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· Handwritten Character Recognition using Multiple Chaotic Chua Attractors in a Neural Associative Memory
Abstract:
Patterns of 40 to 80 Hz oscillation have been observed in the large scale activity (local field potentials) of olfactory cortex \cite{frmnbk} and visual neocortex \cite{gry&sgr}, and shown to predict the olfactory \cite{frmn&bd} and visual \cite{frmn&dik} pattern recognition responses of a trained animal. Similar observations of 40 Hz oscillation in auditory and motor cortex, and in the retina and EMG have been reported. It thus appears that cortical computation in general may occur by dynamical interaction of resonant modes, as has been thought to be the case in the olfactory system The oscillation can serve a macroscopic clocking function and entrain or ``bind" the relevant microscopic activity of disparate cortical regions into a well defined phase coherent collective state or ``gestalt". This can overide irrelevant microscopic activity and produce coordinated motor output. There is further evidence that the oscillatory activity is roughly periodic, but actually appears to be chaotic (nonperiodic) when examined in detail \cite{frmnchs}.
If this view is correct, then oscillatory and chaotic network modules form the actual cortical substrate of the diverse sensory, motor, and cognitive operations now studied in static networks. It must then be shown how those functions can be accomplished with oscillatory and chaotic dynamics. It is our expectation that nature makes good use of this dynamical complexity, and our intent is to search here for novel design principles that may underly the superior performance of biological systems in pattern recognition. These may then be applied in artificial systems to engineering problems.
A neural network cortical model and learning algorithm for associative memory storage of analog patterns, continuous sequences, and chaotic attractors in the same network is described. System performance using many different chaotic attractors from the family of Chua attractors implemented by the Chua hardware circuit is investigated in an application to the problem of real time handwritten digit recognition. Several of these attractors out-perform the previously studied Lorenz attractor system in terms of accuracy and speed of convergence.
In the {\bf normal form projection algorithm}, developed at Berkeley for associative memory storage of dynamic attractors, a matrix inversion determines network weights, given prototype patterns to be stored. There are N units of capacity in an N node network with 3N^2 weights. It costs one unit per static attractor, two per Fourier component of each periodic trajectory, and at least three per chaotic attractor. There are no spurious attractors, and for periodic attractors there is a Liapunov function in a special coordinate system which governs the approach of transient states to stored trajectories.
Unsupervised or supervised incremental learning algorithms for pattern classification, such as competitive learning or bootstrap Widrow-Hoff can easily be implemented. The architecture can be ``folded" into a recurrent network with higher order weights that can be used as a model of cortex that stores oscillatory and chaotic attractors by a Hebb rule.
A novel computing architecture has been constructed of recurrently interconnected associative memory modules of this type. Architectural variations employ selective synchronization of modules with chaotic attractors that communicate by broadspectrum chaotic signals to control the flow of computation.
· Characterization of Spatial Phase Gradients in Neocortical EEG's Yields Segmentation of Perceptual Events and Neural Binding Domains
Abstract:
Arrays of 8x8 electrodes (0.5-0.8 mm spacing) were chronically implanted onto the epidural surface of the prepyriform, visual, auditory, or somatic cortex of NZW rabbits. EEGs recorded during classical conditioning experiments were parsed into overlapping segments of varying temporal length and cross-classified in 64-space relative to the CS- and CS+ contingencies. This analysis yielded post-stimulus temporal epochs (endogenous perceptual events), where the spatial EEG was positively correlated to the stimulus type. Previous experiments in the olfactory bulb by W.J. Freeman et al. demonstrated that each such endogenous event was characterized by an EEG phase gradient in the form of a cone, the apex of which randomly varied in location and sign. The search for temporally stable phase cones within individual records of neocortical EEGs yielded segments with similar cones lasting on the order of 64 ms with average phase velocities of 2.8 m/sec. Commonality of the phase signal (or 'binding') over the array was defined as the oscillation within the half-power range (=B1 cos 45=B0), and was invariant at ~2.0 cm across neocortical experiments. This analytic technique was used to describe the onset and duration of endogenous neocortical activity bound together into perceptual events at temporal epochs, which were positively correlated with stimulus type. Such events could be described as itinerant visitations in wings of a global chaotic attractor maintained by each neocortical region. The precise location, duration, and evolution of individual events may now be determined.
· Explanation, Dynamics and Causes: The Case of the Dungeness Crabs
Abstract:
Philosophical discussions of scientific explanation ask what scientific understanding consist of, and how the structures and methods of science make this type of understanding possible. Widely accepted causal accounts of explanation maintain that science explains by showing how natural phenomena fit into the causal structure of the world. A central thesis of causal accounts is the claim that that information is explanatorily relevant in virtue of its being information about causes. In this paper, I test this claim against examples of "dynamical explanations" from contemporary research in population biology. "Dynamical explanations" are explanations which explicitly appeal to nonlinear dynamical modeling for their force. Specifically, I consider biologists attempts to explain the erratic population cycles of Dungeness crabs. On the basis of these examples, I argue that explanatory information is not always information about causes.
· An Empirical Test of a Cusp Catastrophe Model for Adolescent Alcohol Use
Abstract:
The study was part of a larger survey designed to look at the various risky behaviors middle school adolescents engage in. Subjects were 800 middle school adolescents in a large urban area. For this study the goal was to assess a cusp catastrophe model in predicting adolescent alcohol use. There has been some suggestion in the literature that a catastrophe model may be a more appropriate way to model adolescent risky behavior. The model used in this study is upon based on those variabels used in Fishbein & Ajzen's Theory of Reasoned Action, one of the most commonly used theories in the primary prevention literature, namely attitudes and social norms. The cusp model was tested using Gaustello's Polynomial Regression Technique and the resulting R^2 was then compared to two different linear comparison models as recommended. The results revealed that in fact the cusp castastrophe model did provide a significantly better fit to the data. The statistical and theoretical ramifications of this finding may be of critical importance to the prevention field.
