P.B. Tatarintsev – Ph.D. (Eng.), Associate Professor, Department of Higher Mathematics,  Ugra State University (Hanty-Mansijsk)

E-mail: pbt@ugrasu.ru

Z.I. Molchanova – Ph.D. (Med.), Associate Professor, Department of Neurology and Psychiatry, 

Khanty-Mansiysk State Medical Academy

E-mail: hmgmi2006@mail.ru

I.V. Radysh – Dr. Sc. (Med.), Professor, Аcademician REA, Head of the Department  of Nursing Management Activities, Peoples' Friendship University of Russia (Moscow)

E-mail: iradysh@mail.ru

O.N. Ragozin – Dr Sc. (Med.), Professor, Department of Hospital Therapy, 

Khanty-Mansiysk State Medical Academy

E-mail: oragozin@mail.ru

N.V. Kokorina – Ph.D. (Agricultural), Associate Professor, Department of Ecology  and Nature Management, Ugra State University E-mail: N_Kokorina@ugrasu.ru

The literature review briefly describes the applicability of various mathematical methods of heart rate variability analysis in the time and frequency domains, methods of nonlinear dynamics. Heart rate variability is a manifestation of the variability of adjacent heartbeats. Either heart rate variability expresses in terms of heart rate or in terms of duration of RR intervals, it is a popular subject of physiology research. The aim – a brief historical overview of the development of the concept of heart rate variability, approaches and methods for its quantitative description. The advantages of developing and introducing into clinical practice of circular methods for the heart rate variability analysis described in connection with the quasi-periodicity of the cardiac rate, which is periodic, and in the same time random, close to almost periodic processes. Heart rhythm is partly periodic, partly random, representing a quasiperiodic oscillatory process, similar in its properties to almost periodic processes. Statistical indicators of generally accepted time domain methods for heart rate variability analysis cannot be adequate to the phenomenon under study: their statistical properties depend on the recording time T and the number of heart beats in the sample N. The temporal fluctuations in heart rate show a significant correlation with respiration as a reflection of changes in cardiac autonomic regulation. Despite the lack of certainty in the existing estimates of the contribution of the parasympathetic and the sympathetic divisions of the autonomic nervous system to this variability, a number of mathematical methods for the analysis of heart rate variability have been developed, allowing evaluate the effect of autonomous regulation of the heart. A variety of mathematical methods for analyzing data as a means of assessing for heart rate variability are characterized by unequal potential capabilities, their range of applications is limited. There is the problem of insufficient elaboration of standards. Depending on the capabilities of the data processing method, it is possible to obtain various information about the vegetative regulation of the heart rhythm and the functional state of the organism as a whole. To develop universal indicators of heart rate variability, the magnitude of which would have a clear physiological and clinical interpretation, it is necessary to involve mathematical models of quasi-periodic processes within the framework of the circular approach that are free from these drawbacks.

