Adaptation Noise And Self Organizing Systems

An analytically solvable model is proposed exhibiting 1/f spectrum in any desirably wide range of frequency (but excluding the point f=0). The model consists of pulses whose recurrence times obey an autoregressive process with very small damping.

The dynamics of complex systems in nature often occurs in terms of punctuations, or avalanches, rather than following a smooth, gradual path. A comprehensive theory of avalanche dynamics in models of growth, interface depinning, and evolution is presented. Specifically, we include the Bak-Sneppen evolution model, the Sneppen interface depinning model, the Zaitsev flux creep model, invasion percolation, and several other depinning models into a unified treatment encompassing a large class of far from equilibrium processes. The formation of fractal structures, the appearance of 1/f noise, diffusion with anomalous Hurst exponents, Levy flights, and punctuated equilibria can all be related to the same underlying avalanche dynamics. This dynamics can be represented as a fractal in d spatial plus one temporal dimension. We develop a scaling theory that relates many of the critical exponents in this broad category of extremal models, representing different universality classes, to two basic exponents characterizing the fractal attractor. The exact equations and the derived set of scaling relations are consistent with numerical simulations of the above mentioned models.

This paper contains a summary of mathematical researches of stochastic properties of the long time behavior of a continuously observed (and interactively controlled) quantum--field top. Applications to interactively controlled stochastic computer-graphic dynamical systems are also discussed.

Individuals in groups must often choose between acting selfishly and cooperating for the common good. The choices they make are based on their beliefs on how they expect their actions to affect others. We show that for a broad set of beliefs and group characteristics cooperation can appear spontaneously in non-cooperative groups after very long periods of time. When delays in information are unavoidable the group dynamics acquires a wide repertoire of behaviors, ranging from opportunistic oscillations to bursty chaos, thus excluding the possibility of sustained cooperation.

A learning algorithm for multilayer neural networks based on biologically plausible mechanisms is studied. Motivated by findings in experimental neurobiology, we consider synaptic averaging in the induction of plasticity changes, which happen on a slower time scale than firing dynamics. This mechanism is shown to enable learning of the exclusive-OR (XOR) problem without the aid of error back-propagation, as well as to increase robustness of learning in the presence of noise.

Neural network models offer a theoretical testbed for the study of learning at the cellular level. The only experimentally verified learning rule, Hebb's rule, is extremely limited in its ability to train networks to perform complex tasks. An identified cellular mechanism responsible for Hebbian-type long-term potentiation, the NMDA receptor, is highly versatile. Its function and efficacy are modulated by a wide variety of compounds and conditions and are likely to be directed by non-local phenomena. Furthermore, it has been demonstrated that NMDA receptors are not essential for some types of learning. We have shown that another neural network learning rule, the chemotaxis algorithm, is theoretically much more powerful than Hebb's rule and is consistent with experimental data. A biased random-walk in synaptic weight space is a learning rule immanent in nervous activity and may account for some types of learning -- notably the acquisition of skilled movement.

We present a dynamical theory of asset price bubbles that exhibits the appearance of bubbles and their subsequent crashes. We show that when speculative trends dominate over fundamental beliefs, bubbles form, leading to the growth of asset prices away from their fundamental value. This growth makes the system increasingly susceptible to any exogenous shock, thus eventually precipitating a crash. We also present computer experiments which in their aggregate behavior confirm the predictions of the theory.

Within the framework of a simple model of car traffic on a one-lane highway, we study the probability for car accidents to occur when drivers do not respect the safety distance between cars, and, as a result of the blockage during the time T necessary to clear the road, we determine the number of stopped cars as a function of car density. We give a simple theory in good agreement with our numerical simulations.

A novel mechanism for cell differentiation is proposed, based on the dynamic clustering in a globally coupled chaotic system. A simple model with metabolic reaction, active transport of chemicals from media, and cell division is found to show three successive stages with the growth of the number of cells; coherent growth, dynamic clustering, and fixed cell differentiation. At the last stage, disparity in activities, germ line segregation, somatic cell differentiation, and homeochaotic stability against external perturbation are found. Our results, in consistency with the experiments of the preceding paper, imply that cell differentiation can occur without a spatial pattern. From dynamical systems viewpoint, the new concept of ``open chaos" is proposed, as a novel and general scenario for systems with growing numbers of elements, also seen in economics and sociology.A

We propose a probabilistic cellular automata model for the spread of innovations, rumors, news, etc. in a social system. The local rule used in the model is outertotalistic, and the range of interaction can vary. When the range R of the rule increases, the takeover time for innovation increases and converges toward its mean-field value, which is almost inversely proportional to R when R is large. Exact solutions for R=1 and R=∞ (mean-field) are presented, as well as simulation results for other values of R. The average local density is found to converge to a certain stationary value, which allows us to obtain a semi-phenomenological solution valid in the vicinity of the fixed point n=1 (for large t).