SelfOrganized Criticality in Daily Incidence
of Acute Myocardial Infarction
A.M. SELVAM^{1}, D. SEN^{2} and S.M.S. MODY^{3}
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
Website: amselvam.webs.com/index.html
Website: amselvam.tripod.com/index.html
Telephone: 0910212330846
Fax: : 0910212347825
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 twoyear 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 spacetime fluctuations are
ubiquitous to dynamical systems in nature and have been identified as signatures
of selforganized criticality. The unique quantification for selforganized
criticality presented in this paper is shown to be intrinsic to quantumlike
mechanics governing fractal spacetime 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 twoyear 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 longrange (spacetime) correlations. Longrange spatiotemporal correlations are ubiquitous to dynamical systems in nature and are identified as signatures of selforganized criticality^{ }[2]. The physics of selforganized criticality is not yet identified. Atmospheric flows exhibit selforganized 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 selforganized criticality as intrinsic to quantumlike mechanics governing flow dynamics [49]. 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 selforganized 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^{ }[1011] applicable to all dynamical systems in nature. The model concepts are applied to show that daily incidence of AMI, probably triggered by stressfree and stressful activity cycle corresponding respectively to sleepwake diurnal (night to day) activity rhythm selforganizes to form a broadband spectrum for temporal fluctuations, with universal inverse power law form of the statistical normal distribution. Daily incidence of AMI exhibits selforganized 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 spacetime 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^{ }[49] 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 W^{2}
and length scale R result from integration of enclosed small
scale fluctuations of intensity w*
^{2}
and
length scale r given by the relation
2) The growth of fluctuation pattern follows an overall logarithmic spiral trajectory OR_{1}R_{2}R_{3}R_{4}R_{5} with the quasiperiodic Penrose tiling pattern for the internal structure (Fig. 1). The amplitudes of fluctuations for successive growth stages follow the logarithmic relationship.
3.) The logarithmic spiral can be resolved as an eddy continuum with embedded dominant wavebands R_{o}OR_{1},R_{1}OR_{2},R_{2}OR_{3}, etc. ,the peak periodicities P_{n} being given by
4.) The angular turning dq for successive stages in growth of the logarithmic spiral trajectory is given from Equation(1) as
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 [1516].
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 loglog 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. W^{2} 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
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.
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 quasicontinuous 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 L_{m}
= 2 exp (Cm) where C = .001 and m
= 0, 1, 2,......m. The data series Y_{t} for
the N data points were used. The periodogram estimates the
set of
A_{m}cos (2p
n
_{m}t
 f
_{m})
where A_{m}, n
_{m}
and
f
_{m}
denote respectively the amplitude, frequency and phase angle for the m^{th}
periodicity. The cumulative percentage contribution to total variance was
computed starting from the high frequency side of the spectrum. The period
T_{50}
at which 50% contribution to total variance occurs is taken as reference
and the normalized standard deviation
t_{m} values
are computed as (6).
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 chisquare 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 selforganization of fluctuations gives following model predictions. Spacetime integration of smallscale 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 twoyear 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.
(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) longrange temporal correlations, or fractal structure to temporal fluctuations, namely selforganized 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"^{ }[1516].
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.
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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 
3.065,3.081, 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 
4.175,4.209, 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 
6.450,6.574, 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 
13.412 14.256 15.598 16.186 16.763 17.657 18.712 19.436 
27.144 29.969 
36.276 39.931 
73.563 


Periodicities significant at or less than 5% level
are given in bold letters.
Legend
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 xaxis, the upper half for periods upto 10days 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.