196 research outputs found
Fractional Dynamics and Multi-Slide Model of Human Memory
We propose a single chunk model of long-term memory that combines the basic
features of the ACT-R theory and the multiple trace memory architecture. The
pivot point of the developed theory is a mathematical description of the
creation of new memory traces caused by learning a certain fragment of
information pattern and affected by the fragments of this pattern already
retained by the current moment of time. Using the available psychological and
physiological data these constructions are justified. The final equation
governing the learning and forgetting processes is constructed in the form of
the differential equation with the Caputo type fractional time derivative.
Several characteristic situations of the learning (continuous and
discontinuous) and forgetting processes are studied numerically. In particular,
it is demonstrated that, first, the "learning" and "forgetting" exponents of
the corresponding power laws of the memory fractional dynamics should be
regarded as independent system parameters. Second, as far as the spacing
effects are concerned, the longer the discontinuous learning process, the
longer the time interval within which a subject remembers the information
without its considerable lost. Besides, the latter relationship is a linear
proportionality.Comment: Submitted to 36th Annual Conference of the Cognitive Science Society,
Quebec City, Canada, July 23-26 201
Equivalent continuous and discrete realizations of Levy flights: Model of one-dimensional motion of inertial particle
The paper is devoted to the relationship between the continuous Markovian
description of Levy flights developed previously and their equivalent
representation in terms of discrete steps of a wandering particle, a certain
generalization of continuous time random walks. Our consideration is confined
to the one-dimensional model for continuous random motion of a particle with
inertia. Its dynamics governed by stochastic self-acceleration is described as
motion on the phase plane {x,v} comprising the position x and velocity v=dx/dt
of the given particle. A notion of random walks inside a certain neighbourhood
L of the line v=0 (the x-axis) and outside it is developed. It enables us to
represent a continuous trajectory of particle motion on the plane {x,v} as a
collection of the corresponding discrete steps. Each of these steps matches one
complete fragment of the velocity fluctuations originating and terminating at
the "boundary" of L. As demonstrated, the characteristic length of particle
spatial displacement is mainly determined by velocity fluctuations with large
amplitude, which endows the derived random walks along the x-axis with the
characteristic properties of Levy flights. Using the developed classification
of random trajectories a certain parameter-free core stochastic process is
constructed. Its peculiarity is that all the characteristics of Levy flights
similar to the exponent of the Levy scaling law are no more than the parameters
of the corresponding transformation from the particle velocity v to the related
variable of the core process. In this way the previously found validity of the
continuous Markovian model for all the regimes of Levy flights is explained
Long-lived states of oscillator chain with dynamical traps
A simple model of oscillator chain with dynamical traps and additive white
noise is considered. Its dynamics was studied numerically. As demonstrated,
when the trap effect is pronounced nonequilibrium phase transitions of a new
type arise. Locally they manifest themselves via distortion of the particle
arrangement symmetry. Depending on the system parameters the particle
arrangement is characterized by the corresponding distributions taking either a
bimodal form, or twoscale one, or unimodal onescale form which, however,
deviates substantially from the Gaussian distribution. The individual particle
velocities exhibit also a number of anomalies, in particular, their
distribution can be extremely wide or take a quasi-cusp form. A large number of
different cooperative structures and superstructures made of these formations
are found in the visualized time patterns. Their evolution is, in some sense,
independent of the individual particle dynamics, enabling us to regard them as
dynamical phases.Comment: 8 pages, 5 figurs, TeX style of European Physical Journa
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