· Fractal Memory for Visual Form
Abstract:
We discuss a model of memory for visual form which treats the memory 'trace' as a set of procedures for reconstructing earlier visual experience. The procedures, Barnsley's Iterated Function System (IFS), construct an image from a collection of operators (affine transformations). From this perspective, remembering and imagining are processes whose dynamics are captured by the iterative rules. Changes in memory for visual experience are described as changes in the parameters and weights of the reconstruction operators. The model is used to discuss known phenomena and effects in the empirical literature on memory for visual form.
· The Dynamics of Learning to Automaticity
Abstract:
What are the dynamics of the learning process? Historically the rate of learning has been modeled as an s-shaped (sigmoidal) learning curve, indicating movement from a beginning stage to a mastery stage. The shape of the curve is determined by an underlying differential equation, which states that performance accelerates as knowledge (and/or experience) is accumulated, and performance plateaus as one develops mastery of the studied skill. It is typically assumed that deviations from this deterministic learning curve are random deviations, modeled as normally distributed errors (i.e. white noise).
It is possible, however, that additional knowledge concerning the dynamics of the learning process lurk within the data. Specifically, it is feasible that the learning process may exhibit a form of chaotic behavior. Chaos lies in between randomness and order, and chaotic systems exhibit interesting behavior. The detection of chaotic dynamics has both practical as well as theoretical relevance.
Systems in a chaotic state can exhibit a high degree of sensitivity in one moment and an extreme degree of robustness the next. The output of a chaotic system is point-by-point unpredictable, but forms a recognizable pattern over time if observed properly. One may find that the unstructured, random data historically observed in fact has some (topological) order. The discovery of chaos leads to a rejection of the random hypothesis--this in turn can change our assumptions about the underlying dynamics of the system in question. There are some who hypothesize that systems actually purposefully migrate to such a chaotic state, in that it gives the system maximum flexibility and "learning" capability (Kauffman, 1995).
One way to characterize the temporal dynamics is to calculate the dimension of the system's attractor (its historical trajectory). If it comes out to be fractional, and the system exhibits sensitivity to small changes (this can also be calculated explicitly), then the system may be considered to be behaving in a chaotic state. Numerous different algorithms can be applied to determine with a high degree of confidence whether or not the system is indeed chaotic.
There is some reason to believe the learning process may be chaotic (Goertzel, 1995). In the case of an individual learner, a reason for such chaotic dynamics might be that while the brain is searching for an optimally-chunked pattern by which to store instructions for the task, a chaotic search mechanism would tend to be optimal. A chaotic search mechanism may also lead to "chaotic vibrations" in the outcome of the task.
At the physiological level, chaotic dynamics have been found in numerous studies of neuron-level activity (Freeman, 1992; Mitra & Skinner, 1992; Freeman, 1994). Freeman (1994) states, concerning the utility of chaos: "Certain advantages may accrue by virtue of the breadth of spectrum or immunity to entrapment of the dynamics in limit cycle attractors. The maintenance of set points near separatrixes between different chaotic basins of attraction may, because of sensitivity to initial conditions, allow amplification of small fluctuations into large macroscopic patterns. Perhaps the most compelling advantage may lie in the capacity of chaotic systems to create as well as to destroy information. This property may constitute the key feature by which memories are constructed from the raw materials of sensory input that servives the residues of synaptic changes left behind after traumatic experiences or sleep" (p. 303). Kowalik and Elbert (1994) use measures of "chaoticness" to determine phase transitions in brain activities.
At a psycho-motor task level, Cooney and Troyer (1994) performed a simple symbol-memory test and found response time behaved in a chaotic manner, with low dimensionality. This lies in stark contrast to traditional assumptions about the learning process. Clayton and Frey (1994) however performed a replicate of the study and did not find any evidence of chaos; their analysis indicated the subsequent dynamics were random. At the macro-level of learning, a number of different studies concerning research and development processes have found chaotic (divergent) behavior to be present in the early stages of development, while more orderly behavior (convergent) was present at the later stages of development (Koput, 1992; Jayanthi & Sinha, 1994; Cheng & Van de Ven, 1995). Guastello (1995) has also found chaotic dynamics in a number of different learning environments.
The system in question here is a psychomotor learning task performed to a state beyond mastery, namely automaticity. Automaticity is an important state of learned behavior, where a task can be performed with little or no conscious attention (LaBerge & Samuels, 1974). A skill such as driving is thought to be automatic, as one can simultaneously perform other tasks which require significant conscious attention. In work situations, tasks learned to automaticity can be performed rapidly and with few errors. Conversely, unlearning for an automatic state can be quite difficult. There has been very little empirical work done on learning to automaticity (Flor, 1994).
When testing to see if learning has reached automaticity, one must induce interventions (secondary tasks)--a dual task environment. Before automaticity fully develops, the secondary task will greatly distract from the performance of the primary task. After automaticity matures, however, the secondary task should have no effect on the primary task. In order to understand the dynamics of the response in lieu of these interventions, we propose that another type of nonlinear dynamical model may be appropriate, namely catastrophe models.
Catastrophe models model the phenomenon of discontinuous (abrupt) change. The cusp model, for example, is described by two stable modes of behavior. Change between the two states occurs as a function of the two control parameters, asymmetry (a) and bifurcation (b). When 'b' is low, change is smooth and a function of 'a'. When 'b' is high, change still occurs as 'a' changes, but in this case is more discontinuous and sudden--hence, the cusp. For example, one of Guastello's experiments (1995) shows that when an operator is working with a light load, that the risk of having an accident is a smooth function of the amount of environmental hazards present; when load is high, however, risk is discontinuous and better described by two stable attractors (low risk, high risk) separated by a cusp.
Research Propositions
Previous work has indicated that chaotic dynamics may underlie the basic learning phenomenon. Such dynamics would especially be of potential benefit if the learner is faced with a difficult task in which some level of chunking would be advantageous--for example, in a dual task environment. This leads to the following:
P1: Underlying the sigmoidal learning curve, reaction time behaves chaotically in a duel-task environment, and randomly in a single-task environment.
