332 research outputs found
Phase transitions driven by L\'evy stable noise: exact solutions and stability analysis of nonlinear fractional Fokker-Planck equations
Phase transitions and effects of external noise on many body systems are one
of the main topics in physics. In mean field coupled nonlinear dynamical
stochastic systems driven by Brownian noise, various types of phase transitions
including nonequilibrium ones may appear. A Brownian motion is a special case
of L\'evy motion and the stochastic process based on the latter is an
alternative choice for studying cooperative phenomena in various fields.
Recently, fractional Fokker-Planck equations associated with L\'evy noise have
attracted much attention and behaviors of systems with double-well potential
subjected to L\'evy noise have been studied intensively. However, most of such
studies have resorted to numerical computation. We construct an {\it
analytically solvable model} to study the occurrence of phase transitions
driven by L\'evy stable noise.Comment: submitted to EP
Thouless-Anderson-Palmer equation for analog neural network with temporally fluctuating white synaptic noise
Effects of synaptic noise on the retrieval process of associative memory
neural networks are studied from the viewpoint of neurobiological and
biophysical understanding of information processing in the brain. We
investigate the statistical mechanical properties of stochastic analog neural
networks with temporally fluctuating synaptic noise, which is assumed to be
white noise. Such networks, in general, defy the use of the replica method,
since they have no energy concept. The self-consistent signal-to-noise analysis
(SCSNA), which is an alternative to the replica method for deriving a set of
order parameter equations, requires no energy concept and thus becomes
available in studying networks without energy functions. Applying the SCSNA to
stochastic network requires the knowledge of the Thouless-Anderson-Palmer (TAP)
equation which defines the deterministic networks equivalent to the original
stochastic ones. The study of the TAP equation which is of particular interest
for the case without energy concept is very few, while it is closely related to
the SCSNA in the case with energy concept. This paper aims to derive the TAP
equation for networks with synaptic noise together with a set of order
parameter equations by a hybrid use of the cavity method and the SCSNA.Comment: 13 pages, 3 figure
Low-dimensional chaos induced by frustration in a non-monotonic system
We report a novel mechanism for the occurrence of chaos at the macroscopic
level induced by the frustration of interaction, namely frustration-induced
chaos, in a non-monotonic sequential associative memory model. We succeed in
deriving exact macroscopic dynamical equations from the microscopic dynamics in
the case of the thermodynamic limit and prove that two order parameters
dominate this large-degree-of-freedom system. Two-parameter bifurcation
diagrams are obtained from the order-parameter equations. Then we analytically
show that the chaos is low-dimensional at the macroscopic level when the system
has some degree of frustration, but that the chaos definitely does not occur
without the frustration.Comment: 2 figure
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.
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
Dynamical Probability Distribution Function of the SK Model at High Temperatures
The microscopic probability distribution function of the
Sherrington-Kirkpatrick (SK) model of spin glasses is calculated explicitly as
a function of time by a high-temperature expansion. The resulting formula to
the third order of the inverse temperature shows that an assumption made by
Coolen, Laughton and Sherrington in their recent theory of dynamics is
violated. Deviations of their theory from exact results are estimated
quantitatively. Our formula also yields explicit expressions of the time
dependence of various macroscopic physical quantities when the temperature is
suddenly changed within the high-temperature region.Comment: LaTeX, 6 pages, Figures upon request (here revised), To be published
in J. Phys. Soc. Jpn. 65 (1996) No.
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
Controlling chaos in diluted networks with continuous neurons
Diluted neural networks with continuous neurons and nonmonotonic transfer
function are studied, with both fixed and dynamic synapses. A noisy stimulus
with periodic variance results in a mechanism for controlling chaos in neural
systems with fixed synapses: a proper amount of external perturbation forces
the system to behave periodically with the same period as the stimulus.Comment: 11 pages, 8 figure
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.
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