1. Alieva A.M., Goluhova E.Z., Pinchuk T.V. Variabel'nost' serdechnogo ritma pri hronicheskoj serdechnoj nedostatochnosti (literaturnyj obzor) // Arhiv" vnutrennej mediciny. 2013. № 6 (14). S. 47–52.
2. Baevskij R.M., Ivanov G.G. Variabel'nost' serdechnogo ritma: teoreticheskie aspekty i vozmozhnosti klinicheskogo primeneniya // Ul'trazvukovaya i funkcional'naya diagnostika. 2001. № 3. S. 108–127.
3. Bokeriya L.A., Bokeriya O.L., Volkovskaya I.V. Variabel'nost' serdechnogo ritma: metody izmereniya, interpretaciya, klinicheskoe ispol'zovanie // Annaly aritmologii. 2009. № 4. S. 21–32.
4. Buj M.Z., Taratuhin E.O. Vozmozhnosti metodiki variabel'nosti serdechnogo ritma // Rossijskij kardiologicheskij zhurnal. 2011. № 6 (92). S. 69–75.
5. Vasilenko A.F., Sufianov A.A., SHamurov YU.S., Utochkina I.M., Orlov A.S., Tubaeva T.F. Variabel'nost' serdechnogo ritma pri dvustoronnej glubinnoj stimulyacii subtalamicheskih yader pri bolezni Parkinsona // Sovremennye problemy nauki i obrazovaniya. 2013. № 5. S. 315.
6. Gubareva I.V., Kryukov N.N. Variabel'nost' serdechnogo ritma u bol'nyh s diastolicheskoj disfunkciej levogo zheludochka // Sibirskij medicinskij zhurnal (g. Tomsk). 2012. V. 27. № 3. S. 53–56.
7. Kislenko O.A., Kotlunova N.P., Romanova M.P. Diagnosticheskoe znachenie variabel'nosti serdechnogo ritma u detej grudnogo vozrasta s patologiej serdechno-sosudistoj sistemy // Detskaya bol'nica. 2012. № 3. S. 23−27. 
8. Majstrov A.I., Bogomolov A.V., Alekhin M.D., Zareckij A.P. Matematicheskoe modelirovanie ritmokardiograficheskih signalov dlya standartizacii metodov ih spektral'nogo analiza // Trudy Moskovskogo fiziko-tekhnicheskogo instituta. 2015. T. 7. № 3 (27). S. 116–130.
9. Mihajlov V.M. Variabel'nost' ritma serdca: opyt prakticheskogo primeneniya metoda. Ivanovo. 2002. 290 s.
10. Severin A.E., Rozanov V.V., Torshin V.I., SHCHukin S.I. Odnovremennyj analiz serdechnogo ritma i dyhaniya dlya rasshireniya vozmozhnostej funkcional'noj diagnostiki // Biomedicinskaya radioehlektronika. 2011. № 10. S. 96–102.
11. SHalamova E.YU., Safonova V.R., Ragozin O.N., Radysh I.V., Gudkov A.B., CHizhov A.YA. Vybor koping-strategij i ih vzaimosvyazi s funkcional'nym sostoyaniem u studentov severnogo medicinskogo vuza // Tekhnologii zhivyh sistem. 2016. T.13. № 4. S. 4–16.
12. Akselrod S., Gordon D., Ubel F.A., Shannon D.C., Barger A.C., Cohen R.J. Power spectrum analysis of heart rate fluctuations: a quantitative probe of beat-to-beat cardiovascular control // Science. 1981. V. 213. P. 220–222.
13. Anrep G.V., Pascua W., Rossler R. Respiratory variations in the heart rate I. The reflex mechanism of respiratory arrhythmia // Proceedings of the royal society. 1936. V. 119. P. 197–217.
14. Anrep G.V., Pascua W., Rossler R. Respiratory variations in the heartrate II. The central mechanism of respiratory arrhythmia and the interrelationship between central and reflex mechanisms // Proceedings of the royal society.1936. V. 119. P. 218–230.
15. Bainbridge F.A. The relation between respiration and the pulse-rate // The journal of physiology. 1920. V. 54. № 3. P. 192–202.
16. Bay N.S.Y., Bay B.H. Greek anatomist Herophilus: The father of anatomy // Anatomy & cell biology. 2010. V. 43. № 4. P. 280–283.
17. Bedford E.D. The ancient art of feeling the pulse // British Heart J. 1951. V.13. № 4. P. 423–437.
18. Berntson G.G., Bigger J.T., Eckberg D.L. et al. Heart rate variability: origins, methods, and interpretive caveats // Pyschophysiology. 1997. V. 34. P. 623–648.
