Dynamical properties of the Landau-Ginzburg model with long-range correlated quenched impurities

We investigate the critical dynamics of the time-dependent Landau-Ginzburg model with non conserved n-component order parameter (Model A) in the presence of long-range correlated quenched impurities. We use a special kind of long-range correlations, previously introduced by Weinrib and Halperin. Using a double expansion in \epsilon and \delta we calculate the critical exponent z up to second order on the small parameters. We show that the quenched impurities of this kind affect the critical dynamics already in first order of \epsilon and \delta, leading to a relevant correction for the mean field value of the exponent zComment: 7 pages, REVTEX, to be published in Phys. Rev.

A perturbative approach to non-linearities in the information carried by a two layer neural network

We evaluate the mutual information between the input and the output of a two layer network in the case of a noisy and non-linear analogue channel. In the case where the non-linearity is small with respect to the variability in the noise, we derive an exact expression for the contribution to the mutual information given by the non-linear term in first order of perturbation theory. Finally we show how the calculation can be simplified by means of a diagrammatic expansion. Our results suggest that the use of perturbation theories applied to neural systems might give an insight on the contribution of non-linearities to the information transmission and in general to the neuronal dynamics.Comment: Accepted as a preprint of ICTP, Triest

Time evolution of the extremely diluted Blume-Emery-Griffiths neural network

The time evolution of the extremely diluted Blume-Emery-Griffiths neural network model is studied, and a detailed equilibrium phase diagram is obtained exhibiting pattern retrieval, fluctuation retrieval and self-sustained activity phases. It is shown that saddle-point solutions associated with fluctuation overlaps slow down considerably the flow of the network states towards the retrieval fixed points. A comparison of the performance with other three-state networks is also presented.Comment: 8 pages, 5 figure

Bump formation in a binary attractor neural network

This paper investigates the conditions for the formation of local bumps in the activity of binary attractor neural networks with spatially dependent connectivity. We show that these formations are observed when asymmetry between the activity during the retrieval and learning is imposed. Analytical approximation for the order parameters is derived. The corresponding phase diagram shows a relatively large and stable region, where this effect is observed, although the critical storage and the information capacities drastically decrease inside that region. We demonstrate that the stability of the network, when starting from the bump formation, is larger than the stability when starting even from the whole pattern. Finally, we show a very good agreement between the analytical results and the simulations performed for different topologies of the network.Comment: about 14 page

Statistical mechanics of the multi-constraint continuous knapsack problem

We apply the replica analysis established by Gardner to the multi-constraint continuous knapsack problem,which is one of the linear programming problems and a most fundamental problem in the field of operations research (OR). For a large problem size, we analyse the space of solution and its volume, and estimate the optimal number of items to go into the knapsack as a function of the number of constraints. We study the stability of the replica symmetric (RS) solution and find that the RS calculation cannot estimate the optimal number of items in knapsack correctly if many constraints are required.In order to obtain a consistent solution in the RS region,we try the zero entropy approximation for this continuous solution space and get a stable solution within the RS ansatz.On the other hand, in replica symmetry breaking (RSB) region, the one step RSB solution is found by Parisi's scheme. It turns out that this problem is closely related to the problem of optimal storage capacity and of generalization by maximum stability rule of a spherical perceptron.Comment: Latex 13 pages using IOP style file, 5 figure