Self-Organized Criticality in Daily Incidence of Acute Myocardial Infarction
 
 

A.M. SELVAM1, D. SEN2 and S.M.S. MODY3

1. Indian Institute of Tropical Meteorology, Pune 411 008, India

2. Bombay Hospital, Bombay, 400020, India

3. Wadia Institute of Cardiology, Pune 411 001, India


 




Corresponding author:
 

Dr.(Mrs.)A.M.Selvam

Indian Institute of Tropical Meteorology,

Dr. Homi Bhabha Road, Pashan, Pune, 411 008, India

Email: [email protected]

Website: amselvam.webs.com/index.html

Website: amselvam.tripod.com/index.html

Telephone: 091-0212-330846

Fax: : 091-0212-347825
 
 
 
 
 
 
 
 

Abstract


 


Continuous periodogram power spectral analysis of daily incidence of acute myocardial infarction (AMI) reported at a leading hospital for cardiology in Pune, India for the two-year period June 1992 to May 1994 show that the power spectra follow the universal and unique inverse power law form of the statistical normal distribution. Inverse power law form for power spectra of space-time fluctuations are ubiquitous to dynamical systems in nature and have been identified as signatures of self-organized criticality. The unique quantification for self-organized criticality presented in this paper is shown to be intrinsic to quantumlike mechanics governing fractal space-time fluctuation patterns in dynamical systems. The results are consistent with El Naschie's concept of cantorian fractal spacetime characteristics for quantum systems.
 
 

1. INTRODUCTION


 


The daily incidence of acute myocardial infarction (AMI) during the two-year period June 1992 to May 1994 was obtained from admission records of a premier Institute of cardiology at Pune, India. Continuous periodogram power spectral analysis of the data show a broadband spectrum with embedded dominant wavebands, the bandwidth increasing with period length. Broadband spectra for fluctuations are ubiquitous to dynamical systems in nature [1], such as atmospheric flows, stock market price fluctuations, population growth, spread of infectious diseases, etc. The broadband power spectra exhibit inverse power law f -B where f is the frequency and B the exponent. Inverse power law form for power spectra imply long-range (space-time) correlations. Long-range spatiotemporal correlations are ubiquitous to dynamical systems in nature and are identified as signatures of self-organized criticality [2]. The physics of self-organized criticality is not yet identified. Atmospheric flows exhibit self-organized criticality manifested as the selfsimilar fractal geometry to the spatial pattern concomitant with inverse power law form for spectra of temporal fluctuations, documented and discussed in detail by Lovejoy and his group [3]. A recently developed cell dynamical system model for atmospheric flows predicts the observed self-organized criticality as intrinsic to quantumlike mechanics governing flow dynamics [4-9]. The model predicts the universal inverse power law form of the statistical normal distribution for the power spectrum of fluctuations thereby providing universal quantification for self-organized criticality. The model is based on the concept that cumulative summation (integration) of small scale fluctuations give rise to large scale perturbations generating a hierarchical network, the generation mechanism being dependent only on the intensity of fluctuations and independent of the detailed mechanisms governing the fluctuations. The model is therefore a general systems theory [10-11] applicable to all dynamical systems in nature. The model concepts are applied to show that daily incidence of AMI, probably triggered by stress-free and stressful activity cycle corresponding respectively to sleep-wake diurnal (night to day) activity rhythm self-organizes to form a broadband spectrum for temporal fluctuations, with universal inverse power law form of the statistical normal distribution. Daily incidence of AMI exhibits self-organized criticality with model predicted unique quantification in terms of the statistical normal distribution. Quantumlike mechanical laws may therefore govern fluctuation pattern of AMI incidence. The results are consistent with El Naschie's concept [12] of cantorian fractal space-time fluctuations for of quantum systems

The data used in this study was collected in connection with the dissertation entitled 'A Study of Circadian Rhythm and Meteorological Factors Influencing Acute Myocardial Infarction ' submitted to the university of Pune, India, by Dr. D. Sen, M.B.B.S., in 1995, for the M.D.(Doctor of Medicine) Degree (General Medicine) Branch 1 [13].
 
 

2. MODEL CONCEPTS


 


In summary [4-9] the model is based on Townsend’s [14] concept originally proposed for growth of large eddy structures visualized as envelopes enclosing internal small scale eddy circulations in atmospheric flows. A hierarchical eddy continuum is generated by successive cumulative integration of internal small scale fluctuations, the eddy growth process being dependent only on the intensity and length/time scale of fluctuations and independent of details of mechanisms generating the fluctuations. Large scale fluctuations of intensity W2 and length scale R result from integration of enclosed small scale fluctuations of intensity w* 2 and length scale r given by the relation
 
 
 

                                                                         (1)
The fluctuations self-organize to form a hierarchical eddy continuum. Since the intensity W2 at any large scale is the cumulative integration of enclosed small scale eddies of intensities w* 2, the eddy energy spectrum follows the statistical normal distribution. The square of the eddy amplitude represent the probability. Such a result that additive amplitudes of eddies, when squared, represent probabilities is observed in subatomic dynamics of quantum systems. The growth of fluctuation pattern therefore follows quantumlike mechanical laws. The above visualization for growth of large scale structures from small scale fluctuations result in the following model predictions. 1) The successive values of amplitude W and length scale R follow the Fibonacci mathematical series.