19. Bigger J.T.J., Steinman R.C., Rolnitzky L.M., Fleiss J.L., Albrecht P., Cohen R.J. Power law behavior of RR-interval variability in healthy middleaged persons, patients with recentacute myocardial infarction, and patients with heart transplants //Circulation. 1996. V. 93. P. 2142–2151.
20. Bigger T.J. The Predictive Value of RR Variability and Baroreflex Sensitivity in Coronary Heart Disease // Cardiac Electrophysiology Review. 1997. V. 1. № 1–2. P. 198–204.
21. Boylan M. Galen: on blood, the pulse, and the arteries // J. History of Biology. 2007. V. 40. № 2. P. 207–230.
22. Chess G.F., Tam R.M.K. Influence of cardiac neural inputs on rhythmic variations of heart period in the cat // Am. J. Physiology. 1975. V. 228.  № 3. P. 775–780.
23. Cooley J.W., Tukey J.W. An algorithm machine for the calculation of complex Fourier series // Mathematics of Computation. 1965. V. 19. № 90. P. 297–301.
24. De Jong M., Randall D.C. Heart rate variability analysis in the assessment of autonomic function in heart failure // J. Cardiovascular Nursing. 2005. V. 20. № 3. P. 186–195.
25. Denver J.W., Reed S.F., Porges S.W. Methodological issues in the quantification of respiratory sinus arrhythmia // Biological Psychology. 2007.  V. 74. P. 286–294.
26. Denton T.A., Diamond G.A., Helfant R.H., Khan S., Karagueuzian H. Fascinating rhythm: a primer on chaos theory and its application to cardiology // Am. Heart J. 1990. V.120. № 6. P. 1419–1440.
27. Donders F.C. ZurPhysiologie des nervusvagus // Archivfür die gesamtePhysiologie des Menschen und der Tiere. 1868. V. 1. № 1. P. 331–361.
28. Einthoven W. Über die Form des menschlichenelectrokardiogramms // Pflugers Arch. 1895. № 60. P. 101–103.
29. Farrell T.G., Bashir Y., Cripps T., Malik M., Poloniecki J., Bennett E.D., Ward D.E., Camm A.J. Risk stratification for arrhythmic events in postinfarction patients based on heart rate variability, ambulatory electrocardiographic variables and the signal-averaged electrocardiogram // J. Am. College of Cardiology. 1991. V. 18. № 3. P. 687–697.
30. Fisch C. Centennial of the string galvanometer and the electrocardiogram // J. Am. College of Cardiology. 2000. V. 36. № 7. P. 1737–1745.
31. Floyer S.J. The Physician’s Pulse Watch; or, an Essay to Explain the Old Art of Feeling the Pulse, and to Improve it by Help of the Pulse Watch. London: Published by S.Smith and B. Walford. 1707. 13 p.
32. Floyer S.J. The Pulse Watch. London: Published by J. Nicholson, W. Taylor, and J.H. Clements. 1710. 25 p.
33. Frédéricq L. De l’influence de la respiration sur la circulation. Les oscillations respiratoires de la pressionartérielle chez le chein // Arch. Biol. Paris. 1882. V. 3. P. 55–100.
34. Goldberger A.L., West B.J. Applications of nonlinear dynamics to clinical cardiology // Annals of the New York academy of sciences. 1987. V. 504. P. 195–213.
35. Guevara M.R., Glass L., Shrier A. Phase-locking, period doubling bifurcations and irregular dynamics in periodically stimulated cardiac cells // Science. 1981. V. 214. P. 1350–1353.
36. Guyton A.C., Harris J.W. Pressoreceptor-autonomic oscillation: a probable cause of vasomotor waves // Am. J. Physiology. 1951. V. 165. № 1.  P. 158–166.
37. Hales S. Statistical Essays: Concerning Haemastaticks; or, an Account of some Hydraulick and Hydrostatical Experiments made on the Blood and Blood-Vessels of Animals. London: Published by W. Innys and R. Manby. 1733. 15 p.
38. Hamlin R.L., Smith R.C., Smetzer D.L. Sinus arrhythmia in the dog // Am, J. Physiology. 1966. V. 210. P. 321–328.
