Hiroaki Hara

A temporal model of animal behavior based on a fractality in the feeding of Drosophila melanogaster

We present a new temporal model of animal behavior based on the ethological idea that the internal states of the individual essentially determine the behavior. The internal states, however, are conditioned by the external stimuli. This model, including environmental and internal parameters, predicts a fractal property of the behavior, that is, an inverse power law distribution of the duration. Being consistent with the model, we have found a fractal property of feeding in Drosophila melanogaster: The dwelling time of starved flies on food showed a clear inverse power law distribution. The dependence of the fractal dimension on the intensity of food stimuli has been observed, and the predicted change into an exponential distribution was proved.

Temporal fractal in the feeding behavior ofDrosophila melanogaster

A temporal fractal is clearly shown in the feeding behavior ofDrosophila as a self-similar pattern of locomotive velocity and inverse power law distributions of food dwelling time over the time scale range of 103. The fractality was observed in the dwelling time distribution immediately after the fly was placed to feeding site or on inferior food in a two-choice situation. Fractality may be understood as adaptive, and an intrinsic property of animal behavior that reflects complex information processing in the CNS ofDrosophila.

Riemann scalar curvature of ideal quantum gases obeying Gentile's statistics

The scalar curvature (R) of ideal quantum gases obeying Gentiles statistics is investigated by the method of information geometrical theory. The R value is specified by the fugacity and the maximum number, p, of particles in a state. The lowest case p = 1, corresponds to Fermi-Dirac statistics and the unbounded case, p, to Bose-Einstein statistics. In contrast to R = 0 for ideal classical gases obeying Boltzmann statistics, we find R = (2)1/2/32 for p2 and R = -(2)1/2/32 for p = 1, in 0 which is the classical limit. This means that a quantum statistical character is left in R, in the classical limit. Also, a correlation between the sign of R and a quantum mechanical exchange effect is recognized for 0 and >>1. Furthermore, we obtain results that support the instability interpretation of R proposed by Janyszek and Mrugala.

Memory effects on scale-free dynamics in foraging Drosophila

The fruit fly, Drosophila melanogaster, displays a scale-free behavior in foraging, i.e., the dwell time on food exhibits a power law distribution. The scaling exponent is generally believed to be stable and the significance of the exponent itself with respect to the scale-free behavior remains elusive. We propose a model whereby the scaling exponent of the scale-free behavior of an animal depends on the memory of the individual. The proposed model is based on the premise that animal behaviors are associated with internal states of the animal. The changes in the scaling exponent are derived by considering losing memory as increasing uncertainty, which is expressed in terms of information entropy of the internal states. Predicted model behaviors agree with experimental results of foraging behavior in wild-type and learning/memory Drosophila mutants. The concept of changes in the scaling exponent due to the amount of memory provides a novel insight into the emergence of a scale-free behavior and the meaning of the scaling exponent.

Path integrals for Fokker-Planck equation described by generalized random walks

Path integral representations for Fokker-Planck (FP) equations, described by the random walks (RW) and the generalized random walks (GRW), are given. The GRW is a generalized one from the usual random walks to study non-linear, non-equilibrium processes. The GRW includes some memory effects and couplings through the jumping probabilities. To derive the path integrals of the processes, a transformation of probability, scalings of site (space) and step (time) are performed on the GRW. By a function in the exponent of the path integrals for the FP equation obtained by the RW or the GRW, a “Lagrangian” giving most probable path is introduced. From the Lagrangian, an “effective Hamiltonian” is deduced.

Fractional Brownian motions described by scaled Langevin equation

Abstract An ideal system is considered in simulating the dynamics of a random process. The system is composed of a set of clusters, where a cluster is an aggregation of elements activated in a random manner. The time evolution of each cluster is described by the Langevin equation, which characterizes a family of the Brownian motion. A scaling rule is introduced to the set of the Langevin equations in order to model the complexity of the random system. The response of this system is considered as an illustration of the fractional Brownian motion. Fractal dimension D defined by the scaling constant is related to the Hurst exponent H empirically introduced to specify the fractional Brownian motion as 1 - D = 2H - 2 (D > 0). Solutions of the scaled Langevin equation constitute an orthonormal system of functions. The theory can be developed to describe a more complicated system in which local clusters are characterized by plural scaling rules. A special case is considered where scaling parameters are defined by complex numbers. A complex fractional Brownian motion. The complex Brownian motion in this study is characterized by a complex fractal dimension defined by the scaling parameter. The complex system shows the limit-cycle behavior of the Brownian motion and the instability of the Brownian motion.

Dynamical process of complex systems and fractional differential equations

Behavior of dynamical process of complex systems is investigated. Specifically we analyse two types of ideal complex systems. For analysing the ideal complex systems, we define the response functions describing the internal states to an external force. The internal states are obtained as a relaxation process showing a “power law” distribution, such as scale free behaviors observed in actual measurements. By introducing a hybrid system, the logarithmic time, and double logarithmic time, we show how the “slow relaxation” (SR) process and “super slow relaxation” (SSR) process occur. Regarding the irregular variations of the internal states as an activation process, we calculate the response function to the external force. The behaviors are classified into “power”, “exponential”, and “stretched exponential” type. Finally we construct a fractional differential equation (FDE) describing the time evolution of these complex systems. In our theory, the exponent of the FDE or that of the power law distribution is expressed in terms of the parameters characterizing the structure of the system.

Max-plus analysis on some binary particle systems

We are concerned with a special class of binary cellular automata, i.e. the so-called particle cellular automata (PCA) in this paper. We first propose max-plus expressions to PCA of four neighbors. Then, by utilizing basic operations of the max-plus algebra and appropriate transformations, PCA4-1, 4-2 and 4-3 are solved exactly and their general solutions are found in terms of max-plus expressions. Finally, we analyze the asymptotic behaviors of general solutions and prove the fundamental diagrams exactly.

Scaled Langevin equation for complex systems: New linear scaling relation for weight factor

Abstract A set of scaled Langevin equations is proposed to study a long time tail of correlation functions for two model systems (Type I and Type II). Each system is composed of elements which are grouped into clusters according to dynamical activations for external forces. The clusters in Type I are characterized by linear scaling rules in repetitive operations, whereas the clusters in Type II are characterized by different kinds of linear scaling rules in the repetitive operations. We obtain a new linear scaling relation for a weight factor for the clusters in Type II, as a condition for the correlation function to behave a long time tail.

Solution of the Fokker-Planck equation with spatial coordinate-dependent moments

The solution of the Fokker-Planck equation with spatial coordinate-dependent moments is given in the form of the path integral, involving the conditional probability expressed in terms of the moments at the “postpoint”. It is easily seen that it satisfies the Fokker-Planck equation.