Ernest Lawrence Rossi

The Feigenbaum scenario of the mathematical period doubling sequence from order to deterministic chaos has led to new insights about the nonlinear dynamics of a wide variety of physical and biological systems. Multiple realms from purely mechanical systems, fluid dynamics and weather to the patterns of biological growth and the dynamic of the heart, hormone and brain rhythms have been found to exhibit aspects of the Feigenbaum period doubling sequence. We explore the possibility that the Feigenbaum scenario can be extended to experiences of sensation, perception and human cognition as well. We also review the empirical data that supports the view that the Feigenbaum scenario of the period doubling sequence may portray an important limit in conscious information processing. We conclude that a major function of consciousness may be to transform the nonlinear, irrational and difficult to predict dynamics of unconscious nature into the more linear, rational and predictable psychodynamics that make human experience and social life possible.

Key words: Feigenbaum scenario, chaos theory, information, consciousness, chronobiology

Albert Einstein wrote a letter outlining his philosophy of science a few year before his death in 1955 to his friend Maurice Solovine. The essence of Einstein’s view was illustrated in a simple freehand drawing he made of how one must first take a "creative leap" to formulate a new "axiom." One subsequently deduces a series of consequences from this axiom that may be tested for empirical verification in the "real world." Quaint as this classical view of science may seem in our post-modern age of de-constructivism, it certainly served Einstein well in the development of his revolutionary concepts on all levels from the cosmological to the quantum. In this paper we will follow Einstein’s approach to take a creative leap regarding the significance of Feigenbaum’s scenario as portraying a limit in conscious human information processing. We will review how this hypothesis is supported by existing empirical data and what further research is needed to confirm it. We will conclude with a number of open questions about the relevance of the Feigenbaum period doubling sequence for a deeper understanding of the paradoxes and uncertainties of human experience awaiting exploration in the fields of logic, computer science and quantum physics as well as psychobiology, depth psychology and the social sciences.

This paper first outlines the Feigenbaum scenario of the period doubling sequence and the dynamics of feedback and iteration in the context of the current debate regarding linear and nonlinear dynamics in psychobiology. We then propose a creative leap regarding the role of the Feigenbaum scenario for understanding an important limit in the human capacity for conscious information processing, bringing together previously unrelated data in psychobiology, chronobiology, depth psychology and social systems. The implications of this association between nonlinear dynamics of the Feigenbaum period doubling sequence and the limits of conscious information processing for understanding some of the most famous paradoxes of logic and uncertainty in quantum physics will then be touched upon.

The basic iterative process for generating the Feigenbaum period doubling sequence is to take the answer from an equation and feed it back into the same equation to get an new answer and then keep doing this over and over again. This is the same fundamental process that takes place in all biological and psychosocial systems. Life at all levels from the molecular-genetic to the psychosocial involves doing the same iterative operations over and over again in the process of evolutionary adaptation as well as daily survival. At the molecular-genetic level, for example, gene transcription and translation takes place over and over again in an adaptive feedback loop with the environment. In psychology one stimulus-response unit of behavior is feedback for the next; the experience of one cognition becomes feedback leading to the next etc. The Feigenbaum period doubling sequence is an appropriate model for the life sciences because it inherently models the physical and biological process of feedback with mathematical iteration. Most important was the discovery that there is a well defined path or route which leads from order to chaos that is described as "universal" (Feigenbaum, 1980). It is universal because the same abrupt changes between order and chaos, usually called "bifurcations," can be found in many apparently different equations used to model different processes in nature. When the series of iterated answers to different equations are graphed they all have features in common with the Feigenbaum period doubling sequence as illustrated with the logistic equation in Figure One.

The first long stem coming down from the top of figure one represents a series of solutions that then branches or "bifurcates" as indicated and from each of these two branches we see two more bifurcating again and so on. This is called the "period-doubling regime" of the Feigenbaum period doubling sequence on the path from order to "deterministic chaos." The branches get shorter and shorter as they move down until a threshold is finally reached, that is now called the Feigenbaum point where the parameter "a" is about 3.7 after the fourth bifurcation (so small it is too difficult to see and label in Figure One) where the system falls "appears" to fall into chaos as illustrated as with the dark but structured smudge.