39. Heart rate variability: standards of measurement, physiological interpretation and clinical use. Task Force of the European Society of Cardiology and the North American Society of Pacing and Electrophysiology // Circulation. 1996. V. 5. № 93. P. 1043–1065.
40. Hering E. Über den Einfluss der Athmung auf den Kreislauf. Zweite Mitteilung: Über eine reflectorische Beziehungzwischen Lunge und Herz. Sber Akd Wiss Wien Math-naturwiss Klasse 2 // Abteilung. 1871. V. 64. P. 333–354.
41. Holter N.J. New method for heart rate studies continuous electrocardiography of active subjects // Science. 1961. № 134. P. 1214–1220.
42. Houle M.S., Billman G.E. Low-frequency component of the heart rate variability spectrum: a poor marker of sympathetic activity // Am. J. Physiology. 1999. V. 276. P. H215–H223.
43. Hyndman B.W., Kitney R.I., Sayers B.M. Spontaneous rhythms in physiological control systems // Nature. 1971. № 233. P. 339–341.
44. Hyndman B.W., Gregory J.R. Spectral Analysis of sinus arrhythmia during mental loading // Ergonomics. 1975. V. 18. P. 255–227.
45. Iyengar N., Peng C.K., Morin R., Goldberger A.L., Lipsitz L.A. Age-related alterations in the fractal scaling of cardiac interbeat interval dynamics // Am. J. Physiology. 1996. V. 271. P. R1078–R1084.
46. Janßen J.D., Schanze T. Analysis and classification of ECG-waves and rhythms using circular statistics and vector strength // Current Directions in Biomedical Engineering. 2017. V. 3. № 2. P. 91–94.
47. Katz L.N., Hellerstein H.K. Electrocardiography // In: Circulation of the Blood Men and Ideas. 1982. P. 265–354.
48. Katona P.G., Poitras J.W., Barnett G.O., Terry B.S. Cardiac vagal efferent and heart period in the carotid sinus reflex // Am. J. Physiology. 1970. V. 218. P. 1030–1037.
49. Kay S.M., Marple S.L. Spectrum analysis: a modern perspective // Proceedings of the IEEE. 1981. V. 6. P. 1380–1419.
50. Kleiger R.E., Miller J.P., Bigger J.T.J., Moss A.J. Decreased heart rate variability and its association with increased mortality after acute myocardial infarction // Am. J. Cardiology. 1987. V. 59. № 4. P. 256–262.
51. Kingwell B.A., Thompson J.M., Kaye D.M., McPerherson G.A., Jennings G.L., Esler M.D. Heart rate spectral analysis, cardiac norepinephrine spillover, and muscle sympathetic nerve activity during human sympathetic nervous activation and failure // Circulation. 1994. V. 90. P. 234–240.
52. Kollai M., Mizse G. Respiratory sinus arrhythmia is a limited measure of cardiac parasympathetic control in man // J. Physiology. 1990. V. 424. № 1. P. 329–342.
53. Lombardi R., Sandrone G., Mortara A., Torzillo D., LaRovere M.T., Signorini M.G., Cerutti S., Malliani A. Linear and non-linear dynamics of heart rate variability after acute myocardial infraction with normal and reduced left ventricular ejection fraction // Am. J. Cardiology. 1996. V. 77. № 15. P. 1283–1288.
54. Ludwig C. Beitrage zur Kenntniss des Einflusses der Respriations bewegungen auf den Blutlaufim Aortensysteme // Arch AnatPhysiol Leipzig. 1847. V. 13. P. 242–302.
55. Mäkikallio T.H., Huikuri H.V., Hintze U.., Mitrani R.D., Castellanos A., Myberburg R.J., Moller M. Fractal analysis and time- and frequency-domain measures of heart rate variability as predictors of mortality in patients with heart failure // Am. J. Cardiology. 2001. V. 87. № 2. P. 178–182.
