| Accepted by: International Journal of Engineering Applications of Artificial
Intelligence
Complex Systems: Chaos and Beyond Berlin/Heidelberg: Springer-Verlag, 2001 Christian Storm* and Walter J. Freeman *Corresponding Author: raystorm@uclink4.berkeley.edu
About the authors Walter J. Freeman studied physics and mathematics at M.I.T., English and Philosophy at the University of Chicago, medicine at Yale University (M.D. 1954), internal medicine at Johns Hopkins, and neurophysiology at UCLA. He has taught brain science at Berkeley since 1959, where he is Professor of the Graduate School. He is the author of "Mass Action in the Nervous System" (1975), "Societies of Brains" (1995), "How Brains Make Up Their Minds" (1999), and "Neurodynamics: An Exploration of Mesoscopic Brain Dynamics" (2000). He is a Fellow of the IEEE, cited for his models of machine intelligence based on the nonlinear dynamics governing behavior. Christian Storm received degrees in physics and mathematics at UC Santa Cruz and joined the UC Berkeley Biophysics Group in 1996. His current research focuses on developing state space measures that characterize the stability of chaotic systems to both parametric and input perturbations. Review The study of complex systems is emerging as the science of understanding the shade under nature's umbrella of irreducibility and nonlinearity. The tremendous breadth of this umbrella and the exquisite phenomena it covers both captivate and, at the same time, impede scientists from enjoying the converging efforts of those in other disciplines. In this sense, complex systems research can be likened to a black art with its practitioners using whatever available tool seems to decipher any aspect of the complex phenomena under study empirically instead of being guided by a universal complex systems theory. Kaneko and Tsuda acknowledge in their new book "Complex Systems: Chaos and Beyond" that complexity science is still in its infancy, and they rightly caution that a balance must be struck between the study of individual phenomenon and describing the umbrella itself. Their book strikes this balance by describing some of the "contours of complex systems" with a specific aim toward how we define and characterize networked chaotic systems. In the broadest view their work aims at unveiling the mysteries posed by interacting chaotic systems or networks of chaotic elements. With these networks appearing in phenomena ranging from the beating heart, capillaries, and crystal growth to neural, ecological, and information processing systems, the tools and theories they present to the reader, using the coupled map lattice as their experimental system, are more than timely, they are necessary for basic research in these areas. For example, one species of artificially intelligent machines now under development will undoubtedly employ such a network architecture to process information regarding the environment and its own internal states needed to cope with the unpredictable, chaotic changes in the environment. Such a machine can be envisioned to take advantage of the robust nature of chaos in being fluid and not bound to the finite number of states prescribed by such computational structures as decision trees and template stores, allowing it to explore the infinity of its own state space and those of its surroundings. The liberty bestowed by chaos also calls for rigorous stability analyses to prevent these machines from flying off the handle if perturbed externally by unconstrained input or internally by noise. From Androv and Pontryugin's pioneering work on the structural stability of models to perturbation came the precise idea that the qualitative behavior of a dynamical system should remain unaltered under small perturbations (Arnold, 1982). However, realistic nonlinear models, whether they be of physical systems found in nature or analog circuits engineered for the purposes of computation, must account for the presence of noise and its effect on (in)stability. Regarding 'real-world' systems continually exposed to noise, the authors introduce the notion of noise induced order. Under certain conditions a system will undergo the transition from a chaotic dynamic to an ordered motion upon the application of noise, which they demonstrate quite effectively with the Belousov-Zhabotinsky reaction. This counter-intuitive finding, which dictates that entropy must decrease as a function of increased noise levels, demonstrates how the same stimuli can bring about two contradictory behaviors. In this case, stability and order or instability and disorder. In a step toward a new formulation of stability, they undertake development of the notion of descriptive (in)stability for noisy systems. This theory will be crucial to the engineering community involved in designing machines that rely on noisy spatiotemporal chaos for information processing. One of the topics central to the book in their chapter on "Chaotic Information Processing in the Brain", relies on the theory of neurochaos. To this end, they briefly summarize three alternative theories regarding the genesis of neurochaos. The authors' detailed studies of coupled map lattices, which 'live' in a world of discrete time and space, constitutes a clear model system under the first theory regarding neurochaos, which in the higher levels of a network result directly from its constituents - chaotic 'neurons'. A proliferation of chaotic neuronal models as in cellular neural networks (Chua and Yang, 1988) has promoted this formulation. A second theory comes from studies