Power Laws and Self-Organized Criticality in Cardiovascular Avalanches

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The present manuscript aims to investigate the self-organized criticality behavior of the cardiovascular system. Beat-to-beat pressure and heart rate data from seven healthy subjects in the head-up position were analyzed. The authors developed a three-stage assessment workflow to evaluate the power-law behavior of cardiovascular avalanches using five types of distributions.
The study provides valuable insights into the dynamics of the cardiovascular system, shedding light on the self-organized criticality of this complex system.

Overall, the study is well-written and structured.

The present reviewer suggests that the authors revise the results section. Currently, there is a mix between illustrative examples and general results, what makes the findings complex to follow.

For example, in Section 3.2, the example in Figure 1 should be introduced first, followed by the overall population results. The same applies to Section 3.3: first introduce the illustrative subject in Figure 2, and then present the results for the overall population.  One cannot state general results while refering to an illustrative example, e.g., lines 268-269: “(mean r of seven subjects was 0.96 ± 0.02; Figure 2, Panel D). 

Please provide a clearer explanation of the results in Table 1. How do these results support the presence of a straight-line distribution in only four out of seven patients?

Author Response

Manuscript ID: fractalfract-3522159

Reviewer #1

The present manuscript aims to investigate the self-organized criticality behavior of the cardiovascular system. Beat-to-beat pressure and heart rate data from seven healthy subjects in the head-up position were analyzed. The authors developed a three-stage assessment workflow to evaluate the power-law behavior of cardiovascular avalanches using five types of distributions.
The study provides valuable insights into the dynamics of the cardiovascular system, shedding light on the self-organized criticality of this complex system.

Overall, the study is well-written and structured.

The present reviewer suggests that the authors revise the results section. Currently, there is a mix between illustrative examples and general results, what makes the findings complex to follow.

For example, in Section 3.2, the example in Figure 1 should be introduced first, followed by the overall population results. The same applies to Section 3.3: first introduce the illustrative subject in Figure 2, and then present the results for the overall population.  One cannot state general results while refering to an illustrative example, e.g., lines 268-269: “(mean r of seven subjects was 0.96 ± 0.02; Figure 2, Panel D). 

Thank you for this valuable suggestion. We have revised the Results section to improve clarity by restructuring it so that each illustrative example is presented before the explanation of the corresponding group-level result. This should make the findings easier to follow and enhance the overall coherence of the section.

L278: The quality of the recordings was high, with few artifacts or ectopic beats. An illustra-tive example demonstrates this quality, showing a total of three artifacts or ectopic beats, of which only two occurred during the analyzed period (Figure 1, panel A). In this example, a 1-minute zoom randomly performed on the analyzed period revealed several cardiovascular avalanches of both types, suggesting that numerous such events occurred throughout the entire analyzed period (Figure 1, panel B). Consistent with this example (Figure 1, panel B), numerous cardiovascular avalanches were observed in the recordings from the whole group of subjects.

L301: The illustrative example revealed a Gutenberg-Richter distribution of the subject's vasovagal sequences, indicated by a straight line (Figure 2, panel C). Analysis of the whole group of subjects confirmed this distribution, as the regression coefficient (r) exceeded 0.95 in each case

L339: The Zipf distribution graph of the time intervals between sequences showed a straight line in the illustrative example (Figure 2, panel D). A similar straight line was confirmed in six of the seven subjects during the first stage

L344: The first stage of analysis revealed a straight-line pattern in the Zipf distribution graph of the symbolic analysis from the illustrative example. However, this pattern was limited to a short segment of the data (Figure 2, panel E). A similar straight-line pattern in the Zipf distribution was observed in each subject of the whole group and confirmed by surrogate data analysis, though it was likewise restricted to a short segment of the data, as in the illustrative example (mean r = 0.99 ± 0.00 for the seven subjects, compared to 0.92 ± 0.01 for the corresponding surrogate data; only one surrogate dataset exceeded the threshold of 0.95).

Please provide a clearer explanation of the results in Table 1. How do these results support the presence of a straight-line distribution in only four out of seven patients?

Thank you for pointing this out. Table 1 has now been deleted and replaced by Tables 3 and 4. Table 3 now presents the individual Clauset p-values, supporting or not the presence of a straight-line distribution. Table 4 provides the complementary numerical results of the Clauset statistics, as requested by Reviewer #2. Table 4 is now introduced by the following sentence:

L353: The scaling exponent alpha, determined by Clauset statistics, was in the same range for the Gutenberg-Richter distribution and the first three Zipf distributions (classical, modified, and delta t), but not for the last studied Zipf distribution (symbolic, Table 4).

