Kazuhiro Satoh

Computer Experiment on the Cooperative Behavior of a Network of Interacting Nonlinear Oscillators

The cooperative behavior of a network of interacting nonlinear oscillators is investigated numerically. Such an oscillator-net serves as a phenomenological model for an endogenous circadian pacemaker in organisms. Following two types of network are examined. (1) Ordered oscillator-net: oscillators sit on the square-lattice points and interact with the nearest neighbors. (2) Random oscillator-net: each oscillator is influenced by a certain number of randomly chosen oscillators. Detailed computer experiment is made on the oscillator-net containing one hundred to one thousand oscillators. The self-entraining behavior is found to be very different in the two types of oscillator-net. The sample- and size-dependence are also studied. The relevance of the oscillator-net as a model for the circadian pacemaker is discussed.

Computer Experiment on the Complex Behavior of a Two-Dimensional Cellular Automaton as a Phenomenological Model for an Ecosystem

Numerical studies are made on the complex behavior of a cellular automaton which serves as a phenomenological model for an ecosystem. The ecosystem is assumed to contain only three populations, i.e., a population of plants, of herbivores, and of carnivores. A two-dimensional region where organisms live is divided into square cells and the population density in each cell is regarded as a discrete variable. The influence of the physical environment and the interactions between organisms are reduced to a simple rule of cellular automaton evolution. It is found that the time dependent spatial distribution of organisms is, in general, very random and complex. However, under certain conditions, the self-organization of ordered patterns such as rotating spirals or concentric circles takes place. The relevance of the cellular automaton as a model for the ecosystem is discussed.

Numerical Study on a Coupled-Logistic Map as a Simple Model for a Predator-Prey System

Dynamics of an iterated map of an interval, x n +1 = a x n (1- x n )+ c x n y n , y n +1 = b y n (1- y n )- d y n x n is studied numerically. Such a coupled-logistic map serves as a simple model for a predator-prey system in ecology. A generated sequence { X n } (where X n =( x n , y n )) is attracted to a periodic, or a torus, or a chaotic attractor depending on parameters a , b , c and d . A phase diagram which classifies the attractors in the a - b plane is investigated varying the coupling strength c (where c = d ). It is found that the global structure of the phase diagram is rather simple, whereas the detailed structure of the phase diagram exhibits remarkable self-similarity. Properties of the attractors are empirically studied on the Fourier and fractal-dimension analysis. Initial value dependence of the attractors is also discussed.

Self-Similar Structures in the Phase Diagram of a Coupled-Logistic Map

The Lyapunov exponents of attractors of a coupled-logistic map, x n +1 = a x n (1- x n )+ c x n y n , y n +1 = b y n (1- y n )- c y n x n , are evaluated in the a - b plane for several values of c . The first exponent λ 1 ( a , b ) is graphically shown as a contour map (high-resolution color picture), and thus the detailed structure of the phase diagram, especially its self-similarity (K. Satoh and T. Aihara: J. Phys. Soc. Jpn. 59 (1990) 1184), is revealed precisely.

Single and multiarmed spiral patterns in a cellular automaton model for an ecosystem

The behavior of a cellular automaton model for an ecosystem (K. Satoh: J. Phys. Soc. Jpn. 58 (1989) 3842) is reinvestigated on the hexagonal lattice. It is found that a variety of single and multiarmed spiral patterns are self-organized.

Computer Experiment on Characteristic Modes of Excitation in a Random Neural Network on the McCulloch-Pitts Model

Numerical studies are made out the behavior of a random neural network in which each neuron is coupled to a certain number of randomly chosert neurons. Such a random-net serves as a simple model for an elemental sub-network of the cortex. Neurons are regarded as binary decision elements, and they synchronously update their values in discrete time steps according to a deterministic equation (the McCulloch-Pitts model). It is found that each random-net containing one hundred neurons has only a few kinds of characteristic modes of excitation. Periods of these modes are usually less than ten steps when the number of connections per neuron is two to five. For the random-net containing one thousand neurons, an excited mode is practically aperiodic. When the refractory period is introduced, however, a nearly periodic oscillation takes place in the activity of the network.

Rhythmic Activity in a Random Neural Network Model

A random neural network model is found to exhibit rhythmic activity which is very similar to that of the brain waves. Dimensional analysis suggests that chaotic attractors reconstructed by the embedding method are high-dimensional ones.

Critical Phenomenon in a Neural Network Model: A Localization-Delocalization Transition of Excited Clusters

Numerical study is done on a critical phenomenon in a neural network model of the McCulloch-Pitts type. Such a net, one of excitable media, consists of “neurons” (binary decision elements) each of which randomly sits on a square lattice and is connected to its four neighbors. When the net is activated locally, the “fire” spreads over from the origin according to the deterministic rule. After transient, a self-sustained mode of excitation (time-periodic firing pattern) is established. It is found that a size of the largest excitation tends to diverge as the excitability of the net is increased (a localization-delocalization transition). Numerically evaluated power-law exponents suggest that the criticality of such transition belongs to the same universality class of the percolation transition.

Complex Behavior of a Randomly Connected Predator-Prey Network : a "Survival Game" for a Mass of Competing Species

An artificial survival Game for a mass of competing species is introduced and investigated numerically. It is assumed that (1) each species consists of hunting predators. (2) All the species have their own nests to live. (3) They feed on common preys which are distributed in a one dimensional cellular region. (4) Each species randomly attacks a certain number of cells to feed on preys. The evolutional rule of this randomly connected predator-prey network is given in a highly simplified manner as a set of difference equations. An optimizing rule which facilitates the efficiency of the whole system is also included. In spite of its simplicity, this Game is found to exhibit complex behavior, i.e., its long-term behavior is chaotic, or self-organizing, or critical, depending on parameters involved.

Numerical study on a coupled logistic map as a simple model for three competing species

A coupled logistic map is applied to population dynamics of three competing species. Interaction between three species (labeled by X, Y, and Z) is assumed to be intransitive, i.e., X beats Y, Y beats Z, and Z beats X. Phase diagrams are determined for a circumstance in which the species Z interacts weakly with the XY system. It is found that the intransitive interaction results in a variety of bifurcation diagrams of torus attractors and associated hysteresis phenomena.