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Bios, a process approach to living system theory. In honour of James and Jessie Miller.

INTRODUCTION

Here we present empirical and mathematical studies that identify bios as a generic pattern of creative processes, and explore its generation though computer experiments. Bios embodies the notion of a biological archetype presented at a systems conference in an article entitled 'A cosmic gene? A biological model of complex systems. In honor to James Miller' (Sabelli and Carlson-Sabelli, 1996). As many other health professionals, we encountered systems theory as an approach to comprehensive medical care. The American Psychiatric Association has made a bio-psycho-social approach (Engel, 1980) based on systems science its official policy regarding patient care. Few other professions have adopted a systems perspective. Miller's chapter in the Comprehensive Textbook of Psychiatry presented his concept of biological organ functions as archetypes to understand systems such as families and communities (Miller, 1980). Biological processes portray generic forms or archetypes present at multiple levels of organization. A dramatic example is the Mandala pattern (Figure 1) present in heartbeat series when they are decomposed into complementary opposites (Sabelli, 2000). The Mandala symbol is present in many cultures, and appears in doodling and in small children's drawings, indicating its origin in unconscious, psychobiological processes. It is also generated by some biotic series. Finding an archetypal form in empirical data and in its mathematical model is very significant to a psychiatrically trained scientist. As a psychiatrist, James Miller introduced into systems science a biological and psychological perspective (Miller, 1978). The complementary perspective is physics; Miller gave us a seminal advice: formulate process theory (Sabelli and Carlson-Sabelli, 1989) in terms of physical dimensions. At a more personal level, the collaboration between Jessie and James Miller serves as a model for creative marriage.

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Science was born in ancient Greece as physiology, a rational study of nature that regarded the spontaneous creativity of biological matter as evidence and model for the spontaneous creativity of physical matter. As the physics of matter split from the physiology of life, the focus of attention shifted from change to substance and from creation to determination. As mechanism cannot account for creation, contemporary sciences (statistical mechanics, quantum mechanics, the standard model in cosmology, neo-Darwinian evolutionary theory, standard economics) hold accidental events responsible for innovations. James Miller's general theory of living systems (Miller, 1978) returns to biology as a model for other levels of organization. In the same spirit, we have studied the pattern of heart rate variation as a model for physical and mental processes (Carlson-Sabelli et al., 1994a,b, 1995; Sabelli et al., 1995a,b). We thus propose that fundamental natural processes are both causal and creative, and to identify bios as their prototype (Kauffman and Sabelli, 1998, 2003; Sabelli and Carlson-Sabelli, 1999; Sabelli, 1999b, 2005).

EMPIRICAL IDENTIFICATION OF CREATIVE PHENOMENA

Creative processes are characterized by diversification, episodic pattern, novelty and nonrandom complexity, measurable features that are present in empirical data, mathematical bios generated mathematically by recursions of bipolar feedback (to be described later), and some stochastic processes. These features of creativity are absent in random, periodic or chaotic processes. Measures of causation differentiate biotic from stochastic processes, and show that many empirical processes, suspected up to now to be noise, actually are biotic. Similarly, many processes suspected to be chaotic appear to actually be biotic. Figure 2 illustrates the mathematical generation of bios, the similarity between cardiac and mathematical bios, and their difference with chaos.

[FIGURE 2 OMITTED]

Diversification is quantified as increase variance with time (Sabelli and Abouzeid, 2003). Global diversification measures the increase in variance with duration of the sample. Local diversification measures the increase in variance with increasing embedding (Figure 3). Remarkably, these two widely different procedures provide the same results in most cases. Notwithstanding, local diversification is a better measure of creativity because global diversification may result from trend. Diversification distinguishes three types of processes: (1) Conservative processes (mechanical, random) maintain phase space volume. (2) Attractive processes converge to equilibrium, periodic, or chaotic attractor. (3) Creative processes diversify, i.e. expand their phase space volume. Creative processes are to be distinguished from the self-organization of dissipative structures. Once formed, dissipative structures and chaotic attractors remain unchanged, in contrast to creative processes that continually evolve.

