Transient dynamics for sequence processing neural networks: effect of degree distributions

We derive a analytic evolution equation for overlap parameters including the effect of degree distribution on the transient dynamics of sequence processing neural networks. In the special case of globally coupled networks, the precisely retrieved critical loading ratio $\alpha_c = N ^{-1/2}$ is obtained, where $N$ is the network size. In the presence of random networks, our theoretical predictions agree quantitatively with the numerical experiments for delta, binomial, and power-law degree distributions.Comment: 11 pages, 6 figure

Synchronization and directed percolation in coupled map lattices

We study a synchronization mechanism, based on one-way coupling of all-or-nothing type, applied to coupled map lattices with several different local rules. By analyzing the metric and the topological distance between the two systems, we found two different regimes: a strong chaos phase in which the transition has a directed percolation character and a weak chaos phase in which the synchronization transition occurs abruptly. We are able to derive some analytical approximations for the location of the transition point and the critical properties of the system. We propose to use the characteristics of this transition as indicators of the spatial propagation of chaoticity.Comment: 12 pages + 12 figure

Fractal Dimensions of Confined Clusters in Two-Dimensional Directed Percolation

The fractal structure of directed percolation clusters, grown at the percolation threshold inside parabolic-like systems, is studied in two dimensions via Monte Carlo simulations. With a free surface at y=\pm Cx^k and a dynamical exponent z, the surface shape is a relevant perturbation when k<1/z and the fractal dimensions of the anisotropic clusters vary continuously with k. Analytic expressions for these variations are obtained using a blob picture approach.Comment: 6 pages, Plain TeX file, epsf, 3 postscript-figure

Secure exchange of information by synchronization of neural networks

A connection between the theory of neural networks and cryptography is presented. A new phenomenon, namely synchronization of neural networks is leading to a new method of exchange of secret messages. Numerical simulations show that two artificial networks being trained by Hebbian learning rule on their mutual outputs develop an antiparallel state of their synaptic weights. The synchronized weights are used to construct an ephemeral key exchange protocol for a secure transmission of secret data. It is shown that an opponent who knows the protocol and all details of any transmission of the data has no chance to decrypt the secret message, since tracking the weights is a hard problem compared to synchronization. The complexity of the generation of the secure channel is linear with the size of the network.Comment: 11 pages, 5 figure

Training a perceptron in a discrete weight space

On-line and batch learning of a perceptron in a discrete weight space, where each weight can take $2 L+1$ different values, are examined analytically and numerically. The learning algorithm is based on the training of the continuous perceptron and prediction following the clipped weights. The learning is described by a new set of order parameters, composed of the overlaps between the teacher and the continuous/clipped students. Different scenarios are examined among them on-line learning with discrete/continuous transfer functions and off-line Hebb learning. The generalization error of the clipped weights decays asymptotically as $exp(-K \alpha^2)$/$exp(-e^{|\lambda| \alpha})$ in the case of on-line learning with binary/continuous activation functions, respectively, where $\alpha$ is the number of examples divided by N, the size of the input vector and $K$ is a positive constant that decays linearly with 1/L. For finite $N$ and $L$, a perfect agreement between the discrete student and the teacher is obtained for $\alpha \propto \sqrt{L \ln(NL)}$. A crossover to the generalization error $\propto 1/\alpha$, characterized continuous weights with binary output, is obtained for synaptic depth $L > O(\sqrt{N})$.Comment: 10 pages, 5 figs., submitted to PR

Nonlocal mechanism for cluster synchronization in neural circuits

The interplay between the topology of cortical circuits and synchronized activity modes in distinct cortical areas is a key enigma in neuroscience. We present a new nonlocal mechanism governing the periodic activity mode: the greatest common divisor (GCD) of network loops. For a stimulus to one node, the network splits into GCD-clusters in which cluster neurons are in zero-lag synchronization. For complex external stimuli, the number of clusters can be any common divisor. The synchronized mode and the transients to synchronization pinpoint the type of external stimuli. The findings, supported by an information mixing argument and simulations of Hodgkin Huxley population dynamic networks with unidirectional connectivity and synaptic noise, call for reexamining sources of correlated activity in cortex and shorter information processing time scales.Comment: 8 pges, 6 figure

On the Use of Finite-Size Scaling to Measure Spin-Glass Exponents

Finite-size scaling (FSS) is a standard technique for measuring scaling exponents in spin glasses. Here we present a critique of this approach, emphasizing the need for all length scales to be large compared to microscopic scales. In particular we show that the replacement, in FSS analyses, of the correlation length by its asymptotic scaling form can lead to apparently good scaling collapses with the wrong values of the scaling exponents.Comment: RevTeX, 5 page

A lattice gas model of II-VI(001) semiconductor surfaces

We introduce an anisotropic two-dimensional lattice gas model of metal terminated II-IV(001) seminconductor surfaces. Important properties of this class of materials are represented by effective NN and NNN interactions, which result in the competition of two vacancy structures on the surface. We demonstrate that the experimentally observed c(2x2)-(2x1) transition of the CdTe(001) surface can be understood as a phase transition in thermal equilbrium. The model is studied by means of transfer matrix and Monte Carlo techniques. The analysis shows that the small energy difference of the competing reconstructions determines to a large extent the nature of the different phases. Possible implications for further experimental research are discussed.Comment: 7 pages, 2 figure

Dynamical Replica Theory for Disordered Spin Systems

We present a new method to solve the dynamics of disordered spin systems on finite time-scales. It involves a closed driven diffusion equation for the joint spin-field distribution, with time-dependent coefficients described by a dynamical replica theory which, in the case of detailed balance, incorporates equilibrium replica theory as a stationary state. The theory is exact in various limits. We apply our theory to both the symmetric- and the non-symmetric Sherrington-Kirkpatrick spin-glass, and show that it describes the (numerical) experiments very well.Comment: 7 pages RevTex, 4 figures, for PR