While the Feigenbaum point marks the onset of "deterministic chaos," it is not really random from a statistical point of view. Deterministic Chaos only appears to be random because of the limitations of human conscious perception. It is precisely this "apparent" loss of our ability to see clear and distinct branching pathways, no matter how many times we zoom in and blow up the bifurcating diagram of Figure One, that leads to the first intimation of the Feigenbaum point as illustrating a fundamental limit in human conscious information processing. Before we review the empirical evidence supporting this possible limit in conscious human information processing it is important to summarize another important number often called the Feigenbaum constant.

There is a fundamental ratio that quantifies the period doubling path from order to "apparent" chaos that is found to be true of many different equations when they are iterated. This ratio is called Feigenbaum’s constant which converges to a value of 4.6692... This ratio is calculated by comparing of the lengths of any two successive branches of solutions with the following formula: (Sk - Sk-1) / (Sk+1 - Sk) where S is the length of any branch in Figure One. This Feigenbaum constant is now regarded as being just as fundamental in nonlinear dynamics theory as the number Pi is to geometry (Peitgen et al., 1992). As mentioned previously the Feigenbaum constant is described as "universal" (Feigenbaum,1980) because the same abrupt change between order and apparent chaos after the fourth bifurcation can be found in many different equations provided that they have a parabolic or "hump" structure. Equations containing a sine function or the "tent map," for example, can all be iterated to produce the tree-branching Feigenbaum period doubling sequence illustrated in Figure One. As we shall soon see this parabolic or hump structure is the second intimation of the Feigenbaum point and period doubling sequence as a limit in conscious human information processing.

The typical textbook linear approach to analyzing data in psychology is illustrated on the left side of Figure Two. The straight line cutting through a cluster of data points is the best linear fit, or the best statistical approximation of the whole cluster. Traditionally each of the data points that make up the cluster is said to be a combination of a real psychological factor and experimental error or noise. The new nonlinear dynamics systems approach in psychology, however, proposes that the apparently random deviations of many points around the straight line actually may be the signature of chaos when studied over time. Environmental variables and psychological factors which we cannot control could be responsible for deviations from the straight line that are called noise (Combs and Winkler, 1995). The recent re-examination of the field of classical psychophysics -- the relationships between physical stimuli and human perception such as how bright a light appears to be -- confirms this. Current researchers find that the noise, once thought to be due to errors of measurement, may actually be an important part of the nonlinear dynamics of perception and cognition in self-organizing our view of the world (Gregson, 1992; Guastello, 1995).

The nonlinear relationship between performance and arousal or anxiety illustrated on the right of Figure Two, called the Yerkes-Dodson function (1908), is an example of one the earliest and most well established laws in experimental psychology and studies of human performance under all conditions. Guastello (1995) has updated the significance of this relationship in terms of Thom’s (1972) catastrophe theory of nonlinear dynamics in social organizations and the psychosocial factors in accidents and the work place. This intensively researched relationship between arousal and performance is an important foundation for a new concept of the mindbody relationship in psychotherapeutic work (Rossi, 1996a). It can be seen immediately that this fundamental Yerkes-Dodson function has the basic parabolic or one hump structure of the Feigenbaum period doubling sequence.

Figures 3 through 8 portray the typical data of psychobiological systems on all levels from the cellular-genetic to the cognitive-behavioral that illustrate how the nonlinear feedback dynamics can be interpreted as a series of parabolic or hump structures that are so well modeled by the Feigenbaum period doubling sequence and the Yerkes-Dodson function. One way of understanding such complex dynamical systems is to recognize how they are made up of one or more parts (the parabolic or hump structure of the Feigenbaum period doubling sequence) that can "communicate" with each other over time with feedback loops. These psychobiological feedback loops are the essence of the iterative and recursive cycles of biological processes where they have been summarized as The Unification Hypothesis of Chronobiology (Lloyd & Rossi, 1992, 1993; Rossi, 1996b, 2000, 2002) as follows.

