Adaptation Noise And Self Organizing Systems

The dynamics of velocity fluctuations, governed by the one-dimensional Burgers equation, driven by a white-in-time random force with the spatial spectrum $\overline{|f(k)|^2}\proptok^{-1}$, is considered. High-resolution numerical experiments conducted in this work give the energy spectrum E(k)∝ k −β with β=5/3±0.02 . The observed two-point correlation function C(k,ω) reveals ω∝ k z with the "dynamical exponent" z≈2/3 . High-order moments of velocity differences show strong intermittency and are dominated by powerful large-scale shocks. The results are compared with predictions of the one-loop renormalized perturbation expansion.

High-resolution numerical experiments, described in this work, show that velocity fluctuations governed by the one-dimensional Burgers equation driven by a white-in-time random noise with the spectrum |f(k) | 2 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ∝ k −1 exhibit a biscaling behavior: All moments of velocity differences S n≤3 (r)= |u(x+r)−u(x) | n ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ≡ |Δu | n ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ∝ r n/3 , while S n>3 ∝ r ζ n with ζ n ≈1 for real n>0 (Chekhlov and Yakhot, Phys. Rev. E {\bf 51}, R2739, 1995). The probability density function, which is dominated by coherent shocks in the interval Δu<0 , is P(Δu,r)∝(Δu ) −q with q≈4 .

A class of models for large-scale evolution and mass extinctions is presented. These models incorporate environmental changes on all scales, from influences on a single species to global effects. This is a step towards a unified picture of mass extinctions, which enables one to study coevolutionary effects and external abiotic influences with the same means. The generic features of such models are studied in a simple version, in which all environmental changes are generated at random and without feedback from other parts of the system.

Lattice models describing the spatial spread of rabies among foxes are studied. In these models, the fox population is divided into three-species: susceptible, infected or incubating, and infectious or rabid. They are based on the fact that susceptible and incubating foxes are territorial while rabid foxes have lost their sense of direction and move erratically. Two different models are investigated: a one-dimensional coupled-map lattice model, and a two-dimensional automata network model. Both models take into account the short-range character of the infection process and the diffusive motion of rabid foxes. Numerical simulations show how the spatial distribution of rabies, and the speed of propagation of the epizootic front depend upon the carrying capacity of the environment and diffusion of rabid foxes out of their territory.

We describe and investigate the learning capablities displayed by a population of self-replicating segments of computer-code subject to random mutation: the tierra environment. We find that learning is achieved through phase transitions that adapt the population to whichever environment it encounters, with a learning rate characterized by the environmental variables. Our results suggest that most effective learning is achieved close to the edge of chaos.

In supervised learning, the redundancy contained in random examples can be avoided by learning from queries. Using statistical mechanics, we study learning from minimum entropy queries in a large tree-committee machine. The generalization error decreases exponentially with the number of training examples, providing a significant improvement over the algebraic decay for random examples. The connection between entropy and generalization error in multi-layer networks is discussed, and a computationally cheap algorithm for constructing queries is suggested and analysed.

A simple model of self-organised learning with no classical (Hebbian) reinforcement is presented. Synaptic connections involved in mistakes are depressed. The model operates at a highly adaptive, probably critical, state reached by extremal dynamics similar to that of recent evolution models. Thus, one might think of the mechanism as synaptic Darwinism.

Basic problems in complex systems are surveyed in connection with Life. As a key issue for complex systems, complementarity between syntax/rule/parts and semantics/behavior/whole is stressed. To address the issue, a constructive approach for a biological system is proposed. As a construction in a computer, intra-inter dynamics is presented for cell biology, where the following five general features are drawn from our model experiments; intrinsic diversification, recursive type formation, rule generation, formation of internal representation, and macroscopic robustness. Significance of the constructed logic to the biology of existing organisms is also discussed.

We find that intervals between successive drops from a leaky faucet display scale-invariant, long-range anticorrelations characterized by the same exponents of heart beat-to-beat intervals of healthy subjects. This behavior is also confirmed by numerical simulations on lattice and it is faucet-width- and flow-rate-independent. The histogram for the drop intervals is also well described by a Lévy distribution with the same index for both histograms of healthy and diseased subjects. This additional result corroborates the evidence for similarities between leaky faucets and healthy hearts underlying dynamics.

A new order parameter approximation to Random Boolean Networks (RBN) is introduced, based on the concept of Boolean derivative. A statistical argument involving an annealed approximation is used, allowing to measure the order parameter in terms of the statistical properties of a random matrix. Using the same formalism, a Lyapunov exponent is calculated, allowing to provide the onset of damage spreading through the network and how sensitive it is to minimal perturbations. Finally, the Lyapunov exponents are obtained by means of different approximations: through distance method and a discrete variant of the Wolf's method for continuous systems.