Reviewer 2 Report

Comments and Suggestions for Authors

Please see the attached file.

Comments for author File: Comments.pdf

Comments on the Quality of English Language

Тhe reviewer did not notice any significant gaps in the English language.

Author Response

Manuscript ID: fractalfract-3522159

Reviewer #2

The issue studied in the manuscript is of interest from a scientific point of view. The authors claim novelty by checking the action of power laws under certain conditions in cardiological data. The reviewer accepts this claim, but suggests providing more results in defense of this thesis.

1. It would be good to provide some formulas by which the parameters under consideration are calculated. For example, for the regression coefficient.

We have added several formulas to the revised manuscript, including the one for the regression coefficient. We have also included Clauset’s statistic formulas and formulas describing the identification and sorting of cardiovascular avalanches.

L153; L163; L187; L224; L253; L256

2. The article has a rather medical focus. This assumes a somewhat more indepth medical knowledge of the readers in the field of the cardiovascular system and knowledge of specific medical cardiac terminology. I leave it to the editors to assess whether this manuscript is suitable for this journal from this point of view.

Thank you for your comments. I was approached by the journal to submit this manuscript, which aims to integrate cardiovascular physiology with concepts from complex systems theory. This interdisciplinary approach may require familiarity with both medical and scientific terminology. I trust the editors will assess the manuscript’s suitability accordingly. Moreover, in response to the reviewer’s other recommendations, we have added mathematical formulas in the Methods section and included numerical values in the Results section, along with three additional tables (see answer to comment #7). These improvements should make the manuscript more accessible and aligned with the reviewer’s expectations.

 

3. How exactly do the authors determine whether the unusually high pulse rate registered at times is real, and not, for example, artifacts due to noise effects?

Thank you for pointing this out. We have clarified in the manuscript how unusually high or low pulse rates were evaluated and corrected when necessary.

L124: All detections were manually reviewed by a trained operator, who carefully examined the ECG recording to identify artifacts or ectopic beats. When an unusually high or low pulse rate was observed, the operator checked the corresponding ECG trace: if it showed an artifact, or an ectopic beat, or a missing detection, the erroneous value was corrected by replacing it with the preceding one; otherwise, the value was retained.

 

4. It would be good to explain when which statistic is used and why. For example, why was Clauset’s statistics used for Table 1?

Thank you for your comment. We have clarified in the manuscript that the Wilcoxon test was chosen due to the small sample size and the paired nature of the data.

L240: A Wilcoxon test was chosen because the sample size was small and the data were paired, as each subject’s actual and surrogate data stemmed from the same underlying series.

In addition to clarifying the choice of the Wilcoxon test, we have also explained why Clauset’s method was selected, as it is, to the best of our knowledge, the only established statistical approach for confirming a power law distribution.

L245: Clauset’s method was selected because, to the best of our knowledge, it is the only established statistical method for rigorously testing and validating the presence of a power law distribution.

 

5. Why was the Wilcoxon paired-rank test used to compare real and surrogate data?

Thank you for your question. As mentioned in a previous response, the Wilcoxon paired-rank test was used due to the small sample size and the paired nature of the data, where each subject’s actual data were compared with their own surrogate data.

L240: A Wilcoxon test was chosen because the sample size was small and the data were paired, as each subject’s actual and surrogate data stemmed from the same underlying series.

 

6. The authors declare that they also use the Kolmogorov-Smirnov method. Why did they choose it? Yes, there are a few words written, but it would be good if the justification were written a little better.

Thank you for your comment. In response, we have clarified in the manuscript that the Kolmogorov-Smirnov method was used to estimate the lower threshold xmin by minimizing the distance between the empirical and model distributions, as part of the procedure outlined by Clauset et al. This method is a standard approach in power law analysis, and it was applied here to ensure the accuracy of the model fitting.

L249: The lower threshold xmin ​was estimated using the Kolmogorov-Smirnov method to minimize the distance between the empirical and model distributions, following the procedure outlined by Clauset et al. (2009).

7. It is good to give the numerical results obtained for the statistical tests used.

Thank you for this valuable suggestion. We have added the numerical results for the statistical tests used in the revised manuscript. This has led to the inclusion of three additional tables to present these results clearly and comprehensively.

L324: Table 1. Regression coefficients (r) for the seven subjects across the five studied distributions.

	S1	S2	S3	S4	S5	S6	S7
Gutenberg-Richter	0.994	0.974	0.993	0.972	0.988	0.987	0.978
Zipf (classical)	0.993	0.989	0.999	0.976	0.991	0.990	0.994
Zipf (modified)	0.979	0.947	0.971	0.981	0.990	0.978	0.984
Zipf (delta t)	0.984	0.956	0.988	0.955	0.980	0.877	0.973
Zipf (symbolic)	0.991	0.989	0.982	0.988	0.970	0.998	0.995

Subjects are indicated as S1 to S7. A regression coefficient (r) > 0.95 was used as the cutoff for accepting a distribution as a power law.