"For the first time psychobiological processes of information transduction and communication that encompass behavioral and psychosocial levels become both clearly delineated and firmly based within the molecular, genetic and cellular dynamics of the whole organism. Integrated function at the highest level (i.e. self-conscious thought and creativity) shows the most complex time dependencies, as underlying is a concatenation of nested time domains. Human health, performance, stress and illness show dependence on intricate time structures; both life styles and therapies need to take heed of these new chronobiological insights." (Lloyd & Rossi, 1992, pp. 404-5).

These feedback loops in nested time domains can be understood as the fractal iterative processes which are the essence of the Feigenbaum period doubling sequence. Virtually all living systems, life as we know it, are made up of multiple feedback loops in iterative processes of adaptation. This makes it very difficult to untangle the simple cause-effect relationships that were the ideal of the older classical linear mathematics in the life sciences and psychology. This is why we have such puzzlement about cause and effect in psychology and psychobiology. In fact, now we can say that whenever we are confronted with such iterative fractal feedback loops of circular causation we are entering the area of nonlinear dynamics that is most vividly portrayed by the Feigenbaum period doubling sequence.

Since psychobiological processes utilize iterative information feedback on many levels as portrayed by the Feigenbaum period doubling sequence we naturally wonder whether this bifurcation model of the can illustrate anything interesting and empirically verifiable about human choice points and information processing on the conscious and/or unconscious levels. That is the creative leap that we would like to explore with the Feigenbaum period doubling sequence of the logistic equation illustrated in Figure One that was originally proposed as a model of population dynamics where feedback prevents populations of bacteria, plants and animals from growing infinitely because of environmental limitations of food supplies and space as well as the presence of predators. Can the logistic equation as well as other equations leading to the parabolic or hump structure be used to illustrate any facets of the population dynamics of ideas, awareness or consciousness? Does the limitation in human conscious perception of the Feigenbaum period doubling sequence to the range of 7 to 15 units or pathways noted in Figure One have any empirical foundation in human experience?

A number of classical studies in experimental psychology confirm that seven units or chunks (plus or minus two) is, in fact, a very common limit in human, sensation, perception, memory and performance (Miller, 1956). There are actually an infinity of physical wave lengths that make up the spectrum of ordinary light, for example, yet humans typically "chunk-out" or perceive only seven colors. Traditionally humans are regarded as having five basic senses but if one includes the more subtle varieties of kinesthetic and proprioceptive sensations the human number of conscious senses easily jumps up to seven but probably not much beyond that. Sperling (1960) found that people could remember about seven letters over a 1-second interval. He called this the iconic trace. Neisser (1967) found that the auditory trance, which he called echoic memory, had similar characteristics. Musical tone recognition has a similar range form the familiar octave scale to the more cerebral 12 tone scale. Is the number seven as a typical band-width in human such sensory-perceptual-memory studies and the seven chunks or paths we can see easily in the Feigenbaum period doubling sequence of Figure One simply a coincidence? Or is it another example of the seemingly unreasonable effectiveness of mathematics in modeling human experience in the most unexpected ways? The upper limit of about 15 units in conscious human perception illustrated in the Feigenbaum period doubling sequence of Figure One is supported by Kihlstrom (1980). Kihlstrom found that 15 items was the upper limit for post-hypnotic memory in low as well as highly susceptible hypnotic subjects who apparently achieved that performance limit by different routes.