If learning is a continuous phenomena then the level of chaotic dynamics may change over time. The "level of chaos" is measured by the fractal dimension of the system's attractor. Dimension indicates the number of "state" variables that define system performance. A dimension of three of less would typically be considered a "simple" system. As automaticity matures, one would expect the brain to do less-and-less "searching", thereby decreasing dimensionality:
P2: The dimensionality of the learning system decreases over time.
The relative effect of the distractor task should depend on how much automaticity has been developed. One may formulate that such dependence is smooth and linear, or step-like in structure, and nonlinear. If the change is more discreet, then a nonlinear model, such as a catastrophe model, should outperform a similar linear alternative.
P3: Catastrophe models can be used to best describe task performance in lieu of secondary (intervening) tasks.
Experimental Design
Fifty five college students participated in a learning experiment which involved learning a hand-tapping pattern. The subjects were shown ten numbers on one card (corresponding to left-hand) and ten numbers on another. Subjects had to tap the correct sequence on a keyboard. The keyboard was set-up so as to collect data on response time and errors. All subjects performed the task to mastery in a first session. It was noted that a significant difference existed in tapping stype: simultaneous tappers (using both hands at once) were significantly quicker than sequential tappers.
An experimental group performed a second session, under a dual-task condition, and went to automaticity. The secondary task involved holding a set of instructions in memory until the primary task was finished, and then successfully carrying out the said task. The secondary task was invoked about half the time. A control group performed a second session under the same initial conditions as the first session. Data was collected on the number of errors, and the response time.
Mastery was defined as that point, after an initial practice period, when a subject could perform the primary task three times in a row without cue cards and without error.
Automaticity was defined as that point, after mastery, where two consecutive trials with distractor tasks were performed at or below mastery speed (Flor, 1994).
Results
The full paper will present the dynamical analysis results of nine subjects: three control, three simultaneous tappers, and three sequential tappers. A data analysis protocol will be applied to ensure uniform and objective testing of the propositions. Discussion will follow.
· Examining Sequence Effects using Nonlinear Dynamics
Abstract:
In this paper, nonlinear dynamics is used as a means of describing and testing data sets exhibiting sequence effects. The term sequence effects describes the phenomena in which the dependent measure appears to be systematically influences by prior events such as the stimulus from the previous trial. For example, a particular type of sequence effect known as a repetition effect has faster response latencies to the letter X when the letter X was preceded by another letter X. Although sequence effects have been observed over a wide range of paradigms, the discussion will focus on sequence effects in choice reaction time experiments. There are a number of experimental variables known to systematically influence sequence effects. The question we are examining is whether or not these mean differences in the experimental variables reflect differences in the number of nonlinear dynamic variables (i.e., a change in the dimensionality).
· Plenary Talk -- Multifarious Concepts of Equilibrium and Nonequilibrium in the New Dynamical Sciences
Abstract:
The concept of equilibrium has undergone a far-reaching metamorphosis from its origins in the study of levers in Ancient Greece. This paper traces these changes, particularly in the last hundred years as equilibrium and its corollary, non- or far-from-equilibrium have been employed in dynamics, thermo- dynamics, chaos theory, and complex, adaptive systems theories. The application of equilibrium and far-from-equilibrium in the social sciences will be emphasized.
· Is the Neuron a Phase Portrait?
Abstract:
The close resemblance of stellate and pyramidal neurons to local phase portraits. The chain: dynamical system <-> Pfaffian system <-> Lie derivative. The invariant transformations of form perception and the psychological constancies. Colonnier's tangential structure of the visual cortex: spheroidal (stellate cell) and radial (pyramidal cell) symmetries. The shear effect of brain convolutions and gross brain anatomy. Computer modeling of neuronal morphology. Shepherd's local brain circuit. Neuronal discharge as an Omega Explosion. Generation of visual contours by the exponential map and/or parallel transport.
· Exploring the Dynamical System of an Archetype in the Creation Myth of Valentinian Gnosticism
Abstract:
If it is true that archetypes are strange attractors of the human psyche (Abraham, 1989; van Eenwyk, 1991), then two approaches to their investigation seem most relevant. One, the quantitative approach, analyzes a multiplicity of archetypal images that are assumed to be the iterative output of an archetypal nonlinear equation, and thereby derives the dimensionality and shape of the archetypal attractor and specifies its generating equation(s). The other, the qualitative approach, examines an archetypal narrative, such as a myth or fairy tale, for the characteristics of nonlinear dynamical systems, including bifurcations, dissipative structures, and self-organization. In this examination, characters and events as well as their interactions and contiguities are considered as symbolic representations of the structures and processes of a dynamical system. Identified patterns, such as a series of bifurcations leading to self-organization, can then be compared to similar patterns in the physical sciences--a viable comparison in Jungian psychology as psyche and matter are considered to be two sides of the same coin (von Franz, 19xx). This paper proposes to adopt the qualitative approach to explore the dynamical system of an archetype in a creation myth, specifically, a creation myth of the Valentinian Gnostic sect.
Although chaos and dynamical systems theories were developed after Jung's lifetime (e.g., Ford, 1989; Nicolis, 1989), many of the concepts of the "new sciences" are highly compatible with Jungian psychology. Jung described the human psyche as a self- regulating system, its processes governed by a compensatory relationship between the conscious ego and the collective unconscious and organized by activated archetypes and their interactions. He believed that the "irrepresentable basic form" of the archetype could never be known although its organizing presence could be inferred from the recurrence of motifs in the dreams, myths, art and religion of different individuals, cultures, and eras (1954/1969, p.213). Although Jung referred to archetypes as forms or structures, he also stressed that the archetype is a "dynamic action plan" shaping consciousness, emotions, behaviour and relationships as well as the realm of matter. The concepts and technology of chaos theory hold the promise of providing the means for a more precise delineation of archetypes and their effects.
