176 research outputs found
Oscillator neural network model with distributed native frequencies
We study associative memory of an oscillator neural network with distributed
native frequencies. The model is based on the use of the Hebb learning rule
with random patterns (), and the distribution function of
native frequencies is assumed to be symmetric with respect to its average.
Although the system with an extensive number of stored patterns is not allowed
to get entirely synchronized, long time behaviors of the macroscopic order
parameters describing partial synchronization phenomena can be obtained by
discarding the contribution from the desynchronized part of the system. The
oscillator network is shown to work as associative memory accompanied by
synchronized oscillations. A phase diagram representing properties of memory
retrieval is presented in terms of the parameters characterizing the native
frequency distribution. Our analytical calculations based on the
self-consistent signal-to-noise analysis are shown to be in excellent agreement
with numerical simulations, confirming the validity of our theoretical
treatment.Comment: 9 pages, revtex, 6 postscript figures, to be published in J. Phys.
Pattern-recalling processes in quantum Hopfield networks far from saturation
As a mathematical model of associative memories, the Hopfield model was now
well-established and a lot of studies to reveal the pattern-recalling process
have been done from various different approaches. As well-known, a single
neuron is itself an uncertain, noisy unit with a finite unnegligible error in
the input-output relation. To model the situation artificially, a kind of 'heat
bath' that surrounds neurons is introduced. The heat bath, which is a source of
noise, is specified by the 'temperature'. Several studies concerning the
pattern-recalling processes of the Hopfield model governed by the
Glauber-dynamics at finite temperature were already reported. However, we might
extend the 'thermal noise' to the quantum-mechanical variant. In this paper, in
terms of the stochastic process of quantum-mechanical Markov chain Monte Carlo
method (the quantum MCMC), we analytically derive macroscopically deterministic
equations of order parameters such as 'overlap' in a quantum-mechanical variant
of the Hopfield neural networks (let us call "quantum Hopfield model" or
"quantum Hopfield networks"). For the case in which non-extensive number of
patterns are embedded via asymmetric Hebbian connections, namely,
for the number of neuron ('far from saturation'), we evaluate
the recalling processes for one of the built-in patterns under the influence of
quantum-mechanical noise.Comment: 10 pages, 3 figures, using jpconf.cls, Proc. of Statphys-Kolkata VI
Fluctuation Dissipation Relation for a Langevin Model with Multiplicative Noise
A random multiplicative process with additive noise is described by a
Langevin equation. We show that the fluctuation-dissipation relation is
satisfied in the Langevin model, if the noise strength is not so strong.Comment: 11 pages, 6 figures, other comment
An associative network with spatially organized connectivity
We investigate the properties of an autoassociative network of
threshold-linear units whose synaptic connectivity is spatially structured and
asymmetric. Since the methods of equilibrium statistical mechanics cannot be
applied to such a network due to the lack of a Hamiltonian, we approach the
problem through a signal-to-noise analysis, that we adapt to spatially
organized networks. The conditions are analyzed for the appearance of stable,
spatially non-uniform profiles of activity with large overlaps with one of the
stored patterns. It is also shown, with simulations and analytic results, that
the storage capacity does not decrease much when the connectivity of the
network becomes short range. In addition, the method used here enables us to
calculate exactly the storage capacity of a randomly connected network with
arbitrary degree of dilution.Comment: 27 pages, 6 figures; Accepted for publication in JSTA
Transient dynamics for sequence processing neural networks
An exact solution of the transient dynamics for a sequential associative
memory model is discussed through both the path-integral method and the
statistical neurodynamics. Although the path-integral method has the ability to
give an exact solution of the transient dynamics, only stationary properties
have been discussed for the sequential associative memory. We have succeeded in
deriving an exact macroscopic description of the transient dynamics by
analyzing the correlation of crosstalk noise. Surprisingly, the order parameter
equations of this exact solution are completely equivalent to those of the
statistical neurodynamics, which is an approximation theory that assumes
crosstalk noise to obey the Gaussian distribution. In order to examine our
theoretical findings, we numerically obtain cumulants of the crosstalk noise.