[FIGURE 3 OMITTED]

Episodic patterns (complexes) can be detected in recurrence and wavelet plots. We plot and quantify recurrences with the method of Webber and Zbilut (1994, 1996). The Euclidean norm of vectors of 2, 3, 5, ..., N consecutive terms starting with each term in the series ('embeddings') are compared to each other and if the differences between them is less than 0.1%, they are regarded as isometric, and a recurrence is plotted (Figure 4) and counted (Figures 5 and 6). Recurrences are uniformly scattered in random data and in chaotic series. In contrast, temporal clusters of isometries (complexes) are observed in physiological (Figure 4, left), meteorological and economic series, mathematical bios and coloured noise. Shuffling the data eliminates complexes (Figure 4, right).

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Novelty is also revealed by comparing isometry in the series and shuffled copies. Shuffling the data increases the number of isometries in many empirical time series (Figure 6) such as heartbeat intervals, most economic series, atmospheric and oceanic temperature, and also in mathematical bios (Figure 5) and in stochastic series. We define this phenomenon as novelty (Sabelli, 2000). Periodic series are recurrent at periodic intervals; shuffling the data reduces isometry (Figure 5). Chaotic attractors are neither novel nor recurrent; they have the same number of isometries as their shuffled copies (Figure 5). Repetitive, crystal-like order and biological organization represent opposite departures from randomness. Order is characterized by recurrence; creative organization, by novelty. Novelty, i.e. the generation of change beyond random variability, is an essential feature of creativity; e.g. sexual reproduction produces faster change than random mutation.

Innovations can also be produced by random changes. Stochastic processes generated by random changes can mimic empirical data often as well as biotic models that are generated causally. Whether natural processes are biotic or stochastic must be investigated empirically in every case.

Causation is detected by partial autocorrelation and by consecutive recurrence at low embedding dimensions. Partial autocorrelation measures the association between terms [A.sub.t] and [A.sub.t+k] in a time series when the effects of the intervening time lags [A.sub.t] + 1, [A.sub.t+2], up to [A.sub.k-1] are partialled out (Kendall, 1973). Positive and negative partial autocorrelations are observed in a wide variety of empirical series (Patel and Sabelli, 2003) as well as in biotic series generated mathematically; such correlations are not observed with random walks and for Brownian noise.

Consecutive recurrence at low embedding dimension also indicates causation. To study both simple and complex components of a time series, we plot isometries as a function of the duration of the vector embedding plots, (Figures 5 and 6) [Sabelli et al., 1995a, 1997]. Recurrences are uniformly scattered in random data and in chaotic series. In contrast, they are consecutive (i.e. if vector At is isometric with vector [A.sub.k], then [A.sub.t+1] is isometric with [A.sub.k+1]) in patterned series. Periodic series show consecutive recurrences at periodic intervals (Figure 5). Deterministic chaos and bios show consecutive recurrence at low embedding dimensions. Random data and stochastic series do not.

Deterministic creation occurs when a simple process (cause) generates novel organization; the time series of a creative process may thus be expected to contain both low and high dimensional pattern. This is in fact the case: many empirical series and mathematical bios show a high percentage of consecutive recurrences and high recurrence entropy at low embedding dimensions, indicating deterministic causation, as well as at high dimensions. Stochastic series such as Brownian noise show consecutive recurrence only at high dimensions; they do not contain simple components. Having observed that several processes described as noise show partial autocorrelation and consecutive recurrences, we conjecture that many others will also turn out to be biotic. For instance, it seems unlikely that economic processes result from random changes rather than from the bipolar interaction of supply and demand (Sabelli, 2003c).

Arrangement is an empirically applicable method to measure nonrandom complexity. Complexity has become a major focus of contemporary research, but the term 'complexity' is used to refer to both randomness and to organization. For instance, the complexity of a series is defined algorithmically as the size of the smallest programme that generates it (Chaitin, 2001). This criterion allows measuring the complexity of mathematical but not of empirical processes. Also, algorithmic complexity is maximal for random sequences and lower for biological processes. Algorithmic complexity is a measure of randomness, and randomness is not what is usually meant by complexity, either in ordinary language or in scientific discourse. Thus, algorithmic complexity is not a valid construct of real complexity (Gell-Mann, 1994).