According to von Franz, Jung's closest collaborator, the archetype in its purest form is found in fairy tales. Each fairy tale is "a relatively closed system" (von Franz, 1975, p. 1) whose characters and events symbolize the response of the psychic system to a problematic situation. In a fairy tale, the system with all of its dynamic complexity is already established and, amidst this complexity, the identification of nonlinear archetypal processes is difficult. Myths are considered to be less "pure" than fairy tales due to cultural artefact, but creation myths, particularly those that "begin before the beginning," may be the more revealing prototype of an archetypal pattern. From the perspective of chaos theory, these creation myths describe the evolution of an archetypal system from the undifferentiated homeostasis of equilibrium to the symmetry-breaking perturbation of the primordial matrix and on through the increasing differentiation, coupling and complexity of the unfolding system. Psychologically, creation myths symbolize the origin of consciousness, the process of individuation, and the dynamics of creativity, each closely associated with the archetype of the self, the archetype associated with the self-regulation of the human psyche (von Franz, 1972). In addition, as the creation myth chosen for this study is from a mystical tradition, the findings may also have relevance for the symmetry breaking properties of mystical experiences.
Gnosticism was a religion during the first two centuries of the Christian era until its suppression by the Christian church. Gnostic myths and allegories described the creation of the world, the situation of human beings, and the means of their salvation. Salvation required gnosis, an experiential knowledge of the divine. Many of the Gnostic writings have been destroyed or lost, and those remaining were written by their opponents, the Christian heresiologists. One of these was St. Irenaeus who described in detail the myths of the Valentinian Gnostics, including the creation myth ascribed to Ptolemy, an eminent student of Valentinus (Layton, 1987). In addition to the value of this creation myth for the present study, it is important also to note that the Gnostic literature was very familiar to Jung (e.g., Aion). Gnosticism like its later successor, alchemy, was closely connected with the collective unconscious, and its symbolism and motifs provided validation for his theory of the unconscious, including the archetypes.
I have chosen the Gnostic creation myth of Ptolemy for several reasons. The first is a sense of personal affinity with the Valentinian Gnostic system. This will allow an experiential understanding to guide the inquiry while also recognizing the need to control subjective biases. Second, the structures and processes in the unfolding of this creation myth seem to most clearly parallel the properties of nonlinear dynamical systems. My first intuitive flash of the connection between Gnosticism and chaos theory occurred when I saw the cascading bifurcations diagram in Gleick's book, Chaos, as a parallel to the successive engendering of aeon pairs in the world's creation. I will limit this paper to examining the first part of the myth, which describes the initial perturbation of the unknowable and ineffable "Abyss", the engendering of the aeons and their organization, and the emission of the spiritual Christ and his angels. (These beings continue the creation process resulting in the formation of the material world, familiar to most people from Book of Genesis in the Old Testament.) An added advantage to this myth is that available for comparison are other versions of the Gnostic creation myth (allowing an investigation of sensitive dependence on initial conditions and self-similarity), as well as creation myths from other religions.
The paper will be organized as follows: (1) rationale for the study, (2) presentation of the creation myth, and (3) consideration of the creation myth as the unfolding of a nonlinear dynamical system generated by the activation of an archetype. My hope is that, with the presentation of the paper, audience members who have far a greater knowledge of mathematics and the sciences than myself will recognize parallels from their work and provide input regarding the possible quantification of this archetypal pattern.
References
Abraham, F. (1989). Toward a dynamical theory of the psyche. Psychological Perspectives, 20, 156-166.
Ford, J. (1989). What is chaos, that we should be mindful of it? In P. Davies (Ed.), The new physics (pp. 348-372). Cambridge: Cambridge University Press.
Gleick, J. (1987). Chaos. New York: Penguin.
Jung, C. G. (1969). On the nature of the psyche. The collected works of C.G. Jung: Vol. 8. The structure and dynamics of the psyche (2nd ed., pp. 159-236). (R.F.C. Hull, Trans.) Princeton: Princeton University Press. (Original work published 1954)
Jung, C. G. (1968). The collected works of C. G. Jung: Vol. 9ii. Aion (2nd ed.). (R.F.C. Hull, Trans.) Princeton: Princeton University Press. (Original work published 1951)
Layton, B. (1987). The Gnostic scriptures (pp.276-302).
Nicolis, G. (1989). Physics of far-from-equilibrium systems and self-organisation. In P. Davies (Ed.), The new physics (pp. 316-347). Cambridge: Cambridge University Press.
van Eenwyk, J. R. (1991). Archetypes: The strange attractors of the psyche. Journal of Analytical Psychology, 36, 1-25.
von Franz, M. L. (1972). Creation myths. Dallas, TX: Spring Publications.
von Franz, M. L. (1975). Interpretation of fairytales. Zurich: Spring Publications.
von Franz, M. L. (19xx). Psyche and matter. Dallas, TX: Spring Publications.
· Attitudes as Attractors: Measuring Perceived Self-Efficacy in a Second Language
Abstract:
This theoretical paper suggests that when attitudes are studied as attractors, a crucial bidirectional variable is perceived self-efficacy-- "people's judgments of their capabilities to organize and execute courses of action required to attain designated types of performances" (Bandura, 1986, p. 391). There is now a considerable amount of work applying self-efficacy to writing (Shell, Murphy & Bruning, 1989; Pajares & Johnson, 1994; Zimmerman & Bandura, 1994). However, this has not been linked to chaos and comlexity theory, nor Eiser's study of attitudes as attractors "within a recurring and recognizable pattern" (1994a, p. 161; 1994b).
What is popularly known as self-confidence or self-esteem has been quite precisely defined in many psychological investigations as "perceived self-efficacy." However, the topic is being studied purely within the framework of factorial research, continuing "the overuse and perhaps abuse of the linear model by educational researchers" (Brown, 1975, p. 491; cf. Gilgen, 1995, p. xvi). This paper proposes that the conical pendulum is an excellent metaphor for the relationship between perceived self-efficacy and various aspects of writing competence in a second language.
