|Year : 2021 | Volume
| Issue : 4 | Page : 177-185
Response of heart rate variability and cardiorespiratory phase synchronization to routine bronchodilator test in patients with asthma
Chih-Hsiang Tsou1, Lin-Sen Pon2, Jun-Zhi Liang3, Yun-Hsiang Chan4, Kuan-Jung Chen4, Feng-Shiang Cheng5, Tsair Kao6, Sai-Wei Yang7
1 Division of Chest Medicine, Ren-Ai Branch, Taipei City Hospital, Taipei; Department of Medicine, School of Medicine, National Yang Ming Chiao Tung University, Taiwan
2 Department of Electric Engineering, Chinese Culture University, Taipei, Taiwan
3 Department of Physics, Fu-Jen Catholic University, New Taipei City, Taiwan
4 Division of Chest Medicine, Ren-Ai Branch, Taipei City Hospital, Taipei, Taiwan
5 Department of Education and Research, Taipei City Hospital, Taipei, Taiwan
6 Medical Division, ACME Portable Machines, Inc., New Taipei City, Taiwan
7 Department of Biomedical Engineering, National Yang Ming Chiao Tung University, Taipei, Taiwan
|Date of Submission||13-Mar-2021|
|Date of Decision||23-Jun-2021|
|Date of Acceptance||30-Jun-2021|
|Date of Web Publication||28-Aug-2021|
Dr. Chih-Hsiang Tsou
No.10, Sec. 4, Ren-Ai Rd., Division of Chest Medicine, Ren-Ai Branch, Taipei City Hospital, Taipei; Department of Medicine, School of Medicine, National Yang Ming Chiao Tung University, Taipei
Source of Support: None, Conflict of Interest: None
Heart rate variability (HRV) and cardiorespiratory phase synchronization (CRPS) were employed to study the cardio- and respiratory interactions in patients with asthma receiving inhalation of beta2-agonist (Berotec 200 mcg) for routine bronchodilator test. Both time- and frequency-domain parameters were used to analyze the HRV. A weighted G-index was introduced to study the quality of the CRPS. The HRV parameters, in both the time and frequency domains, exhibited significant changes pointing to a sympathetic activation of the autonomic balance immediately after the inhalation. On the other hand, the CRPS index barely changed throughout the entire process. This indicates that inhalation of beta2-agonist does not alter the CRPS appreciably, and that the CRPS, in contrast to HRV, is relatively stable in response to the inhalation of beta2-agonist in patients with asthma.
Keywords: Asthma, beta2-agonist, cardiorespiratory phase synchronization, heart rate variability
|How to cite this article:|
Tsou CH, Pon LS, Liang JZ, Chan YH, Chen KJ, Cheng FS, Kao T, Yang SW. Response of heart rate variability and cardiorespiratory phase synchronization to routine bronchodilator test in patients with asthma. Chin J Physiol 2021;64:177-85
|How to cite this URL:|
Tsou CH, Pon LS, Liang JZ, Chan YH, Chen KJ, Cheng FS, Kao T, Yang SW. Response of heart rate variability and cardiorespiratory phase synchronization to routine bronchodilator test in patients with asthma. Chin J Physiol [serial online] 2021 [cited 2022 May 21];64:177-85. Available from: https://www.cjphysiology.org/text.asp?2021/64/4/177/324869
| Introduction|| |
The interaction between human cardio- and respiratory systems is sophisticated.,,,, Different patterns of cardiorespiratory coupling may represent various physiologic conditions or indicate different physical disorders. One type of cardiorespiratory interaction is related to cardiac autonomic control, which can be evaluated by analyzing the heart rate variability (HRV). Cardiorespiratory phase synchronization (CRPS) is another type of cardiorespiratory interaction. Its physiologic significance is unclear at present, much less its relation to the cardiac autonomic control.
Asthma is a chronic airway disease, which is characterized by a reversible airflow limitation and episodic attack. Clinically, patients with asthma usually receive bronchodilator test to assess airway reversibility. Besides, asthma attack, due to bronchospasm, can be relieved by inhalation of bronchodilators as well. In addition to bronchodilation, bronchodilators can trigger sympathomimetic reaction, which will stimulate cardiac autonomic activity provoking side effects, such as tachycardia or hand tremor., It has been shown that reduced cardiac autonomic control, represented by specific cardiorespiratory coupling pattern, can reflect the different degrees of severity in patients with asthma.