2) The growth of fluctuation pattern follows an overall logarithmic spiral trajectory OR1R2R3R4R5 with the quasiperiodic Penrose tiling pattern for the internal structure (Fig. 1). The amplitudes of fluctuations for successive growth stages follow the logarithmic relationship.

                                                                     (2)
where Z is the scale ratio equal to R/r and k is the steady state fractional volume dilution of large eddy by turbulent eddy fluctuations.

3.) The logarithmic spiral can be resolved as an eddy continuum with embedded dominant wavebands RoOR1,R1OR2,R2OR3, etc. ,the peak periodicities Pn being given by

Pn = t n(2+t )T                                                            (3)
where t is the golden mean equal to (1+Ö 5)/2 (» 1.618) and T is the diurnal trigger such as the sleep - wake/night - day cycle associated with stress-free and stressful activity rhythms.

4.) The angular turning dq for successive stages in growth of the logarithmic spiral trajectory is given from Equation(1) as

dq                                                                          (4)
The phase angle q at any stage of growth is therefore proportional to the variance from Equation(1)
q µ W2                                                                    (5)
The phase spectrum will therefore represent the variance spectrum.

The successive growth stages of the logarithmic spiral trajectory may therefore be visualized, particularly in traditional power spectrum analysis, as a continuum of eddies with progressive increase in phase.

The association between phase angle, variance and length scale as obtained above at Equations 4 and 5 are intrinsic to the microscopic dynamic of quantum systems and has been identified as Berry’s phase [15-16].

5) The root mean square (r.m.s.) amplitude of fluctuations W and w* (2) represent the standard deviation and also the mean, since each level represents the mean for next stage of eddy growth. The standard deviation of the fluctuations is therefore represented by logZ where Z is the scale ratio representing the ratio of frequencies( or periods or wavelengths).

6) The conventional power spectrum plotted as cumulative percentage contribution to total variance versus the frequency (or period or wavelength )on log-log scale will now represent the cumulative percentage probability on log scale versus the standard deviation on linear scale since earlier (1) it was shown that variance, i.e. W2 distribution corresponding to logZ represents probability densities and also that logZ represents the standard deviation of the fluctuations (2).

Following traditional concepts in statistics, a normalized standard deviation t for logZ distribution can be defined as

t =                                                         (5)
where L is the period in years and T50 the period up to which the cumulative percentage contribution to total variance is equal to 50. LogT50 will correspond to the mean value for the variance W2 distribution which was shown to follow normal distribution (1).

The power spectrum when plotted as cumulative percentage contribution to total variance versus logZ expressed in terms of the normalized standard deviation t (5) will represent the statistical normal distribution.

The above model concepts are dependent on the amplitude W of fluctuations of length (or time) scale R alone and totally independent of the detailed mechanisms underlying the fluctuations. The model predicts that temporal (or spatial) fluctuations of dynamical systems in general self -organize to form the universal inverse power law form of the statistical normal distribution (1 - 6).
 
 

3. DATA AND ANALYSIS


 


The daily incidence of acute myocardial infarction (AMI) for the two year period June 1992 to May 1994, was obtained from admission records of a premier Institute for Cardiology at Pune, India. The power spectrum of AMI incidence (daily) was computed by an elementary but very powerful method of analysis developed by Jenkinson [17] which provides a quasi-continuous form of the classical periodogram allowing systematic allocation of the total variance and degrees of freedom of the data series to logarithmically spaced elements of the frequency ranges (0.5, 0). The peridogram was constructed for a fixed set of 10000(m) periodicities which increase geometrically as Lm = 2 exp (Cm) where C = .001 and m = 0, 1, 2,......m. The data series Yt for the N data points were used. The periodogram estimates the set of Amcos (2p n mt - f m) where Am, n m and
f m denote respectively the amplitude, frequency and phase angle for the mth periodicity. The cumulative percentage contribution to total variance was computed starting from the high frequency side of the spectrum. The period T50 at which 50% contribution to total variance occurs is taken as reference and the normalized standard deviation tm values are computed as (6).
 
 
 

tm = (log Lm / log t50) - 1                                                             (7)


The corresponding phase spectrum was computed as the cumulative percentage contribution to total rotation, i.e. normalized with respect to total rotation. The variance spectrum, phase spectrum and the statistical normal distribution plotted respectively as cumulative percentage contribution to total variance, cumulative percentage contribution to total rotation and cumulative percentage probability are shown in Fig. 2.