Writing is a second language is viewed as a complex system in which second language learners "attend excessively to form and are unable to attend to meaning because their cognitive capacity is already overloaded" (Leki, 1993, p. 22; McLeod & McLaughlin, 1986). The possibility of using phase diagrams to study the relationships between form and meaning in second language writing will be considered, and a number of hypotheses will be suggested.
Because the field is new, much of the initial research needs to be exploratory and hypothesis-generating, rather than confirmatory and hypothesis-testing (cf. Levin, 1992, p. 221). Therefore, this paper asks questions and seeks answers in a manner that is generative and heuristic, rather than reductionist or definitive.
References
Bandura, A. (1986). Social foundations of thought and action: A social cognitive theory. Englewood Cliffs, NJ: Prentice Hall.
Brown, D. J. (1975). Mirror, mirror . . . Down with the linear model. American Educational Research Journal, 12, 491-505.
Eiser, J. R. (1994a). Attitudes, chaos and the connectionist mind. Oxford & Cambridge, MA: Blackwell.
Eiser, J. R. (1994b). Toward a dynamic conception of attitude consistency and change. In R. R. Vallacher & A. Nowak (Eds.), Dynamical systems in social psychology (pp. 197-217). San Diego, CA: Academic Press.
Gilgen, A. R. (1995). Prefatory comments. In F. D. Abraham & A. R. Gilgen (Eds.), Chaos theory in psychology (pp. xv-svii). Westport, CO & London: Greenwood Press.
Leki, I. (1992). Understanding ESL writers: A guide for teachers. Portsmouth, NH: Boynton/Cook, Heinemann.
Levin, J. R. (1992). Single-case research design and analysis: Comments and concerns. In T. R. Kratochwill & J. R. Levin (Eds.), Single-case research design and analysis: New directions for psychology and education (pp. 2130224). Hillsdale, NJ: Lawrence Erlbaum Associates.
McLeod, B., & McLaughlin, B. (1986). Restructuring or automaticity? Reading in a second language. Language Learning, 36(2), 109-123.
Parares, F., & Johnson, M. J. (1994). Confidence and competence in writing: The role of self-efficacy, outcome expectancy, and apprehension. Research in the Teaching of English, 28, 313-331.
Shell, D. F., Murphy, C.C., & Bruning, R. H.(1989). Self-efficacy and outcome expectancy mechanisms in reasing and writing achievement. Journal of Educational Psychology, 81, 91-100.
Zimmerman, B. J., & Bandura, A. (1994). Impact of self-regulatory influences on writing course attainment. American Educational Research Journal, 31, 845-862.
· Implications of Feigenbaum Diagram
Abstract:
We would like to propose a theoretical paper that examines the organizational social time implications of the Feigenbaum Diagram. Recent research has shown that disaster response organizations demonstrate chaotic behavior. The Feigenbaum Diagram traces the evolution of this behavior (resources vs. environmental disorder over time). Sociologists have developed the concept of social time to refer to the unique way that organizations organize themselves around a concept of the future. Research shows that following a disaster an organization's future orientation collapses in to "presentism". This collapse probably occurs at the "edge of chaos." We discuss what this means and make further suggestions about the nature of social time at other points on the Feigenbaum Diagram. Hypothesis are suggested and possible management implications derived.
· A Nonlinear Dynamical Conception of Change and Turbulence in Family Systems
Abstract:
The purpose of this presentation is to review some recent developments in the application of nonlinear models to family interaction. As in many other academic and professional disciplines in the social sciences, recent developments in nonlinear dynamics, such as chaos and catastrophe models, have been slow to take hold in spite of the recognition of their great promise. Recently, however, there have been a number of very cogent application of principles of chaos theory to family interaction, such as Chamberlain (1995), and Ward (1995).
The purpose of the present paper is to discuss the impact of these recent developments on our understanding of family processes, and to consider how such an enhanced understanding may better equip us to readdress some of the more fundamental questions of the field. In the proposed paper, I would like to focus on two of such questions, identified by Ward (1995), namely (1) how change occurs in families interaction in response to unique events, and (2) how to define the difference between order and turbulence in the family.
(1) Change in response to unique events. According to Prigogine & Stengers' (1984) model, the likelihood that change occurs is affected by the distance from a fixed point attractor: far-from-equilibrium systems are more susceptible to structural transitions than near-equilibrium systems. There are varying degrees to which each interactive episode between family members approximates an 'ideal state' of equilibrium (Chamberlain, 1995). Such fluctuations from one interactive episode to the next have little impact on the structure of the family system if this systems is in a state relatively close to equilibrium. If the system is far from equilibrium, there is a greater likelihood that minor fluctuations in the interaction patterns between family members trigger a transformation toward a new equilibrium point. The possibility of conceptualizing change in terms of an effect of minor fluctuations on a far from equilibrium system allows for a better understanding of the connection between the family process and life events literatures in psychology: those life events which typically disrupt families (death of a family member, loss of employment, moving to a new neighborhood) produce far-from-equilibrium conditions as the family is in the process of adapting to a new set of circumstances. In this period of readaptation, minor fluctuations may have a decisive influence on the outcome of this process, while in near- equilibrium conditions, they do not. Some concrete examples will be discussed to illustrate this scenario.
(2) Order and turbulence. The association between behavioral problems in childhood and adolescence, and domestic turbulence has long been acknowledged in the psychological and psychiatric literature. However, this same literature is typically unspecific about which criteria need to be used to distinguish turbulent from non-turbulent families. Traditional family process literature has typically considered order and disorder in terms of gradual rather than qualitative differences. For example, in Olson's Circumplex Model, one of the prevailing family process models (Olson, 1989) distinguishes order and disorder in terms of a quantitative difference in levels of family adaptability and family cohesion. The need to recognize qualitative changes between order and disorder has long been recognized in the family process literature (see e.g. Watzlawick, Weakland & Fish, 1974). However, specifying criteria to distinguish the two states has been a notoriously cumbersome process in traditional literature.