HRV is a measure of cardiorespiratory coupling and has become a useful noninvasive tool to evaluate cardiac autonomic control. Parameters used in the time-domain analysis of HRV, such as the mean heart beat interval (mean RR) and the standard deviation of the heart beat interval (SDRR), can reflect the total sympathetic and parasympathetic activation. For instance, the increase of the mean RR reveals the increase of the HRV, and the SDRR measures how these RR intervals vary with time. On the other hand, parameters used in the frequency-domain analysis can indicate the tendency of sympathovagal balance during the change of physiologic conditions. High-frequency (HF) power of the HRV, which can be augmented by respiration, shows the influence of parasympathetic activation.,
Cardiorespiratory interaction such as respiratory sinus arrhythmia (RSA) can be used as an indicator to identify the depth of anesthesia.,, Therefore, the characterization of sympathovagal balance via HRV can be a good probe to explore the pathophysiology of cardiac autonomic control in patients with asthma. In addition to RSA, CRPS has also been shown to be an important physiological indicator. The CRPS denotes the coupling between cardiac and respiratory systems that manifests itself as an entrainment ratio between heart and respiratory rhythms. With the phase synchronization being present, it can then serve as a measure of the strength of coupling between physiologic systems. An alteration of phase synchronization may provide important information about the coupled physiologic systems. The CRPS in human physiologic systems can be visualized by a synchrogram, though its underlying physiologic significance is unclear. Under specific conditions, such as anesthesia, mediation, and deep sleep status, increased CRPS was noted.,, Likewise, paced breathing pattern also augments CRPS. On the other hand, a reduced CRPS was reported in patients with obstructive sleep apnea, when they are falling asleep. Besides, CRPS seems to be related to the aging process and has the potential to be used to assess the developmental maturity of premature infants.,, Moreover, volume status, the hydration status of human bodies, is also an important factor that influences the CRPS. So far, little research has been done about the effect of inhaled sympathomimetic drugs on the cardiorespiratory coupling in patients with asthma.
The hypothesis of this work is that HRV and CRPS would be influenced by inhalation of bronchodilator in patients with asthma as they received routine bronchodilator response test. In this context, we studied the changes of cardiorespiratory coupling by analyzing the data of HRV and CRPS in patients with asthma during inhalation of bronchodilator (beta2-agonist) in routine test. To study the HRV, we apply both time- and frequency-domain analyses. For the study of the CRPS, we introduce a weighted Γ-index to assess the quality of phase synchronization through the corresponding synchrogram. The details of the weighted Γ-index are given in Appendix A.
| Materials and Methods|| |
The local institutional review board had reviewed and approved the study (TPEIRB-1010724-E). Subjects with asthma were recruited at Taipei City Hospital from 2012 to 2013. They were diagnosed with asthma by the clinic pulmonologists. The subjects, aged between 20 and 65 years, who did not have the following conditions, such as allergy to Berotec inhalation, hypertrophic cardiomyopathy, hyperthyroidism, pregnancy, breastfeeding, or poor communication skills, were enrolled in the study. All of them signed the informed consent before taking pulmonary function testing with bronchodilator test.
The subjects stood taking spirometry in the laboratory and then lay supine on the examination table. The examinations were carried out in the daytime from 9 am to 6 pm. Continuous electrocardiogram (ECG) and respiration signal (end-tidal CO2 [etCO2]) were recorded at the same time using surface ECG leads and nasal cannula sensors, respectively. For each subject, at least 40 min of ECG and etCO2 data were recorded. In the middle of the process, at about 20 min from the start, the subject was asked to take a deep breath. Immediately thereafter, an inhalation of two puffs of beta2-agonist (Berotec 2 puffs [2 × 100 mcg]) from a metered-dose inhaler was given to the subject. After the data recording was completed, the subject was given a pulmonary function test to assess the reversibility of airways.
Data acquisition and signal processing
This study employed portable measuring instruments (BP 508, Colin Co., Nippon, Japan, and CO2SMO Plus, Novametrix Medical Systems, Wallingford, CT, USA) to acquire the ECG and etCO2 signals. The collected signals were digitized and transferred to a personal computer via an analog-to-digital converter at a sampling rate of 250 Hz. The raw data were divided equally into eight segments, each of which contains a 5-min interval of ECG and etCO2 signals, henceforth denoted as Stage 1 through Stage 8. The ECG and etCO2 data were processed with MATLAB, picking R peaks and deducing RR intervals. Data of ectopic beats and arrhythmia were removed manually afterward. Subsequently, time sequence of the RR intervals was converted into instantaneous RR time series and resampled at a sampling rate of 5 Hz. Accordingly, the respiratory signals were downsampled to 5 Hz.