Intuitively, the production of complexity is a fundamental feature of evolution that results from the formation of systems from simple elementary particles to atoms, molecules, organisms and brains. This evolutionary perspective implies that biological and psychological phenomena are the most complex processes known. For this reason we define complexity empirically by examining properties found in biological time series and absent in simpler random, periodic or chaotic data. In this manner, we defined a measure of nonrandom complexity that we call arrangement: the ratio of consecutive isometries (as a percentage of all isometries) over the number of all isometries (as a percentage of all the possible isometries) (Sabelli et al., 1995a). Arrangement is high in a variety of biological series (electroencephalogram, electromyogram, respiration), economic processes (Dow-Jones Industrial Average; prices for crude oil, corn, gold, silver; exchange rates for American, British, Canadian, Danish, Japanese, European currencies [Sabelli (2001)]), DNA sequences and complex literary texts. Arrangement is high for biotic series generated with recursions of bipolar feedback and in stochastic noise. Arrangement is low for process chaos, logistic chaos, random data and periodic series (period 2, sine waves) (Sabelli, 2001). Arrangement is an intuitive measure of complexity because is a function of both order and novelty.

BIOS IN BIOLOGICAL, METEOROLOGICAL, ECONOMIC AND COSMOLOGICAL TIME SERIES

As described above, many empirical series show biotic patterns: heartbeat intervals, electroencephalogram, electromyogram, respiration, economic processes, air and ocean temperature, DNA sequences and complex literary texts. Note that the biotic pattern noted in weather records is not incompatible with the chaotic features discovered by Lorenz (1993) because bios displays all the characteristics of chaos.

We cannot discuss here all these processes, but it is cogent to consider the distribution of galaxies as a function of time because this is the paradigmatic case of creation. As result of the enormous distances involved, and the finite velocity of light, the distribution of galaxies as a function of distance (redshift) is a time series. One can observe directly the pattern of evolution of the universe. This time series displays a biotic pattern (Sabelli and Kovacevic, 2003).

We thus regard the universe as an ongoing process of creative evolution that generates the observed heterogeneity in the distribution of matter from atoms to galaxy superclusters. In contrast, the standard cosmological model assumes that the universe is homogenous and isotropic, and that the overall distribution of matter is a random stationary process (Peebles, 1993). The hierarchical structure of the observed universe is dismissed as local random fluctuations. An alternative model postulates self-similar fractal geometry generated by stationary stochastic processes (Mandelbrot, 1977). These hypotheses can be tested empirically. We analysed the extensive data collected by investigators at the Las Campanas Observatory in Chile (Schechter, 1996) and the Anglo-Australian Observatory (Colless, 1999). Together, these two surveys describe the distribution of galaxies in eight directions. Measures of creative phenomena based on recurrence, wavelet and statistical analyses show novelty, nonrandom complexity, diversification, asymmetry, and short-lived patterns (Sabelli and Kovacevic, 2003). These creative features indicate bios and rule out chaos and randomness. As bios is an expansive process generated by bipolar feedback (see below), these results also suggest to us a mechanism for the expansion of the universe.

MATHEMATICAL BIOS

Amongst mathematical processes, only recursions of trigonometric functions are known to generate bios. These recursions represent bipolar feedback, i.e. feedback that is at times positive and at times negative (Sabelli, 2003a). The process recursion (Figure 2)

[A.sub.t+1] = [A.sub.t] + k * t * sin([A.sub.t])

generates (1) convergence to one steady state ('equilibrium'); (2) a cascade of bifurcations producing period 2, 4, 8 ... [2.sup.N]; (3) chaos with a prominent period 4 that remains within boundaries; (4) bios, an expanding chaotic trajectory; and (5) steep positive or negative linear shifts from one biotic level to another.

BIOS AND CHAOS

Bios is aperiodic, deterministic and sensitive to initial conditions as chaos, yet bios cannot be reduced to chaos: Chaos does not show diversification, novelty, nonrandom complexity or episodic patterns. Mathematical bios allows one to show other essential differences: irreversibility (Sabelli, 2003b), global sensitivity to initial conditions (Kauffman and Sabelli, 2003) and expansion. Also, bios has lower entropy than chaos (Sabelli, 2003b), in line with the hypothesis that biological processes have lower entropy than their environment, as proposed by Schrodinger (1945) and by Prigogine (1980).