It is seen that variance and phase spectra follow each other closely and also the statistical normal distribution. The "goodness of fit" of variance spectrum and phase spectrum to statistical normal distribution is within 95% level of significance as determined by the standard statistical chi-square test [18].

The dominant wavebands identified as those for which normalized variance is greater than or equal to 1.0 are shown in Fig.3a plotted in the conventional manner, i.e. normalized variance versus logarithm of period in days. Fig.3b shows the cumulative percentage contribution to total variance and cumulative normalized phase for each dominant wave band. The peak periodicities corresponding to each dominant waveband is listed in Table 1.
 

4. DISCUSSION AND CONCLUSIONS


 


A general systems theory for self-organization of fluctuations gives following model predictions. Space-time integration of small-scale fluctuations give rise to an overall logarithmic spiral trajectory with the quasiperiodic Penrose tiling pattern for the internal structure. The logarithmic spiral trajectory can also be resolved as an hierarchical eddy continuum with progressive increase in phase. The eddy continuum has embedded in it dominant wavebands, the bandwidth increasing with period length. The dominant peak periodicities are functions of the golden mean and the primary triggering cycle of stress - free and stressful activity cycle associated with sleep - wake (night to day) rhythm. Since cumulative integration of enclosed small scale fluctuations results in large scale fluctuations, the eddy energy spectrum follows inverse power law form of statistical normal distribution according to Central Limit Theorem. The square of the eddy amplitude or, the variance represents the probabilities. Such a result that additive amplitudes of eddies, when squared, represent probability densities is observed in the subatomic dynamics of quantum systems such as the electron or photon. The dynamics of spacetime fractal fluctuation pattern formation therefore follows quantumlike mechanical laws, consistent with El Naschie's concept of cantorian fractal characteristics for quantum systems [12].

Continuous periodogram power spectral analysis of daily incidence of acute myocardial infarction for the two-year period June 1992 to May 1994, reported at an Institute for Cardiology in Pune, India show that the following dynamical characteristics of AMI variability are consistent with model predictions summarized above.

(1) The spectrum is broadband with embedded dominant wavebands, the bandwidth increasing with period length (Fig. 3).

(2) The dominant peak periodicities (Table 1) closely correspond to model predicted values (3) 2.24, 3.62, 5.85, 9.47, 15.33, 24.80, 40.13, 64.92 corresponding respectively to n values ranging from -1 to 6.

(3) The spectrum follows the universal inverse power law form of statistical normal distribution (Fig.2) which signifies (a) quantumlike mechanics for the dynamics of AMI incidence (b) long-range temporal correlations, or fractal structure to temporal fluctuations, namely self-organized criticality [5] .

(4) The phase spectrum closely follows the variance spectrum, for the total spectrum and also within each dominant waveband (Figs. 2 - 3). The close association between phase, variance and period length is a feature intrinsic to quantum systems and identified as "Berry’s phase" [15-16].

Self-organized criticality, namely long-range spatiotemporal correlations exhibited by dynamical systems in nature, is a signature of quantumlike mechanics governing the dynamics of fractal space-time pattern evolution. Universal spectrum of AMI day to day variability implies prediction of total pattern of fluctuations. The results are consistent with El Naschie's concept of cantorian fractals characteristics for quantum systems.
 

ACKNOWLEDGMENTS



The authors are grateful to Dr. A. S. R. Murty for his keen interest and encouragement during the course of the study. The authors are indebted to Professor M. S. El Naschie for inspiration and guidance in this new field of research. Thanks are due to Mr. R. D. Nair for typing the paper.
 
 

REFERENCES

 

1. Schroeder, M. ,Fractals , Chaos and Powerlaws, W.H.Freeman and Co., N.Y., 1991.

2. Bak, P.C., Tang, C. Wiesenfeld, K, Self-organized criticality. Phys. Rev. A. 1988, 38, 364 - 374.

3. Tessier, Y., Lovejoy, S., Hubert, P., Schertzer, D., Pecknold, S., Multifractal analysis and modelling of rainfall and river flows and scaling, causal transfer function. J. Geophys. Res., 1996, 101(D21), 26427-26440.

4. Mary Selvam, A. Deterministic chaos, fractals and quantumlike mechanics in atmospheric flows. Can. J. Phys. 1990, 68, 831-841.

5. Mary Selvam,A., Universal quantification for deterministic chaos in dynamical systems. Applied Math. Modelling 1993, 17, 642-649.

