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

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

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

Crossover Scaling Functions in One Dimensional Dynamic Growth Models

The crossover from Edwards-Wilkinson ($s=0$) to KPZ ($s>0$) type growth is studied for the BCSOS model. We calculate the exact numerical values for the $k=0$ and $2\pi/N$ massgap for $N\leq 18$ using the master equation. We predict the structure of the crossover scaling function and confirm numerically that $m_0\simeq 4 (\pi/N)^2 [1+3u^2(s) N/(2\pi^2)]^{0.5}$ and $m_1\simeq 2 (\pi/N)^2 [1+ u^2(s) N/\pi^2]^{0.5}$, with $u(1)=1.03596967$. KPZ type growth is equivalent to a phase transition in meso-scopic metallic rings where attractive interactions destroy the persistent current; and to endpoints of facet-ridges in equilibrium crystal shapes.Comment: 11 pages, TeX, figures upon reques

Derivation of Hebb's rule

On the basis of the general form for the energy needed to adapt the connection strengths of a network in which learning takes place, a local learning rule is found for the changes of the weights. This biologically realizable learning rule turns out to comply with Hebb's neuro-physiological postulate, but is not of the form of any of the learning rules proposed in the literature. It is shown that, if a finite set of the same patterns is presented over and over again to the network, the weights of the synapses converge to finite values. Furthermore, it is proved that the final values found in this biologically realizable limit are the same as those found via a mathematical approach to the problem of finding the weights of a partially connected neural network that can store a collection of patterns. The mathematical solution is obtained via a modified version of the so-called method of the pseudo-inverse, and has the inverse of a reduced correlation matrix, rather than the usual correlation matrix, as its basic ingredient. Thus, a biological network might realize the final results of the mathematician by the energetically economic rule for the adaption of the synapses found in this article.Comment: 29 pages, LaTeX, 3 figure

Analysis of CDMA systems that are characterized by eigenvalue spectrum

An approach by which to analyze the performance of the code division multiple access (CDMA) scheme, which is a core technology used in modern wireless communication systems, is provided. The approach characterizes the objective system by the eigenvalue spectrum of a cross-correlation matrix composed of signature sequences used in CDMA communication, which enables us to handle a wider class of CDMA systems beyond the basic model reported by Tanaka. The utility of the novel scheme is shown by analyzing a system in which the generation of signature sequences is designed for enhancing the orthogonality.Comment: 7 pages, 2 figure

The triangular Ising antiferromagnet in a staggered field

We study the equilibrium properties of the nearest-neighbor Ising antiferromagnet on a triangular lattice in the presence of a staggered field conjugate to one of the degenerate ground states. Using a mapping of the ground states of the model without the staggered field to dimer coverings on the dual lattice, we classify the ground states into sectors specified by the number of ``strings''. We show that the effect of the staggered field is to generate long-range interactions between strings. In the limiting case of the antiferromagnetic coupling constant J becoming infinitely large, we prove the existence of a phase transition in this system and obtain a finite lower bound for the transition temperature. For finite J, we study the equilibrium properties of the system using Monte Carlo simulations with three different dynamics. We find that in all the three cases, equilibration times for low field values increase rapidly with system size at low temperatures. Due to this difficulty in equilibrating sufficiently large systems at low temperatures, our finite-size scaling analysis of the numerical results does not permit a definite conclusion about the existence of a phase transition for finite values of J. A surprising feature in the system is the fact that unlike usual glassy systems, a zero-temperature quench almost always leads to the ground state, while a slow cooling does not.Comment: 12 pages, 18 figures: To appear in Phys. Rev.