Central to the phenomenon of bios is expansion, which is absent in chaotic attractors, but is characteristic of natural creative processes (consider, for instance, the relation between biological, social and economic growth and development). Biotic expansion is a form of deterministic diffusion, a well-studied phenomenon (Arnold, 1965; Chirikov et al., 1985; Geisel and Nierwetberg, 1982; Korabel and Klages, 2002). The essence of bios, however, is creativity, not diffusion. It is in fact possible to generate mathematically bounded bios (homeobios) with a number of different recursions such as

[A.sub.t+1] = [A.sub.t] + sin([A.sub.t-1] * [J.sub.t]) - cos([A.sub.t-1]/[J.sub.t]),

[B.sub.t+1] = [B.sub.t] + g * sin([B.sub.t]) - 0.01 * ([B.sub.t] - 31 * [pi]),

[C.sub.t+1] = [C.sub.t] + g * sin([C.sub.t]) - 0.001 * ([C.sub.t] - [C.sub.1]), or

[D.sub.t+1] = [D.sub.t] + sin(J * [D.sub.t]) + sin(J/[D.sub.t])

where [J.sub.t] = k * t, and k, g and J are constants. The time series generated by any of these recursions show diversification without diffusion (as measured by the mean squared displacement). Homeobios may provide a simple model for physiological processes that are both creative and maintained within homeostatic boundaries.

To describe bios as a sequence of transitions from one attractor to another is to miss the fundamental difference between creative processes that continually create something new and trajectories that converge to a stable state, whether equilibrium or chaos. Creative processes do not have attractors.

FACTORS THAT FOSTER BIOS AND CREATIVITY

The possibility of generating bios with simple recursions provides an opportunity to investigate what factors foster creativity. Computer experiments indicate that the production of bios in mathematical recursions requires:

(1) Two-dimensional bipolarity: Trigonometric recursions produce bios; unipolar feedback, such as in the logistic equation, produce only chaos.

(2) High energy: Bios occurs only at relatively high gain g; at lower gain, these recursions produce only chaos.

(3) Symmetry: The generation of bios requires a relative symmetry of opposites, but a slight asymmetry (q ~0.01) fosters its emergence at lower gain (Sabelli and Kauffman, 1999). This can be demonstrated with the recursion

[A.sub.t+1] = [A.sub.t] + g * (q + sin [A.sub.t])

in which q (which ranges between 1 and -1) provides asymmetry. Complex bios occurs only when the opposite components of bipolar feedback are only slightly asymmetric (q is near 0); when the asymmetry is increased, the pattern becomes increasingly linear, reaching steady state at the extreme of unipolarity (q = 1 or q = -1). At intermediate degrees of asymmetry, one observes bios with a linear trend, which we call parabios (Sabelli and Kauffman, 1999). Parabios is observed in many economic processes.

(4) Conservation: Recursions with a conserve term [A.sub.t] produce bios. Recursions without a conserve term such as

[A.sub.t+1] = k * t * sin([A.sub.t])

produce only chaos (Sabelli, 2003b).

New computer experiments demonstrate two other factors, (5) diversity and (6) bidimensional opposition.

(5) Diversity: Recursion of trigonometric functions produces feedback that is not only bipolar and bidimensional, but also extremely diverse. Diminishing the diversity by truncating the number of digits in the sine function abolishes bios without preventing the generation of periodicity and chaos (Figure 7). This is a new finding that we are just beginning to explore.

[FIGURE 7 OMITTED]

(6) Bidimensional opposition: Diversity can be enormously reduced by generating the time series with a continually varying gain [g.sub.t] = k * t when k is large (for instance, k = 1). These experiments reveal the intimate connection between bios and opposition.

Chaos is associated with triads. Three bodies produce chaos (Poincare) and period 3 implies chaos, as vividly described by the famous article that gave its name to chaos (Yorke and Li, 1975). Sarkovskii's theorem demonstrates that period 3 implies all other periodicities, chaos and infinitations (Sarkovskii, 1964; Peitgen et al., 1992). This is probably because period 3 represents the end of a causal chain that starts with a single state and a cascade of bifurcations. Notably, the threeness of chaos is revealed in logistic time series by a prominent period 3 and by the triadic distribution of rise and fall (Sabelli, 2005). Periods 3 and 6 are present in chaotic series generated by bipolar feedback, but in them period 4 is even more prominent. Such chaos is followed by bios.

