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Sabelli HC, Carlson-Sabelli L, Zbilut J, Patel M, Messer J, Walthall K and Tom C.  Cardiac entropy in coronary and schizophrenic patients, and the process concept of entropy as symmetry.  Cybernetics and Systems`94.  2: 967-974, R. Trappl (Ed.), World Scientific Publ. Company, Singapore, 1994.

CARDIAC ENTROPY IN CORONARY AND SCHIZOPHRENIC PATIENTS                 AND THE PROCESS CONCEPT OF ENTROPY AS SYMMETRY

       H. Sabelli, L. Carlson-Sabelli, J. Zbilut, M. Patel, J. Messer, K. Walthall, C. Tom.

                  Rush University, and University of Illinois, Chicago, IL 60612, USA.

                                                              ABSTRACT

                               A unifying theory of processes offers a new concept of entropy as symmetry, not disorder, and a methodology to study entropy in complex processes. Shannon's entropy was measured with the recurrence method. In model mathematical distributions, there is a gradient of entropy from (0) disorder (randomness) to (1) unimodal order (Bell's normal distribution) to (2) symmetric opposition (sine wave). Electropsychocardiographic studies indicate that awakening decreases entropy and patterning, increasing complexity; conversely, patients with coronary artery disease or schizophrenia demonstrate increased entropy and order, and lesser complexity. This potential gradient of entropy suggests that entropy maximization directs processes from disorder to symmetry. This cosmic asymmetry may drive evolution.

            This article presents a process concept of entropy as symmetry developed within the context of biology, and empirical studies of entropy in cardiological data. Both the thermodynamic concepts and the medical methodology are based on a biologically-based theory of processes and information.^1,2 Shannon's theory of communication³ relates information to entropy; although others define information as negative entropy, entropy and information increase together in Shannon's equation. Neurophysiological processes are paradigmatic examples of energetic processes that produce simultaneously information and entropy. The brain must be studied from a thermodynamic perspective, as it consumes over 20% of metabolic energy, and has an extremely high rate of free energy flux density (150,000 ergs/sec^-1 gm^-1, in contrast to 2 ergs/sec^-1gm^-1 in the sun.⁴ Conceptualizing psychological processes as flows of psychological energy (libido), Freud modelled psychodynamics after the closed system thermodynamics of his time, postulating the conservation of libido. Only in simple, closed and isolated systems near equilibrium is energy conserved; complex systems far from equilibrium, such as living organisms, are open to exchanges, and hence total energy may vary. Thermodynamics was revolutionized by a change in perspective from consideration of isolated and closed systems, which exist only hypothetically, to a focus on processes, i.e. open, interacting and evolving systems.⁵ Yet it has not solved the fundamental contradiction among the three current theories of processes: thermodynamics postulates involution toward resting equilibrium (Clausius) and disorder (Boltzmann); mechanism postulates reversibility and hence the conservation of information; and evolutionary theory describes the emergence of complexity from simpler origins. Statistical Mechanics allows for potential reversibility, and it must explain the tendency to maximize entropy as the result of initial conditions, which are both arbitrary and untestable, as there is nothing in the equations of mechanics or of probability to account for it.⁶ Thus, it provides a scenario in which mechanism and thermodynamics can coexist, but it fails to explain why either evolution or irreversibility occur. Likewise, Schrödinger's⁷ proposal to regard living organisms as pockets accumulating free energy and exporting entropy allows for biological evolution without contradicting classic thermodynamics, but fails to explain why also the physical universe evolves.