The present paper proposes that families who seek equilibrium toward a chaotic attractor rather than a fixed point attractor can be defined as turbulent families. If interaction 'gravitates' toward multiple attractor points rather than a single attractor point, the confounding of roles and relationships occurs which many believe contributes significantly to the psychopathological development of children and adolescents. Some concrete examples of family interaction will be discussed to illustrate the proposed distinction between order and turbulence, and it will be discussed how this distinction facilitates interpretation of interactional episodes in the nuclear family.
REFERENCES
Chamberlain, L. (1995) Strange attractors in patterns of family interaction. In R. Robertson & A. Combs (Eds.) Chaos Theory in Psychology and the Life Sciences. Hillsdale, NJ: Erlbaum.
Olson, D. H. (1989) Circumplex model of family systems VIII: family assessment and intervention. Binghamton, NY: Haworth.
Prigogine, I. & Stengers, I. (1984) Order out of chaos: Man's new dialogue with nature. New York: Bantam.
Ward, M (1995) Butterflies and bifurcations: can chaos theory contribute to our understanding of family systems, Journal of Marriage and the Family, 57, 629-638.
Watzlawick, P., Weakland, J. & Fish, R. (1974) Change: problem formation and problem resolution. New York: Norton.
· Application of Probability Automaton Nets and Representation Theory to Modeling of Human Groups
Abstract:
The model of human institutions was realized by a network of locally interacting automatons which are connected by a system of relations and endowed with the ability to react. To obtain more common property of the nets of probability automatons proposed isageneraldefinition of the automaton nets. It means that we use not only a group of permutations of the finite set of the automaton states but a group measure - preserving trasformations of a smooth manifold or of linear spaces, including the Hilbert space and so on. Inthis report described is the connection between the nets of the automatons with relations and the representation theory. It helps us to investigate the properties of the general nets and gives us a good tool to define and to examine the homogeneous and heterogeneous nets of automatons, i. e. nets of personalities.
Key words: nets of automatons, group and ring representations, Markov chains, social processes. Formulation of the problem. Actually, the situation we shall investigate is the following. Nets of the automatons with relations are definited by: a graph of relations, a set of states of the relations between members of the group; a set of automaton states (now it isa linear space or a manifold with invariant measure) and an algebraic group of the measure- preserving transformations or operators of the linear space (let's call it the group of reactions); a choice function which connects a state of relation with a subset of the group of reactions (for details of the case when the sets of states are finite and the group of reactions is the group of permutations, see ref. [2] ).
The dynamics of the states of a finite net of automatons was deter- mined before by a family of conditional probabilities ([1,2]) but in the models, which will be presented in this paper, the states dyna- mics is controlled by a set of the stochastic matrices with the integer entries 0 and 1. This matrices will be called the control matrices. Sup- poseall the graph of relations edges are marked by the elements belon- ging to the reactions group in accordance with the given system of reac- tions and the choice functions([2]). For the full group of relations the system elements marking a graph of relations can be written in the form of a square matrix. The dynamics of the system after n steps is determined by a product of n matrices applied to the initial state of the net, where every matrix is a tensor product of the control matrix and thesquare matrix of the transformations (every entry in this matrix is a linear transformation of the linear space). If given is a system of the condi- tional probaqbilities on the set of control matrices, then a Markov chain will be defined on the tensor product matrices and therefore the dynamics of the network state will be defined.
Method of proving, result and discussion.
Hence , we have a linear representation of the set of control matrices by the set of tensor-product matrices. But the set of control matrices isn't a group in general and can't be immersed in a group. To apply the representations theory in general case we have to find a minimal linear algebra (over real, complex or Galois fields) which includes the given set of tensor-product matrices. Then we have to find a linear representation of the obtained algebra. To examine the dynamics of the net we have to find the structure of the minimal algebra (the definition and theory see in [6]). For example, for the network with three automa- tons, minimal algebra has one subgroup and one ideal. For larger networks obtained is a partial description of the structure of the minimal algebras. But getting the structures of the tensor-product minimal algebras isn't an easy problem even when the reactions group is finite.
If all control matrices have non-zero determinants we can include the control matrices in a minimal group: first we have to find this group and then the tensor-product minimal group. Here our problem transforms into a pure problem of the repersentation theory of a group (see [3,4]). In this report we used only the finite-dimensional representations of the group and canonical equivalents of an irreducible unitary representation in the space of polynomials ([3]).
References
1. Kovchegov V.B. A principle of maximum nonergodicity for modelling of the human groups by nets of probability automatons . Proceeding of the 14-th IMACS World Congress on Computational and Applied Mathematics ,July 11 - 15, 1994, Georgia Institute of Technology , Atlanta, Georgia , USA ,pp. 787 -790
2. Kovchegov V.B. A model of dynamics of group structures of human institutions . Journal of Mathematical Sociology , 1993 , Vol. 19 (1)
3. Zhelobenko D.P The compact Lie groups and their representations , Moscow , " Nauka " , 1970
4. Varadarajan V.S. Lie groups , Lie algebras and their representations , Englwood Cliffs ,Prentice - Hall Inc., 1974
5. Naimark M.A. Normed Rings , Moscow ," Nauka " ,1968
· Schizophrenia, Dynamics and Self-Organization
Abstract:
The search for a comprehensive view of schizophrenia over the last 15 years has led to converging heuristic models. These models try to integrate biological, psychological and social factors and propose complex interactions. In the last few years, there has been growing enthusiasm about merging these models with dynamical systems approaches. As a conceptual contribution we will discuss the notion of schizophrenia as a dynamical disease. While this view has been very stimulating on the conceptual level, empirical research in this domain turns out to be nontrivial and arduous. We will present studies using psychopathological and functional ratings of patients in different treatment settings, including treatment of acute psychosis and vocational rehabilitation. Generally, we propose ways in which dynamical hypotheses and empirical measures can be linked more specifically and precisely.