56. Mäkikallio T.H., Koistinen J., Jordaens L., Tulppo M.P., Wood N., Golosarky B., Peng C.K., Goldberger A.L., Huikuri H.V. Heart rate dynamics before and after spontaneous onset of ventricular fibrillation in patients with healed myocardial infarcts // Am. J. Cardiology. 1999. V. 83. № 6. P. 880–884.
57. Malik M., Xia R., Odemuyiwa O., Staunton A., Poloniecki J., Camm A.J. Influence of the recognition artefact in the automatic analysis of long-term electrocardiograms on time domain measurements of heart rate variability // Medical & biological engineering & computing. 1993. V. 31. P. 539–544.
58. Malliani A., Pagani M., Lombardi F., Cerutti S. Cardiovascular neural regulation explored in the frequency domain // Circulation. 1991. V. 84. P. 482–492.
59. Moody G.B., Mark R.G., Goldberger A.L. PhysioNet: a research resource for studies of complex physiologic and biomedical signals // Computers in Cardiology. 2000. V. 27. P. 179–182.
60. Myers G.A., Martin G.J., Magid N.M., Barnet P.S., Schaad J.W., Weiss J.S., Lesch M., Singer D.H. Power spectral analysis of heart rate variability in sudden cardiac death: comparison to other methods // IEEE Transactions on Biomedical. 1986. V. BME-33. № 12. P. 1149–1156.
61. Pagani M., Lombardi F., Guzzetti S., Sandrone G., Rimoldi O., Malatto G., Cerutti S., Malliani A. Power spectral density of heart rate variability as an index of symptho-vagal interactions in normal and hypertensive subjects // J. Hypertension. 1984. V. 2. P. 383–385.
62. Peñáz J., Honzikova N., Fisher B. Spectral analysis of resting variability of some circulatory parameters in man // PhysiologiaBohemoslovaca. 1978. V. 27. P. 349–357.
63. Pikkujämsä S.M., Mäkikallio T.H., Sourander L.B., Räihä I.J., Räihä P., Skyttä J., Peng C.K., Gold P.S.K., Huikuri H.V. Cardiac interbeat interval dynamics from childhood to senescence: comparison of conventional and new measures based upon fractals and chaos theory // Circulation. 1999. V. 100. P. 393–399.
64. Preiss G., Iscoe S., Polosa C. Analysis of periodic breathing pattern associated with Mayer waves // Am. J. Physiology. 1975. V. 228. P. 768–774.
65. Pomeranz M., Macaulay R.J.B., Caudill M.A. Assessment of autonomic function in human by heart rate spectral analysis // Am. J. Physiology. 1985. V. 248. P. H151–H153.
66. Thayler J.F., Yamamoto S.S., Brosschot J.F. The relationship of autonomic imbalance, heart rate variability and cardiovascular disease risk factors // International J. Cardiology. 2010. V. 141. № 2. P. 122–131.
67. Traube L. Überperiodisshe Thätigkeits-Ausserungen des vasomotorischen und Hemmungs Nervenzentrums // Centrablatt Med. Wiss. 1865. V. 56. P. 880–885.
68. Tulppo M.P., Kiviniemi A.M., Hautala A.J., Kallio M.S.T., Mäkikallio T., Huikuri H.V. Physiological background of the loss of fractal heart rate dynamics // Circulation. 2005. V. 112. P. 314–319.
69. Saul J.P., Albrecht,P., Berger R.D., Cohen R.J. Analysis of long-term heart variability: methods, 1/f scaling and implications // In: Computers in Cardiology. Silver Springs, MD: IEEE Computer Society Press. 1987. P. 419–422.
70. Sayers B.M. Analysis of heart rate variability // Ergonomics. 1973. № 16. P. 17–32.
71. Weinberg C.R., Pfeifer M.A. An Improved Method for Measuring Heart-Rate Variability: Assessment of Cardiac Autonomic Function // Biometrics. 1984. V. 40. № 3. P. 855–861.
72. Woo M.A., Stevenson W.G., Moser D.K., Middlekauff H.R. Complex heart rate variability and serum norepinephrine levels in patients with advanced heart failure // J. Am. College of Cardiology. 1994. V. 23. № 3. P. 565–569.