            Pasteur proposed an integrative concept of evolution, including both physical and biological development from the simple to the complex, in his notion of cosmic asymmetry (see later). This concept have led us to a process reformulation of thermodynamics: [1] the tendency towards the maximization of entropy is a manifestation of a structurally determined cosmic asymmetry that directs processes from disorder to symmetry, thereby driving evolution ^2,8,9. Symmetry includes not only disorder and uniformity but also complex attractors and structures capable of catalyzing change.^2,8,9 Process theory ^1,2 provides a theoretical framework to integrate mechanical, evolutionary and entropy-generating processes as three coexisting aspects of evolution, as well as a practical method to study processes in evolution. It postulates that all processes exhibit universal forms that can be described as algebraic forms (asymmetry, symmetry, bifurcation) corresponding to the three foundations of mathematics according to Bourbaki (lattice, group and topological theory) and as numerical order (0,1,2,3...) as portrayed by the Fibonacci series.^10,11 Processes thus have common features that repeat at all levels of organization, and also spontaneously create new and more complex forms. The simple forms are universal, and have temporal priority; complex forms have local supremacy. Hence processes must be studied from the double perspective of their simpler and their complex components. Here we apply this idea by studying both simple mathematical time series and time series generated by complex biological processes.

Experimental study: Based on Eckmann's ¹² recurrence method for the investigation of hidden patterns in natural processes, Zbilut and co-workers ¹³ have developed a method to measure the entropy of time series. This provided us with an opportunity to test the process hypotheses. To model the three simplest patterns of nature, we used the following computer-generated distributions: (0) rectangular pseudo-random numbers for randomness; (1) Bell's normal distribution for unimodality; and (2) sine waves for opposition. As models for complex processes, we studied twenty-four hour recordings of the electrocardiogram were obtained from 3 control, 3 schizophrenic, and 5 coronary artery diseased persons, as the time series of cardiac intervals reveal complex patterns that reflect the dynamics of neurophysiological processes (see references in  ¹⁴). We include psychiatric and cardiac patients as representative of dysfunctions at higher and lower levels of integration. Data were processed as described in a companion article¹⁴ with the recurrence method.^12-14 Practical considerations lead one to measure entropy at low embeddings, yet we also studied much higher embedding dimensions because neurophysiological processes are likely to be high dimensional. Recurrences: Isolated recurrences are apparently randomly scattered in stochastic (high dimensional) processes. Periodic processes (sine wave and normal distributions) have

higher % recurrences than aperiodic ones (random numbers). During wakefulness, recurrences were low, while during sleep they were high. Coronary illness and schizophrenia also increased significantly the number of recurrences.

Patterned recurrences: Plots of patterned data reveal line segments parallel to the diagonal, which are few or absent in plots of random numbers. The % of recurrences which are diagonally adjacent with no intervening white space ("patterned recurrences") is high in determined systems (sine wave). During sleep, R-R interval data resembles the intermediate values of patterning observed in normal distributions; wakefulness decreases patterning, while coronary illness and schizophrenia increase it.

               Table 1. Entropy and organization measured with the recurrence method.

               (mean ± S.D., 10 embeddings, 7000 windows, 3 samples/subject, 3 subjects)

Entropy is measured by counting the number of line segments and distributing them over integer bins of a histogram according to their length (which is inversely proportional to the largest positive Lyapunov component.¹² Shannon's entropy = - Σ P_i log₂ (P_i) is measured (in bits of information) by taking P₁ as individual bin probabilities of all non-zero bins greater than or equal to the shortest line segment. As shown in table 1, entropy was lowest for random numbers (maximal disorder), low for the normal distribution and for normal awake subjects, increased during sleep, was higher for cardiac patients, even higher for schizophrenic subjects, and largest for sinusoidal patterns (highest order).

The median embedding dimension, the number of embeddings required for 50% of the recurrences to be patterned (embedding fifty, E₅₀) was consistently smaller during sleep, and for schizophrenics and coronary patients. The number of embeddings represents the space in which to study a process,the E₅₀ is the number of embeddings required to describe deterministically half of its components, and hence as an estimate of complexity.