in the rabbit and rat olfactory system, by which it has become evident that the chaos found in mesoscopic neuronal activity is independent of the individual neurons and, instead, results from multiple feedback loops with nonlinear gains among neuronal populations (Freeman, 2000). Chaos arises through interactions that constrain and channel the microscopic noise, so that chaos consumes noise and thrives on it, while using it to organize the brain's perceptions of the world. A third class of theory is based on self-organization and the stochastic fluctuations found in the trajectories of the state variables of nodes, as seen in the models of Hopfield (1986), Parisi (1986), and Amit (1986), which also demonstrate the importance of noise in both inducing chaos and stabilizing it. The consequence of destabilization in phase space remains a critical and ill understood problem in the study of extended chaotic systems. In their book the authors propose that a major instability throws a system into a noisy or turbulent macroscopic state destroying all features of the state space attractors, whereas a minor instability is not strong enough to destroy the attractors, thereby allowing for the asymptotic capture of the system into an attractor from its basin. The more interesting and biologically plausible case is the gray area somewhere between capture and turbulence or, in a sense, order and disorder. In this regime, Kaneko and Tsuda propose that destabilization occurs continuously over unstable manifolds, which they call a collection of 'attractor ruins'. Each 'ruin' resembles a saddle node that attracts in some directions but leads to the onset of instability even as it is accessed. In one of these newly formed unstable directions their system seeks another attractor ruin, then others, which as a set controls an itinerant trajectory. Here an attractor ruin represents a memory and the linkage between such memories is the trajectory through the set. Given a word, a phrase follows. Given a first line, the poem follows. For Proust, the taste of a madeleine was the trigger for a memory cascade. But this is not like the now legendary example of Lorenz' butterfly in Brazil causing a tornado in Texas, or the example, in the warmth of a New England summer night, of a June bug that follows an unpredictable trajectory, free to wander from light to light in search of a mate. Although feasible mathematically and possibly pertinent to robust chaotic information processing, this model departs from biological brains and human experience owing to a certain rigidity in the implied chains of association. An alternative to their view of the itinerant trajectory of destabilization in phase space is the notion of an adaptive and modifiable attractor landscape, which is latent in the brain of an animal or human that is searching for information leading to a goal (Freeman, 2000). The brain maintains a state of expectancy in which instability is manifested in a high-dimensional phase akin to turbulence to support the search for a stimulus that isn't present. The range of possibilities is latent in an attractor landscape that has been formed through experience with multiple classes of stimuli, having a basin of attraction for each learned class. Each act of search creates the landscape. With the arrival of a sought stimulus the system condenses, leading to a capture in a dynamical pattern in stable lower dimensional attractor. If the stimulus does not arrive, the sensory system is captured in the basin of an attractor that denotes "not yet". In either case, the landscape collapses, analogously to the 'ruin', and a new act of search occurs. This high dimensional phase of turbulence is key, for it allows the system to be at once everywhere with respect to the lower dimensional attractor landscape, thereby allowing for the rapid transition amongst distinct memories. An analogy is a 2-D surface of hills and valleys, each basin having a 1-D limit cycle at its base, with limited access by driving over surface roads across boundaries (separatrices) but with rapid access by taking flight in 3-D. Another analogy is a tesseract (a 4-D cube) where each face is a 3-D cube like a house with 8 rooms. Unlike walking through a 3-D house with limited routing, the trajectory through a tesseract gives access to every room directly from every other room. Learning then has the effect of modifying the thresholds of a pathway between rooms or, more specifically, the topology of this attractor landscape, as in creating a new attractor with its attendant basin in coming to identify a new class of stimuli. These and other complex system models are emerging in both academe and industry to shed light on unsolved mysteries and solve a multitude of technical problems. The founding principles that govern these systems are still to be fully developed. In "Complex Systems: Chaos and Beyond", Kaneko and Tsuda have well formulated some of these principles and shown us how to use powerful tools to tie extended chaotic systems together and open them to useful applications in the future. Their book is highly recommended for all students and researchers in theoretical neuroscience and neural-based cognitive scientists. Arnold, V.I., 1982. Geometrical Methods in the Theory of Ordinary Differential Equations. Springer, New York. Amit, D.J., 1986. Neural networks - achievement, prospects, difficulties. Int. Symp. on The Physics of Structure Formation (Tubingen) Chua, L.O., Yang, L., 1988. Cellular neural networks: theory. IEEE Trans.
Circ. Syst. 35 (10), 1257-1272 |