6. Selvam A. M. and Suvarna Fadnavis, Signatures of a universal spectrum for atmospheric interannual variability in some disparate climatic regimes. Meteorology and Atmospheric Physics , 1998, 66, 87-112. (http://xxx.lanl.gov/abs/chao-dyn/9805028)

7. Selvam A. M. and Suvarna Fadnavis, Superstrings, cantorian-fractal space-time and quantum-like chaos in atmospheric flows. Chaos, Solitons and Fractals, 1998 (in Press)

8. Selvam A. M. and Suvarna Fadnavis, Cantorian fractal spacetime, quantum-like chaos and scale relativity in atmospheric flows. Chaos, Solitons and Fractals, 1998. (in Press)

9. Mary Selvam, A., Quasicrystalline pattern formation in fluid substrates and phyllotaxis, In: Jean, R. V. and Barabe, D.,(eds.),Symmetry in Plants. World Scientific Series No 4. in Mathematical Biology and Medicine, World Scientific, Singapore 1998,795-805.

10. Peacocke, A. R., The Physical Chemistry of Biological Organization, Clarendon Press, Oxford, U.K, 1989.

11. Klir, G. J., Systems science : a guided tour, J. Biological Systems 1992, 1, 27-58.

12. El Naschie, M. S., Penrose tiling, semi-conduction and Cantorian 1/fa spectra in four and five dimensions, Chaos, Solitons and Fractals ,1993, 3(4), 489-491.

13. Selvam, A. M., Sen, D. and Mody, S. M. S., A Study of Circadian Rhythm and Meteorological Factors Influencing Acute Myocardial Infarction, Research Report number RR-073, April 1997, Indian Institute of Tropical Meteorology, Pune 411008, India.

14. Townsend, A. A., The Structure of Turbulent Shear Flow. Cambridge University Press, London, U.K., 1956.

15. Berry, M. V., The geometric phase. Sci. Amer. 1988, Dec., 26-32.

16. Maddox, J., Can quantum theory be understood ? Nature 1993, 361, 493.

17. Jenkinson, A. F., A powerful elementary method of spectral analysis for use with monthly, seasonal or annual meteorological time series. (U.K. Meteorol. Office) Met. O 13 Branch Memorandum No. 57, 1-23, 1977.

18. Spiegel, M. R., Statistics. McGraw-Hill Book Co., N.Y., 359 pp. 1961.
 
 
 
 

Table 1 Dominant peak periodicities (days)


2-3
3-4
4-6
6-12
12-20
20-30
30-50
50-80
120-200
200-300
2.006,2.014,2.022,
2.030,2.042,2.055,
2.067,2.082,2.090,
2.122,2.128,2.136,
2.151,2.158,2.186,
2.199,2.217,2.228,
2.246,2.280,2.301,
2.317,2.326,2.352,
2.373,2.411,2.440,
2.448,2.460,2.477,
2.517,2.527,2.548,
2.565,2.578,2.623,
2.636,2.649,2.660,
2.673,2.689,2.721,
2.735,2.757,2.785,
2.810,2.824,2.838,
2.901,2.933,2.963
3.035,3.050,
3.065,3.081,
3.146,3.181,
3.197,3.213,
3.248,3.274,
3.307,3.341,
3.357,3.418,
3.439,3.484,
3.544,3.576,
3.604,3.688,
3.714,3.759,
3.785,3.866,
3.916,3.964,
3.991
4.101,4.134,
4.175,4.209,
4.268,4.394,
4.514,4.550,
4.596,4.670,
4.880,4.944,
5.044,5.079,
5.151,5.208,
5.388,5.535,
5.636,5.687,
5.750,5.872,
5.972
6.099,6.342,
6.450,6.574,
6.694,7.023,
7.251,7.442,
7.669,8.441,
8.569,8.707,
8.856,8.990,
9.135,9.357,
9.499,9.652,
9.916,10.383,
10.763,11.637,
11.895
12.124
13.412
14.256
15.598
16.186
16.763
17.657
18.712
19.436
23.598
27.144
29.969
32.367
36.276
39.931
64.531
73.563
162.73
245.70

 

Periodicities significant at or less than 5% level are given in bold letters.
 
 
 

Legend

Fig. 1. The quasiperiodic Penrose tiling pattern

Fig. 2. Variance and phase spectra.The statistical normal distribution is also shown in the Figure.

Fig. 3. The periodogram is plotted in two sections, on logarithmic scale for the periods(days) on the x-axis, the upper half for periods upto 10-days and the lower half for periods 10 - 100 days. Each section contains the following two displays:

    (a). The power spectrum plotted as normalized variance versus period (days) for dominant wavebands(normalized variance >= 1.0).
    (b) cumulative percentage contribution to total variance versus cumulative normalized phase for each dominant waveband, demonstrating Berry’s phase

 
 
 
 
 
 
 
 
 
 
 

Fig. 1


 











 

Fig. 2.


 













 
 

Fig. 3.