Heart rate variability study
The study of the variation of RR intervals, known as the HRV, can be carried out with time- and frequency-domain analyses. In this work, we applied both the time-domain analysis and the frequency-domain analysis to the RR interval and power spectrum of the RR intervals, respectively. For time-domain analysis, the mean RR interval and the standard deviation of RR interval were calculated. For frequency-domain analysis, the power spectrum of RR intervals was divided into three bands: the very-low-frequency band (0–0.04 Hz), the low-frequency band (0.04–0.15 Hz), and the HF band (0.15–0.4 Hz). Current knowledge of the power spectrum shows that the HF band power can be modulated by parasympathetic activation, mainly from respiratory influence, whereas the low-frequency band power is primarily influenced by sympathetic and parasympathetic activation. The power ratio of the low-frequency band to the HF band (LF/HF) reflects the sympathovagal balance. In practice, the power spectrum was obtained by applying the fast Fourier transform (FFT) to the RR interval time series. Further parameters used in this work are the normalized power of the power spectrum of the RR intervals in the low-frequency band LFn = LF/(TP − VLF) and that in the HF band HFn = HF/(TP − VLF), where TP denotes the total power of the spectrum in the entire frequency range (0–0.4 Hz) and VLF represents the power of the very-low-frequency band.
Cardiorespiratory phase synchronization
It is well known that there is a coupling between cardiac and respiratory rhythms in human bodies.,, However, the analytic relationship between these two systems has yet to be clarified. RSA is a well-known cardiorespiratory coupling. Unlike RSA, the CRPS, also a cardiorespiratory coupling, is less understood. The CRPS manifests itself as the fact that cardiac and respiratory rhythms tend to oscillate with a repeating sequence of relative phase angles. Shäfer et al. proposed to visualize the phase synchronization by a synchrogram. It has been shown that the CRPS is more prominent in certain activities, such as Zen meditation and deep anesthesia, and in specific groups such as athletes., On the contrary, CRPS decreases in the arousal stage and in patients with obstructive sleep apnea. In this study, we followed Shäfer's method using synchrogram to visualize the CRPS during inhalation of beta2-agonist in patients with asthma. To quantify the degree of the CRPS, we employed the sliding window method and introduced a weighted Γ-index to unveil the dynamics between cardiac and respiratory oscillatory systems. The details of the algorithm are described in Appendix A.
Statistical analysis was carried out using the SPSS 11.0.1 software (SPSS Inc., Chicago, IL, USA). The resulting values were presented as mean ± standard deviation and summarized in [Table 1]. Wilcoxon signed-rank test was used to assess the significance of comparison between two continuous dependent variables. P < 0.05 was considered statistically significant.
|Table 1: Parameters of heart rate variability and Γ-index of cardiorespiratory phase synchronization at different stages (total n=48)|
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| Results|| |
A total of 48 eligible subjects with asthma were enrolled in the study. There were 25 men and 23 women. Their mean age, body weight, and body height were 47.21 ± 13.78 years, 65.29 ± 13.26 kg, and 163.52 ± 7.85 cm, respectively. Their spirometric data before and after the inhalation of Berotec were as follows. Before inhalation of Berotec, the forced expiratory volume in 1 s (FEV1) was 2.39 ± 0.92 L, the forced vital capacity (FVC) was 3.14 ± 1.06 L, and FEV1/FVC was 0.75 ± 0.11. After the inhalation, they were FEV1 = 2.50 ± 0.90 L, FVC = 3.18 ± 0.99 L, and FEV1/FVC = 0.77 ± 0.10.
