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 ($\xi_i^{\mu}=\pm 1$), 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 $p$ of patterns are embedded via asymmetric Hebbian connections, namely, $p/N \to 0$ for the number of neuron $N \to \infty$ ('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 $[0,2\pi)$ 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.