We verify that the third- and fourth-order cumulants are equal to zero, and
that the crosstalk noise is normally distributed even in the non-retrieval
case. We show that the results obtained by our theory agree with those obtained
by computer simulations. We have also found that the macroscopic unstable state
completely coincides with the separatrix.Comment: 21 pages, 4 figure
Thermodynamic properties of extremely diluted symmetric Q-Ising neural networks
Using the replica-symmetric mean-field theory approach the thermodynamic and
retrieval properties of extremely diluted {\it symmetric} -Ising neural
networks are studied. In particular, capacity-gain parameter and
capacity-temperature phase diagrams are derived for and .
The zero-temperature results are compared with those obtained from a study of
the dynamics of the model. Furthermore, the de Almeida-Thouless line is
determined. Where appropriate, the difference with other -Ising
architectures is outlined.Comment: 16 pages Latex including 6 eps-figures. Corrections, also in most of
the figures have been mad
Toward Generalized Entropy Composition with Different q Indices and H-Theorem
An attempt is made to construct composable composite entropy with different
indices of subsystems and address the H-theorem problem of the composite
system. Though the H-theorem does not hold in general situations, it is shown
that some composite entropies do not decrease in time in near-equilibrium
states and factorized states with negligibly weak interaction between the
subsystems.Comment: 25 pages, corrected some typos, to be published in J. Phys. Soc. Ja
Associative memory storing an extensive number of patterns based on a network of oscillators with distributed natural frequencies in the presence of external white noise
We study associative memory based on temporal coding in which successful
retrieval is realized as an entrainment in a network of simple phase
oscillators with distributed natural frequencies under the influence of white
noise. The memory patterns are assumed to be given by uniformly distributed
random numbers on so that the patterns encode the phase differences
of the oscillators. To derive the macroscopic order parameter equations for the
network with an extensive number of stored patterns, we introduce the effective
transfer function by assuming the fixed-point equation of the form of the TAP
equation, which describes the time-averaged output as a function of the
effective time-averaged local field. Properties of the networks associated with
synchronization phenomena for a discrete symmetric natural frequency
distribution with three frequency components are studied based on the order
parameter equations, and are shown to be in good agreement with the results of
numerical simulations. Two types of retrieval states are found to occur with
respect to the degree of synchronization, when the size of the width of the
natural frequency distribution is changed.Comment: published in Phys. Rev.
The path-integral analysis of an associative memory model storing an infinite number of finite limit cycles
It is shown that an exact solution of the transient dynamics of an
associative memory model storing an infinite number of limit cycles with l
finite steps by means of the path-integral analysis. Assuming the Maxwell
construction ansatz, we have succeeded in deriving the stationary state
equations of the order parameters from the macroscopic recursive equations with
respect to the finite-step sequence processing model which has retarded
self-interactions. We have also derived the stationary state equations by means
of the signal-to-noise analysis (SCSNA). The signal-to-noise analysis must
assume that crosstalk noise of an input to spins obeys a Gaussian distribution.
On the other hand, the path-integral method does not require such a Gaussian
approximation of crosstalk noise. We have found that both the signal-to-noise
analysis and the path-integral analysis give the completely same result with
respect to the stationary state in the case where the dynamics is
deterministic, when we assume the Maxwell construction ansatz.
We have shown the dependence of storage capacity (alpha_c) on the number of
patterns per one limit cycle (l). Storage capacity monotonously increases with
the number of steps, and converges to alpha_c=0.269 at l ~= 10. The original
properties of the finite-step sequence processing model appear as long as the
number of steps of the limit cycle has order l=O(1).Comment: 24 pages, 3 figure
First order phase transition in a nonequilibrium growth process
We introduce a simple continuous model for nonequilibrium surface growth. The
dynamics of the system is defined by the KPZ equation with a Morse-like
potential representing a short range interaction between the surface and the
substrate. The mean field solution displays a non trivial phase diagram with a
first order transition between a growing and a bound surface, associated with a
region of coexisting phases, and a tricritical point where the transition
becomes second order. Numerical simulations in 3 dimensions show quantitative
agreement with mean field results, and the features of the phase space are
preserved even in 2 dimensions.Comment: 7 figures, revtex, submitted to Phys. Rev.
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