· Dynamics of Structural Transformation in Entrepreneurial Firms: Using Catastrophe Theory to Model Self Organizing Processes of Organizational Emergence
Abstract:
The basic idea is to use catastrophe theory to model emergent structural transformations that would result from self-organization. Self-organization theory can be designed as a three-stage process theory: 1) dynamic resonance; 2) turbulence at a threshold; and 3) self-referenced self-organization. An empirical test of self-organizing processes would hinge on modeling the structural shift that occurs in the third stage. Here, a simple cusp catastrophe model offers an excellent empirical method to test whether a self-organizing change has occurred. Hypotheses for this test are generated for rapidly growing entrepreneurial firms, and the whole piece is framed from the Nonlinear Dynamic Systems approach.
· The application of a systems dynamic model for borderline personality disorder: towards the implementation of effective acute and long term clinical interventions
Abstract:
In [will look up]'s model of BPD dynamics, interpersonal distance is viewed as a single linear parameter similar to tempurature and people modulate this distance towards a comfortable ideal (point attractor) by means of a control mechanism analogous to a thermostat/ heater/ air conditioner system. Individuals with BPD have a specific disruption in their control mechanism, leading to a local repellor at the point of healthy interpersonal distance that others are attracted to. Based on this model, I will propose that BPD dynamics are mathmetically chaotic, that this presents a simple model for a common pathogenisis of BPD as well as a conceptually simple model for a therapeutic stance. Specific examples of interventions will be discussed, as well as the integration of this stance into well established theory and practice of the treatment of BPD.
· Advances in Game Theory and Their Application to Psychotherapy
Abstract:
Since its development in the first half of the 20th century, game theory has been used as an analytic and strategy development tool in many branches of the social sciences. With the availability of accesable computer power, Axelrod and others have created much more sophisticated analyses with the use of multiple iterated game series. This allows for the testing of such important psychological processes as the development of trust and the outcome of the use of specific defensive reactions in differing situations. This presentation shows how the recent body of game theory work, particularly the Prisoner's Dilemma and the Diner's Dilemma scenerios may be used as a paradigm for clinical psychotherapy in a variety of contexts. Using this tool, there will be an analysis of interpersonal conflict as a social source of Axis II and certain Axis I disorders. Clinical interventions will be described as derived from different elements of well established techniques but recontextuallized to the new paradigm. This will be presented from the perspective of a variety of specific therapeutic disciplines. Finally, there will be a discussion of how this approach may be integrated into emerging concept of Holland and others' Complex Adaptive Systems Theory.
· Is Psychoanalysis ``The Chaos Game?" Patient Transference Development as an Iterated Function: An Analogy to Barnsley's Affine Transformation Geometry Abstract:
Affine transformation geometry describes transfering a pattern from one set of boundaries to a different set of boundaries. Barnsley developed a dynamic algorithm for achieving this, frequently used in computer image compression schemes. This branch of geometry will be compared with the dymamics of psychoanalytic treatment. Interestingly, this model predicts what is known as a contractive affine transformation for an individual's concept of his/her self. This suggests that ones ego may be fractal in nature. Implications of this model for psychological health and illness as well as the nature of psychoanalysis will be discussed.
· Using Models of Dynamic Chaos in Political Psychology
Abstract:
The processes of political life of a society are characterized by the constantly colliding necessary and casual phenomena. The struggle of political parties, national movements, results of national referendums demonstratea celebration of the synergetic world, in which the accident is not something collateral, minor, and on the contrary, steady, characteristic property, condition of existence and development of public system. Just as the nature of the phenomena of self-organizing is explained in physics by presence of millions of atoms, in biology by millions cells, cooperating with each other by a nonlinear image, dynamics of political consciousness is a consequence of nonlinear interaction of individual consciousness of making its millions of the people. Using a synergetic metaphor, it is possible to tell, that the consciousness of each individual "functions" by a principle nonlinear oscillator. The processes of self-organizing of public consciousness submit to general laws of developing and formation. Hence, is lawful to expect, that the theory of attractors and catastrophes can be used for the description of functioning of political consciousness. Objects of our research were 20 political parties in Russia, as signs of value sets creating a structure of political mentality in society. Two interrogations of representative members of political parties were conducted during two the most critical moments of modern history of Russia: the summer 1991 and the summer 1993. By the method of semantic differential the political spaces were constructed, being operational model for studying political mentality. Discrete differential equations and Lamerey's diagrams were used as operational model of dynamic chaos for studying political consciousness. On the basis of the analysis of the empirical results of the changing of political positions after interviewing members of political parties were constructed theoretical regression curve, enabling to predict results of possible changes in political sets, positions of consensus in the society in the relations of various political problems.
· Group Dynamics: Notation and Simulation
Abstract:
We examine the evolution of the dynamics of a group formed by several psychosomatic patients, during a sampling period of six months. The notation method in use was the so-called 'Bion's Grid'. The history, the general features and our specific implementation of this observational instrument are examined. We used some of the main features of the Grid to build cellular automata which could simulate some dynamic evolutions of this group. The relationships between notations and simulations are discussed."
· Plenary Talk -- Death of a Paradigm Crisis
Abstract:
The kinds of control that operate in psychological and related systems are discussed and cast into a unified mathematical form, the Discrete Control Equation (DCE). This can apply to a variety of human and animal "systems" and tasks, whether the framing language is that of homeostasis, cognitive error correction, coupled biological oscillators, motor control, or adaptive change. Applicability of the DCE to the temporal structure of human movement is then examined in detail. Given certain plausible boundary conditions, the equation produces various standard and generalized forms of the circle map class governing relative phase in motor coordination, and hence generates well-documented nonlinear "dynamical" motor phenomena such as behavioral attractors, phase transitions, critical slowing, etc. Under certain other plausible boundary conditions, the DCE produces the stochastic timing models of the "cognitive" approach, accommodating hierarchical effects, open- and closed-loop conditions, and unilimb and multilimb movement. A number of predictions are identified from the approach, and a review of experimental evidence supports the claim that the current formulation is an integrated generalization and improvement of existing cognitive and dynamical approaches.