As chaos is associated with triads, bios is associated with tetrads. Computer experiments show that two orthogonal pairs of opposites generate bios. Using the battery of tests required to detect bios (Figure 8), we find complexes in both recurrence and wavelet plots, novelty, high consecutive recurrence, high recurrence entropy, and arrangement, in time series constructed by recursions in which we substitute the sine wave by only two orthogonal pairs of opposite values (Figure 8). A cross of opposites is included in the circularity of the trigonometric functions that generate bios. Bipolar feedback recursions display a prominent period 4. Biotic series display a tetradic pattern of rise and fall (Sabelli, 2005). This relation between bios and tetrads also is a new finding that we are currently exploring. It relates bios to the theory of complementary opposites that plays a central role in systems theory (Bai and Lindberg, 1998; Capra, 1975; Sabelli, 1989, 1998; Xu and Li, 1989). The interaction of complementary opposites produces creativity (Sabelli, 2001).

[FIGURE 8 OMITTED]

BIOTIC FEEDBACK AS A GENERIC NATURAL PROCESS

Steady states, bifurcation, periodicity, chaos and bios are regularities observed in physical, biolo

gical and human processes. They may be regarded as generic forms or archetypes. Life is full of forms that travel across time, space and species. Arm, leg, fin and wing all derive from the same origin, and have the same fundamental structure, modified to perform different functions. Biologists say that they are homologous, meaning that they have the same cosmic form, the same 'logos' (as in 'logic' and bio-'logy'). Homology is not confined to life forms. The continuity of evolution requires that the same fundamental forms must be expressed at physical, biological and psychological levels of organization. While elementary particles can be found only at the simplest level of organization, elementary forms recur fractally at all levels. For instance, asymmetry, discovered by Pasteur in biomolecules, is a feature of all fundamental processes, from elementary physics (Haldane, 1960) and cosmological evolution (Anderson and Stein, 1987) to biological, social and psychological processes and structures. Pasteur's asymmetry illustrates the concept of cosmic form--'cosmic' means universal, ordered and beautifying (as in 'cosmetic'). There is a profound homology between nature and mind. This is evident in the facts that mathematics, a product of human thought, so splendidly describes reality. Because we do it so 'naturally', we fail to realize a most astounding occurrence: the certainty with which we can discover facts about nature by manipulating numbers generated in our minds. Mathematical calculations have much to say about the real world, allowing us even to determine with surprising accuracy interplanetary travel. Through mathematics, the human mind must undoubtedly tap into fundamental physical processes. This correspondence of thought with reality is understandable, because brain functioning is the product of animal evolution (Vandervert, 1988). Adopting an evolutionary perspective, systems theory interprets the similarities between processes at different levels of integration as homologies (Bertalanffy, 1968).

There is profound homology between fundamental mathematical structures (Bourbaki, 1946) and fundamental mental structures (Beth and Piaget, 1961); they are also homologous to action, opposition and formation in physical and biological processes (Sabelli, 1989, 1999a,b, 2001, 2003b). This extends the range of applicability of Miller's Living Process Theory.

Miller describes 20 functional components that are common to all systems. At a more abstract level, steady state ('equilibrium'), bifurcation, periodicity, chaos, bios and shifts are also generic forms. They repeat at each level of organization, producing a fractal structure. Bipolar feedback generates all of these forms. The diversity of natural and human environments furnishes synergic and antagonistic responses to the output of any system (Francois, 1997). Feedback may thus be expected to be bipolar. We suspect that biotic feedback processes operate at every level of organization (Sabelli, 2003a) and are instrumental in generating creativity.

ACKNOWLEDGEMENTS

We are thankful to the Society for the Advancement of Clinical Philosophy, to Mrs. Maria McCormick for her invaluable support for this research, and to Drs. Arthur Sugerman, Louis Kauffman, Joseph Messer, Minu Patel, Lazar Kovacevic, Aushra Abouzeid and Jerry Konecki, for collaboration in various aspects of this research as listed in the references.

Note added in proof:

The calculation of recurrence isometries can be performed with Bios Data Analyzer (Sugerman et al., 2005 in Sabelli, 2005).

Sugerman A, Sabelli H, Patel M, Kauffman L, Kovacevic L. 2005. Bios Data Analyzer. CD-ROM in Bios: a Study of Creation. World Scientific.

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Hector Sabelli * and Linnea Carlson-Sabelli

Chicago Center for Creative Development, and Rush University, USA

* Correspondence to: Hector Sabelli, Chicago Center for Creative Development, 2400 N, Lakeview, Unit 2802, Chicago, IL 60614. E-mail: Hector_Sabelli@rush.edu
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