Entropy and organization of psychocardiological processes: Elsewhere (see references in ¹⁴ we discuss the clinical implications of these findings regarding schizophrenia and coronary illness, and the contributions that the illness themselves, and of their treatments, to the observed changes. Here we shall focus on the thermodynamic implications of the observed data. The cardiac entropy of the normal wakeful person coincided with that of a normal distribution at low embeddings; this is to be expected, as a Bell's normal distribution provides an approximate portrait of a variable oscillating about a equilibrium (homeostasis). However, cardiac patterns were much more organized than normal distributions, as shown by the striking difference in the median embedding dimensions. In both cardiac and psychiatric illness as well as during sleep, there was an increase in both entropy and order (% of recurrences and patterned recurrences) --at variance with the view of entropy as disorder--, and a decrease in complexity (E₅₀). The fact that 10 to 100 embeddings are required for 50% of recurrences to be patterned in cardiac data suggest to us that we are dealing with processes of high dimensionality. As classic dynamics attempted to reduce all processes to manifestations of a unidimensional tendency to equilibrium, entropy and disorder, traditional biology assumed a unidimensional flow towards homeostasis (point attractor). We now recognize the importance of periodic attractors (such as biological clocks), of non-periodic chaos (with fractal dimensions), and of higher dimensional dissipative⁵ and autocatalytic¹⁵ structures. In the case of cardiac activity, the high dimensionality of its timing does not indicate beat to beat modulation by a large number of independent factors (thermal, respiratory and hormonal control, etc.), because all these processes are components of patterned activities integrated in the central nervous system. Time graphs of recurrences¹⁴ indicate that specific patterns of cardiac activity accompany specific activities and emotions. Cardiac behavior is a necessary component of behavior because changes in heart timing is one of the processes through which the central nervous system adjusts the circulatory system to distribute the energy supplies (oxygen, metabolites) required for performance, behavior consists of well organized, patterned processes, some relatively long such as sleep, some goal directed such as nutrition and sexual behavior. Each of these patterns of activity includes physiological, behavioral and subjective patterns of function. Cardiac timing would thus be organized by higher level neurophysiological processes (supremacy of the complex^1,2), which are both highly patterned and entropy- producing.

            Process theory postulates that evolution is a process of dimensiogenesis: catastrophes and other bifurcations generate dimensions, not just new forms. Thus the interaction of processes creates complexity, moving them from the low dimensional attractors described by non-linear dynamics to the higher dimensions of biological matter; to the "hearty" dimensions, if the pun be forgiven, of an organ that gives the energy supply to the brainy body of a person; and to the infinite dimensions of the cosmic attractor of the universe.² The elusive concepts of quality and complexity may be conceived as dimensions, and fitted within the context of a numerical order of nature. Physical dimensions are universal, but dimensions can multiply locally. Biological, social, and psychological dimensions are as real as physical dimensions. We cannot as yet identify these complex dimensions in the way in which we can recognize the physical dimensions of time and space, but by comparing recurrent plots obtained at a wide range of number of embeddings, we can estimate the dimensions of the process under study.

A process interpretation of entropy: Adopting Shannon's definition of entropy in terms of information, we observed that entropy was lowest for random numbers (maximal disorder), low for the normal distribution and for normal awake subjects, increased during sleep, was higher for cardiac and for schizophrenic subjects, and largest for sinusoidal patterns (highest order). There is an entropy gradient from random disorder to a sine wave (a symmetric alternation of opposites). The existence of this gradient implies that processes that maximize entropy create order. We thus formulate three hypotheses:  

