentrations might be involved. Fifth, inputs to the respiratory network from the external world might be ordered as a fractal rather than random sequence of events. Sixth, fractal fluctuations in respiratory pattern might arise from changes in the level of wakefulness and attention. Finally, it also remains to be determined whether the long-range correlations observed in the present study reflect deterministic chaos within the respiratory system.
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TOPABSTRACTMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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ACKNOWLEDGMENTS |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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FOOTNOTES |
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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.
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REFERENCES |
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