· Paradox and Psychological Health: Four easy Steps Beyond Homeostasis
Abstract:
This paper addresses the problem of applying dynamical systems models to "normal" and healthy physiological, psychological, and psychosocial functioning. It presents a theoretical program and several specific examples for the revision of "homeostasis" and "dynamic equilibrium" models of healthy systems, paying attention to what happens when various forms of non-linearity and coupling come into play.
While this knowledge is often ignored, it is well known that the relationships between most psychological and behavioral characteristics and healthy outcomes (e.g., personal or social well-being, "Quality of Life") are non-linear. Thus, a revision of older models of healthy functioning that emphasize linear relationships between "virtues" and health is warranted. Notions of health that define health in terms of "moderation" represent early models grappling with nonlinearity, albeit static ones.
A second key change in models of health involves taking systems views of psychosocial functioning seriously and focusing on systems including multiple reciprocal causal paths. Healthy biological and psychological systems involve the interactions of multiple coupled processes, both excitatory and inhibitory ones.
A third change involves considering health as a dynamic and temporally changing process, dependent on environment and context. Earlier models that emphasize "healthy adaptation" include both open systems and dynamic elements. However, it is important to assess the degree to which current models tend to imply that health is a steady state or at best a "dynamic steady state" (e.g., a state of "dynamic tension".)
Finally, the paper is interested in the difficult and very current problem of attending to the simple and minimal conditions for complexity to occur in self-regulating systems.
The paper develops an application of a coupled-optimizer or predictive agent model to a simple self-regulating system that can represent "healthy" and "normal" functioning. The paper develops a four-loop model of a dynamic psychological or physiological system, composed of two excitatory and two inhibitory loops. The model is developed in the context of examining processes of stress and coping, and theoretical and empirical work identifying each of the four loops will be described. The model provides the opportunity to explore the question of how relatively simple systems -- composed of subsystems already well-studied in the psychological literature -- are minimally capable of complex behavior.
This discussion will be referred to the current literature on the possibly chaotic characteristics of observed or measured physiological processes. This literature has tended to account for complex biological functions (e.g., apparently "noisy" EEG power spectra) by emphasizing the relationship between fairly complicated biological structures (e.g., the possibly fractal branching of physiological structures) and possibly complex or chaotic behavior, observed at a physiological (in this case, neuroelectrical) level.
However, complicated structures are not required for complex behavior. The paper's description of a conceptually simple "coupled-homeostat" model will allow an examination of the relationship between simple structures and complex functions. I believe that the model sketched out in the paper applies well to a variety of psychosocial phenomena in addition to stress processes, as well as appearing to have some desirable dynamic properties. I am hopeful that the discussions at the conference will allow further development of similar models and exploration of the associated mathematics and possible modeling (or simulation) techniques.
· The Psychoanalytic Flirtation with Chaos: A Potentially Dangerous Liaison
Abstract:
Its continuing search for improved models of psychic reality, body-mind interfaces and the psychotherapeutic process has led psychoanalysis to flirt with nonlinear dynamic systems theory. I believe that this liaison is proving problematic. Though chaos theory can provide psychoanalysis with useful insights about itself, its other and their relationship as well as providing models that link psyche and soma, some psychoanalysts, apparently lost in the passion of their projections, have assigned propositions to chaos theory that are based more upon what they hope to find there than upon what chaos theory actually does or can say. Conversely, they have denied or selectively interpreted what chaos theory does say in an apparent attempt to maintain metapsychological and practical homeostasis. This tendency to use chaos theory as a self-object, rather than as an object in its own right, prevents both from coming to know itself versus its other and developing object relationships through which each can evolve through coevolution with the other.
Here, I will consider 1., what psychoanalysis seeks to maintain via its projection onto chaos theory, 2., what chaos theory brings to their meeting that psychoanalysis seeks to deny and 3., the inevitable consequences for psychoanalysis of it's embracing chaos theory as object. I will explore these questions by considering Robert M. Galatzer-Levy's recent JAPA article (in which he attempts to interpret some elements of psychoanalysis in terms of chaos theory while retaining knowledge to the psychoanalyst that it is impossible for him to have given the halting problem) and Esther Thelen's reading of infant development in which she posits "deep" attractors as ballast for maintaining developmental trajectories, warding off the consequences of environmental perturbation, and fundamentally containing individual differences.
Finally, I will examine the nature and consequences of the chaotic (sense 2) terror that must inevitably face the psychoanalyst when he approaches patients without the tools retained by Galatzer-Levy and Thelen and will suggest how to deal with that occurrence for the benefit of both analyst an patient.
· Fractal Functionality and Differential Psychology
Abstract:
Functionality is the property of system's units to execute different functions at the system's community. Usually one unit has disposition for executing of some function's types and one function can be executed by different sets of units. A unit starts to execute some function when it has more high rank on the formal characteristics that are necessary for this function.
Unit's dissociation on the different functional groups within a system has some general tendencies that repeat at the different levels of complexity and at the systems of different nature. Such property of self-similarity at the functional differentiation processes can be named a fractal functionality.
The model of the main formal parameters that are at the foundation of unit's functional disposition was developed. Such parameters are: potential, diffusion, stage of system's stability, types and ranking of super-system's attractors, that system moves by. Non-constant character of most of them provides to the dynamical features of individuality's system. This model was applied at the psychology of individual differences. The report consists concrete estimations of the main personality's characteristics by indicated parameters.
· The Cusp Catastrophe Applied to Cognitive Performance
Abstract:
We use the cusp catastrophe as a guide for exploring performance in a cognitive task--lexical decision. In a lexical decision task, subjects categorize single presented letter-strings as "words" or "nonwords." The splitting variable is the experimenter's choice of similarity between words and nonword stimuli (i.e., the difficulty of the discrimination). The normal variable is the subject's on-line (trial-by-trial) perception of the similarity between words and nonwords (i.e., the perceived difficulty of the discrimination). So far, we have observed a hysteresis effect in correct "word" response times at the "difficult" setting of the splitting variable, but no hysteresis at the "easy" setting. When hysteresis is observed, we also observe sudden jumps in correct "word" response times.