[Figure 1] shows the results of a representative case. The instantaneous RR interval, instantaneous respiratory interval, and the frequency ratio of the heart rhythm to the respiratory rhythm (fh/frsp) are presented respectively in the upper, the middle, and the lower panels of [Figure 1]a. As can be seen, the mean RR is around 0.7 s, mean respiratory interval around 4 s, and the frequency ratio around 5.5. A decrease in RR interval immediately after the inhalation of Berotec was detected. Thereafter, the RR interval returns gradually to the previous state. The upper panel of [Figure 1]b shows the HRV parameters of RR interval deduced from the ECG at Stage 4, right before the inhalation of Berotec. Applying FFT to the time series of the RR interval, it gives the power spectrum, as shown in the lower panel of [Figure 1]b. The HRV parameters deduced from [Figure 1]b are the following: mean RR = 703.54 ms, SDRR = 33.49 ms, LF = 48 ms2, HF = 137 ms2, LF/HF = 0.35, LFn = 20.26%, and HFn = 57.40%. [Figure 2] depicts a cardiorespiratory phase synchrogram of a separate case with different n:m (m = 1, 2, 3) ratios, where n and m refer to the cardiac cycles and the respiratory cycles, respectively. A transient disrupted CRPS in the n:2 and n:3 synchrograms was noted soon after inhalation of Berotec. After that, they restored to the patterns exhibited prior to the inhalation, similar to the behaviors observed in the instantaneous RR interval as described earlier.
|Figure 1: (a) Time series of heart beat interval (upper panel), time series of respiratory interval (middle panel), and ratio of heart beat frequency to respiratory frequency (lower panel) in a representative case during receiving routine bronchodilator test. Event mark (red line) denotes the start of the inhalation of Berotec. (b) Heart rate variability analysis at stage 4 (5-min segment), shortly before inhalation of Berotec, in a representative case. Upper panel: Cardiotachogram of RR intervals for the time-domain analysis. Lower panel: Power spectrum of the RR intervals for the frequency-domain analysis. Blue line denotes 0.04 Hz mark. Green line denotes 0.15 Hz mark. Red line denotes 0.4 Hz mark.|
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|Figure 2: Synchrograms of a separate representative case. Here, three n:m ratios are shown, where n and m refer to the cardiac cycles and the respiratory cycles, respectively. The X-axis denotes the time-axis (second). The Y-axis denotes the number of respiratory cycles (m). The red vertical line denotes the event mark of the inhalation of Berotec. Upper panel: m = 1. Middle panel: m = 2. Lower panel: m = 3.|
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Heart rate variability and cardiorespiratory phase synchronization
The time- and frequency-domain parameters of HRV and the CRPS index are summarized in [Table 1]. Statistically significant changes of mean RR intervals are noted at Stage 2 and Stage 3, as compared with Stage 1 and Stage 2 (P = 0.0001 and 0.025, respectively). Similar changes of SDRR are noted at Stage 4, as compared with Stage 3 (P < 0.001). After inhalation of Berotec, statistically significant changes of SDRR are noted at Stages 6, 7, and 8, as compared with Stage 4 (P < 0.001, <0.05, and <0.001, respectively). These results are illustrated in [Figure 3]a.
|Figure 3: Cardiorespiratory coupling parameters at different stages in patients with asthma. (a) Time-domain analysis of heart rate variability. Upper panel: the average of the RR intervals. Lower panel: the standard deviation of the RR interval. (b) Frequency-domain analysis of heart rate variability. Upper panel: normalized power of the low-frequency band and normalized power of the high-frequency band. Lower panel: the power ratio of the low-frequency band to the high-frequency band of heart rate variability and the reduced G-index. aP < 0.05 as Stage 2 to Stage 4 compared with the previous stage, and Stage 5 to Stage 8 compared with Stage 4. bP < 0.01 as Stage 2 to Stage 4 compared with the previous stage, and Stage 5 to Stage 8 compared with Stage 4. cP < 0.001 as Stage 2 to Stage 4 compared with the previous stage, and Stage 5 to Stage 8 compared with Stage 4.|
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In the frequency-domain analysis of HRV, before inhalation, we observed significant changes of LF at Stage 4, as compared with that at Stage 3 (P = 0.045). There are also significant changes of HF at Stage 2, as compared with Stage 1 (P = 0.012). After inhalation, statistically significant decreases of HF power were noted at Stage 6 and Stage 7, as compared with Stage 4 (P < 0.01 and < 0.05, respectively). A statistically significant increase of LFn power was found at Stage 5, as compared with that at Stage 4 (P = 0.025). No significant change of HFn was noted in the whole process, as shown in the upper panel of [Figure 3]b. Nevertheless, we noticed a clear elevation of LF/HF at Stage 5, as compared with Stage 4 (2.15 ± 2.32 vs. 1.53 ± 1.34, P = 0.032). On the contrary, no significant change in the Γ-index was found during the entire process, as can be seen in the lower panel of [Figure 3]b.