 

L328: Table 2. Wilcoxon test results for the comparison of actual and surrogate time series across the five studied distributions.

	n	W	p
Gutenberg-Richter	7	-28	0.0156
Zipf (classical)	7	-28	0.0156
Zipf (modified)	7	-28	0.0156
Zipf (delta t)	6	-21	0.0313
Zipf (symbolic)	7	28	0.0156

The number of subjects is n. One subject did not pass the first stage of evidence for Zipf (delta t) distribution and was not included in the comparisons. W is the Wilcoxon W, and p is the p-value. Statistical significance was set at p < 0.05.

 

L332: Table 3. Clauset statistic p-value for the seven subjects across the five studied distributions.

	s1	s2	s3	s4	s5	s6	s7
Gutenberg-Richter	0.027	0.010	0.089	0.401	0.001	0.473	0.592
Zipf (classical)	0.540	0.122	0.261	0.092	0.447	0.190	0.874
Zipf (modified)	0.508	0.385	0.353	0.420	0.398	0.438	0.944
Zipf (delta t)	0.860	0.060	0.030	0.570	0.000	0.000	0.100
Zipf (symbolic)	0.000	0.000	0.041	0.000	0.000	0.000	0.000

A p-value > 0.05 indicated that the distribution was compatible with a power law.

 

L360: Table 4. Scaling exponent determined by Clauset statistics across the five studied distributions studied.

	a
Gutenberg-Richter	7.46 ± 0.99
Zipf (classical)	8.46 ± 0.90
Zipf (modified)	7.03 ± 0.34
Zipf (delta t)	6.36 ± 1.58
Zipf (symbolic)	4.53 ± 0.66

Values are the mean of the seven subjects.

 

8. The authors provide some evidence in defense of the thesis they have put forward. But this is the result of a very small number of records. It is questionable whether generalized assessments can be made.

Thank you for this comment. We believe that this point overlaps with point 10, which raises a similar concern regarding the small sample size and the generalization of our findings. As explained in our response to point 10, we have added a new paragraph to highlight the value of our beat-to-beat recordings despite the small sample size, and we have cited additional studies supporting the presence of power laws in the cardiovascular system. This reinforces the evidence for self-organized criticality and addresses the concern about generalizability.

9. Why did the authors expect "multiple cardiovascular avalanches" during the test? What gave rise to this expectation?

In response to your comment regarding the expectation of "multiple cardiovascular avalanches," we have modified the manuscript to emphasize that, as shown in Figure 1, panel B, numerous cardiovascular avalanches were observed in the recordings from the whole group of subjects, without assuming prior expectation.

L284: Consistent with this example (Figure 1, panel B), numerous cardiovascular avalanches were observed in the recordings from the whole group of subjects.

 

10. The presence of self-organized criticality. According to the reviewer, more evidence should be presented in defense of this statement. Additional assessment of more parameters, for example? The thesis is defended with a small number of records and a small length of the studied section of the record.

Thank you for this valuable comment. In response, we have added a new paragraph to strengthen the evidence supporting the presence of self-organized criticality (SOC) in the cardiovascular system. Specifically, we have highlighted the value of our beat-to-beat recordings, which lasted more than 40 minutes in healthy subjects in the upright position — a rare and difficult condition to achieve for both physiological and ethical reasons. This underscores the quality and significance of the data despite the small sample size. Additionally, we have cited several previous studies that have identified power laws in the cardiovascular system, further supporting the presence of SOC. These cumulative findings provide a more robust foundation for our interpretation.

L488: Although the sample size in our study was small (n = 7), the beat-to-beat heart rate and blood pressure recordings were valuable, as they lasted more than 40 minutes while the subjects, who were healthy, were in the upright position. Such recordings are rare and difficult to obtain for physiological and ethical reasons. These prolonged recordings allowed us to precisely characterize the power laws associated with self-organized criticality in the cardiovascular system. Importantly, the evidence supporting the presence of self-organized criticality in the cardiovascular system is reinforced by several previous studies that have identified power laws, albeit without being able to describe them in detail (Yang et al., 2003; Rodriguez et al., 2015; Fortrat & Gharib, 2016; Rivera et al., 2020; Fortrat; 2020; Fortrat & Ravé, 2023). The cumulative sample size across these studies is becoming substantial, thereby mitigating the limitation of our smaller cohort.