1. Energetic asymmetry: all is action (where action = energy x time). Different classes of energy can transform into each other in a quantitative fashion. Matter is also a form of energy according to Einstein's equation. Information has an energetic equivalent; the entropy cost to obtain one bit of information has been calculated.¹⁶ Nature is thus made of one substance, energy. Thus everything spontaneously changes and exchanges; everything is an open process, and all processes move in a uniform direction, namely, the direction of time.  Time is the asymmetry of change. Energy is a universal asymmetry of nature. It has been generally accepted that fundamental physical interactions are symmetric ("the conservation of parity") and that time-reversible laws govern microscopic physics; cosmic symmetry is considered as a fundamental principle.¹⁷ In contrast, Pasteur¹⁸ postulated the principle of cosmic asymmetry.  As the asymmetry of biomolecules could not arise from symmetric structures and processes, Pasteur reasoned, the most fundamental physical entities must be asymmetric.  This hypothesis has been confirmed by a number of discoveries, beginning with the non-conservation of parity in beta decay.¹⁹ The unified theory of electromagnetism and weak interactions indicates that asymmetry is not limited to nuclear processes; asymmetry exists also in atoms and molecules. The optical rotation of atoms has been demonstrated empirically,²⁰ and may explain biomolecular asymmetry.²¹ String theory postulates that the most elementary components of nature are asymmetric, line-like rather than point-like.  Other naturally occurring asymmetries include the asymmetric preponderance of matter over anti-matter; the unidirectionality of causation; the time-asymmetric collapse of the wave function in quantum mechanics; the spontaneous maximization of entropy (second law of thermodynamics); the role of highly asymmetric, non-equilibrium states in the thermodynamics of open processes;⁵ the violation of gauge symmetry by superfluids; the lack of time symmetry in magnets;²² and biological asymmetries, such as the ionic asymmetry across plasma membranes, brain left-right asymmetries, and other anatomical asymmetries, including social asymmetries of class, sex, race, nationality.^23,2 Expanding the first law of thermodynamics: energy, information, and matter are forms of asymmetry, and they can transform into each other in a quantitative fashion. What is the physical interpretation of energy as asymmetry?  While differing in detail, modern theories picture the void as a "vacuum state," in rapid and random, hence symmetric, flux²⁴ from which energy and matter may arise through a random quantum fluctuation.²⁵  According to Bohm, in what is called a vacuum, there are "zero-point" fluctuations, and matter is a set of small waves in the immense "ocean" of the vacuum state. Thus, the energy of "empty space" is immensely greater than that contained in matter. As heat, thought to be a substance by Carnot, turned out to be a motion, so energy-matter, thought to be a substance, may be an asymmetric fluctuation within the symmetric vacuum flux.  Others assume that primordial energy was symmetric, and explain the evolution of the universe as a series of symmetry breakings.^22,26 No explanation is provided for any of these postulated events of symmetry breaking. Symmetry breaking and the origin of the universe are conceived as two separate problems. One may consider the more parsimonious hypothesis that the spontaneous creation of energy from the void, described as a quantum fluctuation, was in itself the breaking of the symmetry of the vacuum flux. Energy would be the shared asymmetry of all that exists. The existence of this primordial and universal asymmetry would account for the generation of additional asymmetries (evolution). Bifurcating energy and asymmetric information govern catastrophes, the most elementary bifurcations. Asymmetry does not disappear as processes maximize entropy because structures conserve in their organization the asymmetry of the processes that formed them. Further, symmetric equilibrium is always partial and local. The asymmetry and universality of energy imply continuing evolution, in contrast to the views that processes tend to rest or that they are reversible. Prigogine⁵ argued that reversible mechanical processes can never explain irreversible phenomena, that irreversibility is true at all levels or at none. He thus proposed to change the microscopic laws of physics, introducing an intrinsic indeterminism.  Pasteur's cosmic asymmetry represents an alternative modification of the mechanics of fundamental physical entities, a determined asymmetry.  Einstein believed that there was a universal order more fundamental than quantum uncertainty. We propose that time asymmetry is that universal order more fundamental that quantum uncertainty, but allowing uncertainty, rather than forcing a deterministic mechanism. As a result, the universe is a uni-verse, i.e. a unidirectional flow. The second law makes explicit its direction as a tendency toward symmetry.       