| Discussion|| |
To exam the effect of beta2-agonist inhalation in asthmatic patients on HRV and CRPS, HRV parameters and the Γ-index were compared at stages before and after the inhalation. Significant changes in HRV parameters (SDRR, HF, LFn, and LF/HF) were found at stages after beta2-agonist inhalation as compared with before inhalation. After inhalation, significant increases of LFn and LF/HF at Stage 5 are followed by significant decreases of SDRR, and HF at Stage 6 and Stage 7 without significant change in the CRPS Γ-index. These results suggest that autonomic balance is shifted toward sympathetic activation. Remarkably, inhalation of beta2-agonist in subjects with asthma did not alter the CRPS noticeably. This is a novel finding regarding the interaction between beta2-agonist and CRPS. It reveals that the effect of inhalation of beta2-agonist on the CRPS may not be significant.
Immediately before and after inhalation of Berotec (between Stages 4 and 5), the fact that our HRV analysis revealed almost no changes of time-domain parameters (mean RR and SDRR) but significant changes of frequency-domain parameters (LF/HF and LFn) indicates that the whole process is toward sympathetic activation. However, previous studies showed that different beta2-agonists have different effects of cardiac autonomic control on asthmatic patients. For example, Eryonucu et al. reported that Berotec (fenoterol) inhalation does not have any effect on sympathetic activation (mean RR, SDRR, and HF) in asthmatic patients undergoing regular treatment. On the other hand, Zahorska-Markiewicz et al. showed that treatments using both salbutamol and terbutaline increase not only HF but LF and LF/HF as well. Similar HRV studies regarding the effect of beta-agonists on cardio-autonomic control also revealed different findings in asthmatic patients.,, We infer that the main difference between our study and the previous studies may stem from different population groups (asthmatic patients without versus with treatment), different measures of HRV (LF/HF, LFn vs. mean RR, SDRR, HFn), and using different drugs.
Besides, we found that some changes of mean RR, SDRR, LF, and HF before inhalation of Berotec are noticeable from [Table 1]. Since these are real-world data, the most possible reason for these noticeable changes of parameters before inhalation may come from the small sample size (n = 48). Due to this small sample size, the effect of certain confounding factors, such as age, gender, diabetes, hypertension, and disease status, may have more impact on the parameters of HRV. These findings need further investigation in the future.
Human cardiac and respiratory systems interact in a complicated manner. They influence each other under different conditions. As mentioned earlier, higher degree of CRPS is found in high-performance athletes and in people during meditation, sleep, or anesthesia, whereas reduced CRPS is observed in patients with obstructive sleep apnea while sleeping.,, In other words, CRPS seems to be related to the status of general well-being, deep meditation, anesthesia, and physical maturity. In contrast to sensible changes shown in various HRV parameters after inhaling Berotec, no significant change in CRPS was found at all stages. Even if the perturbation with deep breathing and drug inhalation not being strong enough to trigger the shift of the CRPS index could not be definitively ruled out, the result of no significant change in the CRPS Γ-value, as opposed to HRV parameters, suggests that the underlying mechanisms of HRV are different from those of CRPS in response to the inhalation of Berotec. This may further imply that HRV parameters instead of CRPS Γ-index are more sensitively influenced by the perturbation of inspiration and the beta2-sympathomimetic effect on cardiorespiratory coupling. Moreover, almost all Γ-values obtained being approaching to or <0.5 throughout the entire process do agree with the patients' consciousness that is clear and alert; none of them were falling asleep or in deep mediation. This indicates that the patients' mental status is hardly disturbed by the perturbation of inhaling Berotec. It may also be one of the reasons that the CRPS index exhibits irrelevant change under transiently but significantly sympathetic activation.
In this study, there are some limitations. First, normal healthy subjects were not enrolled as the control group for comparison to investigate the effect of asthma on HRV and CRPS. Second, there is a lack of introducing placebo-drug group for comparison to eliminate the effect of deep breathing on HRV and CRPS. Further studies are necessary to clarify the effects of the disease itself and of the breathing on CRPS. Finally, our study sample size is small, and confounding factors such as mental status evaluation, disease severity, and comorbidity (diabetes, hypertension, etc.) were not conducted. A large sample size study with confounding variables' evaluation should be conducted to highlight these variables' impact on HRV and CRPS.
| Conclusion|| |
In this study, a significant change in the HRV parameters without simultaneously an appreciable change in the CRPS Γ-value was found after inhalation of Berotec. It implies that the CRPS, in contrast to HRV, is relatively stable in response to the inhalation of beta2-agonist in patients with asthma.