An intriguing indication that seven is an important limit in complex human conscious functioning was discussed recently by Hofstadter (1995) who reports that since early youth he played and practiced extensively with anagrams. How many arbitrarily arranged letters can one juggle in one’s mind simultaneously until they suddenly rearrange into a meaningful word? He says, "Six letters, yes, but ten, definitely not." The same range is evident in juggling physical objects as well. The world record for juggling is nine balls. Ronald Graham, chief scientist at AT&T laboratories is a gifted juggler who reports that he can juggle six balls consistently and sometimes seven "playing around" (Horgan, 1997).

Further evidence of the significance of seven in complex human conscious information processing is in the extensive use of subjective rating scales that involve human judgments about everything from sense-perceptions to emotional experience. Research documents that most rating scales used in psychological assessment and personality testing (Caprara et al, 1997), for example, can be optimized by requesting discrimination on scale of about seven points. Subjects are asked to judge how happy or satisfied they are with a particular situation or life experience on a five to seven point scale that runs something like "1. Extremely Satisfied, 2. Satisfied, 3. Neutral, 4. Unsatisfactory, 5. Extremely Unsatisfactory." Recently developed cognitive-behavioral approaches for assessing a patient’s response to psychotherapy use a seven point "Validity of Cognition" scale to check on how much conviction a patient has that they have been helped as well as a ten point "Subjective Units of Discomfort Scale" to check on how much their pain or psychological discomfort has changed during therapy (Rossi, 1996a; Shapiro, 1995). Research is now needed to confirm the relevance of the Feigenbaum point and period doubling sequence in this apparently optimal perceptual, performance and subjective judgment limit range of 7 to 15 chunks of conscious experience on the path from order to chaos in human awareness. Freeman’s (1995) research on the nonlinear dynamics of the sensory-perceptual dynamics of smell, for example, would be an important test case.

At present we can only speculate about a number of open questions about the explanatory power of the Feigenbaum point and period doubling sequence in human experience. We can only wonder whether the significance of the number seven in gambling and many ancient mantic and mystical belief systems could have the same source in the limitations in our conscious sensory-perceptual-cognitive limits of awareness that is illustrated in the bifurcation diagrams of the Feigenbaum period doubling sequence. Seven items are about as much as we usually can hold in consciousness so we feel we know them and we are comfortable with them. When consciousness has to juggle more than seven items, dimensions or levels, we experience the stress of the heightened arousal of our neuroendocrine system and ultimately failure as documented by the extensive data supporting the Yerkes-Dodson law discussed above. On a cognitive level human cognition seems to become chaotic, dark, fearful, unconscious and perhaps unreal when we must hold in memory and manipulate more than 7 to 15 factors at one time.

The universality in the appearance of the Feigenbaum period doubling sequence in many complex systems have led to a number of highly speculative views about the possible significance of the Feigenbaum point for psychology, sociology and the humanities in general. Merry (1995, p. 37) suggests, for example, that the Feigenbaum point (about 3.7) is where systems cascade into chaos "where infinite choices create a situation in which freedom has no more meaning." We could generalize this to say that emotions, imagery, behavior and cognition and psychosomatic symptoms that have lost their meaning have somehow fallen into the chaotic regime within experiential space where our sense of reality teeters off the edge of comprehension or rationality. This suggests that beyond the Feigenbaum point inner subjective experience may fall into a sense of what is called unreality. Put another way, the Feigenbaum point may signal the division between primary process (irrational) versus the secondary processes (rational, ego processes) as defined in psychoanalysis. In this sense, the Feigenbaum point could represent a limit of our sense of voluntary ego control over our mental experience and behavior. The physicist uses the route to chaos as a way of describing turbulence in nature (fast moving water flowing over rocks, air turbulence behind an airplane etc.). Likewise the psychologist could describe the experience of confusion, disorientation and unreality as the turbulence of the mind moving past the Feigenbaum point into deterministic chaos.

Current cognitive science has developed these ideas further with research that documents how human ideas are grounded in sensory-motor experience and metaphor.  This view is expressed by Lakoff and Núñez (2000).