2. Entropy as symmetry: universality of opposition. The law of the maximization of entropy may be reformulated as a universal and order-generating asymmetry, namely the tendency of processes toward the symmetry of opposites, which includes complex attractors and structures, not only disordered equilibrium.Every change generates its opposite, so processes increase their internal symmetry, tending toward attractors and forming structures. Modern dynamics describes the morphology of processes in terms of attractors which are low dimensional, stable trajectories toward which processes tend once transients die away. An attractor, is a stability, a balance or symmetry between opposites. As the term equi-librium (equal forces) indicates, a point attractor is an equality or symmetry of opposite forces.  Periodic attractors, the chaotic attractors, and dissipative structures also represent forms of symmetry, in that they are constructed by the balance of opposite forces, alternating in predominance such as in periodic or aperiodic processes, or counterbalancing each other in structures. A delicate balance of forces erects a cathedral or makes an enzyme. Thus the balance of opposites can maintain ordered structure, accelerate change and enhance complexity, rather than arrest change or erode structure. Point, periodic and chaotic attractors serve to model involutionary, mechanical and evolutionary processes respectively, as three coexisting, complementary and opposite types of change. The maximization of disorder represents an asymmetric flow toward equilibrium point attractors. Only near equilibrium, closed systems flow toward resting equilibrium.⁵ Far from equilibrium, highly asymmetric processes tend to more complex periodic and chaotic attractors, can generate novel and complex dissipative structures.  Hence, the spontaneous tendency of processes to flow toward attractors cannot be described as a tendency toward disorder, but always represents a tendency toward symmetry. The formulation of the second law in terms of symmetry accommodates both processes of aggregation (greater complexity) and degradation (the dissipative production of simplicity). One may thus redefine the second law as a maximization of symmetry resulting from the generation of opposite processes. Evolution never achieves the full symmetry of resting equilibrium or disorder. The flow toward symmetry produces multiple types of order, from simplification toward homogeneity and disorder to the formation of complex structures. As free energy decreases and structures are formed, information are both created and destroyed. Energetically-coded information decreases with the maximization of entropy; structural information increases with conversion of energy into matter, the synthesis of heavier atoms from hydrogen in the core of stars, chemical combinations on the surface of the planet, and the origin and evolution of living organisms. All these processes create more complex structures and hence information, although, as energetic processes, they increase entropy. The maximization of entropy (involution) and the production of information (evolution) are two opposite and inseparable aspects of the same process of evolution (Heraclitus' enantiodromia). 

3. Co-creative organization: the generation of complexity via the interaction of opposites The formation, reformation and destruction of complex patterns and structures is the necessary consequence of tridimensional interactions between relatively high intensity opposites at critical points. All natural processes create material structures, i.e. asymmetric and tridimensional patterns of energy and information.  The formation of matter from energy in the evolution of the universe, the spontaneous formation of condensation structures, the formation of dissipative structures in chaotic attractors, the spontaneous synthesis of inorganic molecules, the generation of living organisms, the evolutionary increase in the number, diversity and complexity of species, the development of varied social cultures, and the psychological processes of individuation--all illustrate the spontaneous creation and destruction of structures through a multiplicity of different processes. Although there are obvious differences between dissipative structures and atomic, molecular and astronomical structures, the latter also are dissipative and creative; Prigogine's model for the spontaneous creation of ordered dissipative structures in chaotic processes far from resting equilibrium may shed some light upon the formation of subatomic and atomic structures from radiation. Stars illustrate the creative and dissipative nature of inorganic structures formed by the asymmetric distribution of matter. Energy and matter are transformed into each other, but, in the evolution of the universe, there has been a net formation of matter from energy. This suggests that energy tends to spontaneously form matter more readily than matter decays to form energy. Since energy is asymmetric flow, and matter is a structure maintained by the equilibrium of opposing forces of attraction and repulsion, this example is paradigmatic of the tendency of asymmetric energy flow to produce more symmetric and more complex structures that create and conserve information. The corresponding increase in complexity compensates, at least in part, for the loss of information resulting from the decay of free energy. Since the transformation of free energy into bound energy is the fundamental step in the formation of structures, it is possible that information may be conserved in a manner analogous to the conservation of energy postulated by the first law of thermodynamics.  Biological organisms and other local concentrations of complexity are assumed to arise from the concomitant production of complexity and entropy, rather than from the reduction of entropy as it is exported to the environment. In summary, the recurrence method allows one to measure the entropy of mathematical cosmic forms and of the forms of actual processes in the cosmos. Through a simple calculation it reveals the direction of the maximization of entropy, from disorder to certain forms of order.

Acknowledgements: We thank Ms. María McCormick for her indispensable support.

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