Financial support and sponsorship
This study was supported by grants from the Department of Health, Taipei City Government (Project No. 10102-62-091).
Conflicts of interest
There are no conflicts of interest.
| Appendix A|| |
To study the quality of the cardiorespiratory phase synchronization, we employed the sliding window method and introduced a weighted Γ-index in the analysis of the cardiorespiratory synchrogram (CRS).
Consider the case of n:m cardiorespiratory synchronization where there are n heartbeats within m consecutive respiratory cycles. In this study, the heartbeat information was obtained by the electrocardiograms (ECGs) and the respiratory signals were from the end-tidal capnography. While the onset of the expiration phase was used to denote the start of the respiratory cycle, the series of R-peaks were regarded as the time sequence of the heartbeat, as shown in Figure A1. We followed the convention and assumed that each complete respiratory cycle corresponds to a phase change of 2π. Accordingly, the phase of the inception of the r-th respiratory cycle can be assigned as φ(tr) = 2π(r−1) For an arbitrary instant of time tr≤t<tr+1 linear interpolation was used to calculate the instantaneous phase as
Consequently, for the R-peak in the k-th heartbeat taking place at tk, we relate the phase of this heartbeat to φ(tk).
In the CRS representation, the respiratory phases are wrapped into an interval of [0, 2πm] and the normalized phases of the heartbeats as a function of time are plotted between 0 and m radians. By calculating the normalized cyclic-relative phases, defined as
There will be n nearly horizontal stripes within m respiratory cycles appeared in the n:m phase synchronized CRS. [Figure 2] shows typical n:m (m = 1, 2, 3) synchrograms.
While synchrogram provides a visual tool capable of identifying synchronization of two coordinated systems and distinguishing between different periods of synchronization, even for noisy and nonstationary data, it will be beneficial to have an index that can serve as an indicator reflecting the degree of synchronization quantitatively between the two systems. To this end, the γ-index was used to characterize the strength of synchronization. In this regard, the cyclic phase is calculated according to
whose values lie in the range of 0 to 2nπ. The γ-index based on the first Fourier mode of the intensity distribution of ψn, m is, then, given by
where M is the total number of heartbeats. This index has an advantage of free of parameters in the analysis of the strength of the synchronization because the calculation does not involve the distribution itself. According to the definition, γn,m varies from 0 to 1 with γn,m = 1 corresponding to an extremely high degree of n:m synchronization and γn,m = 0 a completely desynchronized case over the entire time span of interest.
In the case that the physical activity can be divided into different stages, such as the sleep stages and the various stages in the current study, the γ-values within a specific stage and that during the transition from one stage to the other are essential to reveal in detail the cardiorespiratory coordination. To be able to follow the time-varying degree of synchronization, the sliding window method was employed to elucidate the temporal variation of the γ-values. A sliding window consists of three consecutive respiratory cycles that is moved every step one respiratory cycle forward over the entire series of respiratory time sequences. Each window overlaps the adjacent one with two respiratory cycles. Firstly, the γ-index is computed for every preselected n:m coordination for each window. Then, the weighted Γ-index is calculated by
where N is the total number of the respiratory cycles, ti corresponds to the length of the i-th sliding window, and T is the total duration of the data record. In this study, Γ was used to illustrate the goodness of synchronization over a specific time interval. Practically, in a highly n:m synchronized case, this particular n:m dominates over the other preselected n':m'. Thus, we regard γn, m ≥ 0.75 as the case of high n:m synchronization. Accordingly, Γ was reduced to
Here, the reduced Γ takes into account merely the γ-value in the i-th window subject to γn,m ≥ 0.75 For evenly distributed respiratory cycles and strictly n:m synchronization, Γ approaches the value of 3(N – 2)/N. This sets the upper limit of the Γ-value to be 3 for sufficiently large value of N, which is the case of the current study, typically N ≈102. On the other hand, for completely desynchronization, Γ is zero, since γn,m = 0 for every preselected n:m pair in this situation.
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[Figure 1], [Figure 2], [Figure 3]