L431: Recent findings have provided evidence that cardiovascular dynamics display features consistent with self-organized criticality (Yang et al., 2003; Rodriguez et al., 2015; Fortrat & Gharib, 2016; Rivera et al., 2020; Fortrat; 2020; Fortrat & Ravé, 2023).

Fortrat, J.O.; Gharib, C. Self-Organization of Blood Pressure Regulation: Clinical Evidence. Front. Physiol. 2016, 7, 113.

Fortrat, J.O. Zipf’s Law of Vasovagal Heart Rate Variability Sequences. Entropy 2020, 22, 413.

Fortrat, J.O.; Ravé, G. Autonomic Nervous System Influences on Cardiovascular Self-Organized Criticality. Entropy 2023, 25, 880.

Rivera, A.L.; Toledo-Roy, J.C.; Alejandro Frank, A. J. Symmetry and Signs of Self-Organized Criticality in Living Organisms. Phys.: Conf. Ser. 2020, 1612 012024, DOI 10.1088/1742-6596/1612/1/012024

Rodríguez, J.; Prieto, S.; Correa, C.; Mendoza, F.; Weiz, G.; Soracipa, Y.; Velásquez, N.; Pardo, J.; Martínez, M.; Barrios, F. Physical mathematical evaluation of the cardiac dynamic applying the Zipf-Mandelbrot law. J. Mod. Phys. 2015, 6, 1881–1888.

Yang, A.C.; Hseu, S.S.; Yien, H.W.; Goldberger, A.L.; Peng, C.K. Linguistic analysis of the human heartbeat using frequency and rank order statistics. Phys. Rev. Lett. 2003, 90, 108103.

11.The Results section should be expanded, with the authors providing other results. Apart from the data in Table 1, no other numerical results are presented.

Thank you for your suggestion. We have expanded the Results section by providing additional numerical results to strengthen the presentation of our findings. In response to your comment, we have added three new tables (Tables 1, 2, and 3; see response to comment #7) that present detailed numerical data, complementing the information previously shown in Table 1. These additions provide a more comprehensive overview of our results and support the key points discussed in the manuscript.

12. Are there other similar studies on avalanches in cardiology that use power laws? If the authors claim novelty on this issue, it would be good to have more results to support these claims.

Thank you for this insightful comment. To address it, we have added a paragraph discussing recent evidence that arrhythmic episodes in the cardiovascular system follow a power law, which underscores the self-organized criticality underlying cardiovascular function. This supports the novelty of our findings by highlighting that power laws have been observed in other cardiovascular phenomena, reinforcing the relevance of our approach to studying avalanches in cardiology.

L431: Recent findings have provided evidence that cardiovascular dynamics display features consistent with self-organized criticality (Yang et al., 2003; Rodriguez et al., 2015; Fortrat & Gharib, 2016; Rivera et al., 2020; Fortrat; 2020; Fortrat & Ravé, 2023).

L447 Finally, the cardiovascular system is also susceptible to rhythm anomalies that can lead to life-threatening conditions, known as cardiac arrhythmias. Recent evidence has shown that these arrhythmic episodes follow a power law, highlighting the self-organized criticality underlying cardiovascular function (Shahrbabaki et al., 2025).

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

L153: correct for HR instead of BP in the second equation

L226: under-scripts for x and y

L248: explain “displ object”

L310-340: it is mentioned that Table 3 shows that Clauset’s statistical method confirms the straight-line distribution in 4 out of seven patients. However, in line 1 of Table 3, only 3 p-values are greater than 0.05…

Author Response

L153: correct for HR instead of BP in the second equation

Thank you for pointing out this typo. We have corrected "BP" to "HR" in the second equation.

L226: under-scripts for x and y

Thank you for pointing out this typo. We have properly formatted the subscripts for x_i and y_i.

L248: explain “displ object”

Thank you for pointing this out. We have clarified the term "displ object" by specifying that it refers to the implementation of a discrete power-law model in the poweRlaw R package. The revised sentence now reads: "A discrete power-law model (implemented as a displ object in the poweRlaw R package) was fitted to the cardiovascular avalanche or symbolic sequence frequencies."

L310-340: it is mentioned that Table 3 shows that Clauset’s statistical method confirms the straight-line distribution in 4 out of seven patients. However, in line 1 of Table 3, only 3 p-values are greater than 0.05…

Thank you for highlighting this point. We have carefully reviewed Table 3 and confirm that four p-values are greater than 0.05, consistent with the statement in the text. Specifically, the p-values for subjects S3 (0.089), S4 (0.401), S6 (0.473), and S7 (0.592) are all above the 0.05 threshold.