    I now propose that the Feigenbaum point marks the transition between the explicit realm of conscious choice and behavior and the implicate realm of deterministic chaos on the unconscious level (Rossi, 1996a, 1997a, 1998b, 2000a).  The Feigenbaum point is the limit of “subitizing” – the ability to tell at a glance whether there are 1, 2 or 3 objects in conscious (explicit) perception (Lakoff and Núñez, 2000).  These 1, 2 or 3 objects in conscious perception correspond to the first, second and third bifrucation of the Feigenbaum Scenario as illustrated previously in Figure one.  Peitgen et al. (1992) comment on the implicate (unconscious) and previously unknown aspect of how numbers operate in the iterative replays of nonlinear dynamics.

This evolving view of the Feigenbaum point in psychology suggests that regime of deterministic chaos in the brain-mind may serve as a model of what is called the unconscious. This leads to a number of open questions. Is that why during those brief moments of introspection as we are falling asleep or awakening we occasionally glimpse what from our conscious perspective seems to be a confused plethora of inchoate images, feelings, thoughts and what not? Are there circumstances when consciousness gets caught in the chaotic realm so that the person experiences a dissociation and/or a sense of identity loss? Is this what some cultural and spiritual traditions call a "loss of soul?" Obviously such questions will be a rich area for exploring new ways of conceptualizing the foundation of human consciousness and psychological experience in a unified dynamics that shares the same universal principles that govern other complex systems in mathematics, physics, biology and ecology.

Recently a number of investigators have developed new approaches to the study of consciousness and mind-brain relationships that are consistent with our effort to bridge the gap between rational and nonlinear dynamics in human experience. Shawe-Taylor (1996), for example, have integrated theory and data from artificial neural networks and cognitive therapy to outline a theory of consciousness as a linear phenomenon. Their theory provides an opportunity to deepen our understanding of at least one major function of consciousness: mediating the transition between the nonlinear and the linear. It is close to what Freud called rationalization. The Shawe-Taylor theory proposes that the essential function of consciousness is to transform the nonlinear dynamics of the unconscious into a conscious system of linear relationships.

Why did the linearizartion or rational function of consciousness evolve? Nonlinear processes are usually difficult and, indeed, most often impossible to predict. We could hypothesize that consciousness evolved to linearize bounded bits and pieces of nonlinear nature so that the organism could predict important ranges of experience necessary for survival. Linearization is involved in transducing the nonlinear universe into sufficiently predictable patterns - just predictable enough to insure survival. A predatory animal like a fox, for example, has to evolve just enough linear sensory-perceptual-motor organization to predict and track the nonlinear trajectory of its pray. Just as the mathematical process of linearization is always bounded within certain narrow limits in which it is effective, so rational consciousness is also bounded in its linearization and capacity to predict nonlinear nature. When we attempt to step beyond the narrow limits of the linearization function of our current level of human consciousness we fall into conundrums and paradox. This leads to the deep epistemological speculation that the boundary conditions of the linear, rational world view of western consciousness are being revealed by many of the most famous paradoxes of our century: Bertran Russell’s paradoxes of logic and Gödel’s Incompleteness Theorem, Turing’s Halting Problem, the Uncertainty Principle in quantum physics and the nonlinear dynamics of deterministic chaos in the Feigenbaum scenario of the period doubling sequence.

The deep correspondence between the nonlinear dynamics of the Feigenbaum period doubling sequence and the empirical data of psychobiology, chronobiology, experimental psychology and the psychodynamics of psychoanalysis suggest they all derive from a common archetypal foundation of iterative feedback processes. The Feigenbaum point may represent an important limit in conscious information processing that brings together previously scattered data that has been summarized in the unification hypothesis of chronobiology. A major function of consciousness may be to temporarily transform bits and pieces of the nonlinear, irrational, and difficult to predict dynamics of unconscious nature into the more linear, rational and predictable psychodynamics that make human experience and social life possible.

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Originally published as:  

Rossi, E. (1998).  The Feigenbaum Scenario as a Model of the Limits of Conscious Information Processing.  Biosystems, 46, 113-122