The path-integral analysis of an associative memory model storing an infinite number of finite limit cycles

It is shown that an exact solution of the transient dynamics of an associative memory model storing an infinite number of limit cycles with l finite steps by means of the path-integral analysis. Assuming the Maxwell construction ansatz, we have succeeded in deriving the stationary state equations of the order parameters from the macroscopic recursive equations with respect to the finite-step sequence processing model which has retarded self-interactions. We have also derived the stationary state equations by means of the signal-to-noise analysis (SCSNA). The signal-to-noise analysis must assume that crosstalk noise of an input to spins obeys a Gaussian distribution. On the other hand, the path-integral method does not require such a Gaussian approximation of crosstalk noise. We have found that both the signal-to-noise analysis and the path-integral analysis give the completely same result with respect to the stationary state in the case where the dynamics is deterministic, when we assume the Maxwell construction ansatz. We have shown the dependence of storage capacity (alpha_c) on the number of patterns per one limit cycle (l). Storage capacity monotonously increases with the number of steps, and converges to alpha_c=0.269 at l ~= 10. The original properties of the finite-step sequence processing model appear as long as the number of steps of the limit cycle has order l=O(1).Comment: 24 pages, 3 figure

Generating functional analysis of complex formation and dissociation in large protein interaction networks

We analyze large systems of interacting proteins, using techniques from the non-equilibrium statistical mechanics of disordered many-particle systems. Apart from protein production and removal, the most relevant microscopic processes in the proteome are complex formation and dissociation, and the microscopic degrees of freedom are the evolving concentrations of unbound proteins (in multiple post-translational states) and of protein complexes. Here we only include dimer-complexes, for mathematical simplicity, and we draw the network that describes which proteins are reaction partners from an ensemble of random graphs with an arbitrary degree distribution. We show how generating functional analysis methods can be used successfully to derive closed equations for dynamical order parameters, representing an exact macroscopic description of the complex formation and dissociation dynamics in the infinite system limit. We end this paper with a discussion of the possible routes towards solving the nontrivial order parameter equations, either exactly (in specific limits) or approximately.Comment: 14 pages, to be published in Proc of IW-SMI-2009 in Kyoto (Journal of Phys Conference Series

Coupling Unifications in Gauge-Higgs Unified Orbifold Models

Supersymmetric gauge theories, in higher dimensions compactified in an orbifold, give a natural framework to unify the gauge bosons, Higgs fields and even the matter fields in a single multiplet of the unifying gauge symmetry. The extra dimensions and the supersymmetry are the two key ingredients for such an unification. In this work, we investigate various scenarios for the unification of the three gauge couplings, and the Yukawa couplings in the Minimal Supersymmetric Standard Model (MSSM), as well as the trilinear Higgs couplings \lambda and \kappa of the Non-Minimal Supersymmetric Standard Model (NMSSM). We present an SU(8) model in six dimensions with N=2 supersymmetry, compactified in a T^2/Z_6 orbifold which unifies the three gauge couplings with \lambda and \kappa of NMSSM. Then, we present an SU(9) model in 6D, which, in addition, includes partial unification of Yukawa couplings, either for the up-type (top quark and Dirac tau-neutrino) or down-type (bottom quark and tau lepton). We also study the phenomenological implications of these various unification scenarios using the appropriate renormalization group equations, and show that such unification works very well with the measured low energy values of the couplings. The predicted upper bounds for the lightest neutral Higgs boson mass in our model is higher than those in MSSM, but lower that those in the general NMSSM (where the couplings \lambda and \kappa are arbitrary). Some of the predictions of our models can be tested in the upcoming Large Hadron Collider.Comment: 29 pages, 4 figure

Linear Complexity Lossy Compressor for Binary Redundant Memoryless Sources

A lossy compression algorithm for binary redundant memoryless sources is presented. The proposed scheme is based on sparse graph codes. By introducing a nonlinear function, redundant memoryless sequences can be compressed. We propose a linear complexity compressor based on the extended belief propagation, into which an inertia term is heuristically introduced, and show that it has near-optimal performance for moderate block lengths.Comment: 4 pages, 1 figur

Synapse efficiency diverges due to synaptic pruning following over-growth

In the development of the brain, it is known that synapses are pruned following over-growth. This pruning following over-growth seems to be a universal phenomenon that occurs in almost all areas -- visual cortex, motor area, association area, and so on. It has been shown numerically that the synapse efficiency is increased by systematic deletion. We discuss the synapse efficiency to evaluate the effect of pruning following over-growth, and analytically show that the synapse efficiency diverges as O(log c) at the limit where connecting rate c is extremely small. Under a fixed synapse number criterion, the optimal connecting rate, which maximize memory performance, exists.Comment: 15 pages, 16 figure

Error correcting code using tree-like multilayer perceptron

An error correcting code using a tree-like multilayer perceptron is proposed. An original message \mbi{s}^0 is encoded into a codeword \boldmath{y}_0 using a tree-like committee machine (committee tree) or a tree-like parity machine (parity tree). Based on these architectures, several schemes featuring monotonic or non-monotonic units are introduced. The codeword \mbi{y}_0 is then transmitted via a Binary Asymmetric Channel (BAC) where it is corrupted by noise. The analytical performance of these schemes is investigated using the replica method of statistical mechanics. Under some specific conditions, some of the proposed schemes are shown to saturate the Shannon bound at the infinite codeword length limit. The influence of the monotonicity of the units on the performance is also discussed.Comment: 23 pages, 3 figures, Content has been extended and revise

The Cavity Approach to Parallel Dynamics of Ising Spins on a Graph

We use the cavity method to study parallel dynamics of disordered Ising models on a graph. In particular, we derive a set of recursive equations in single site probabilities of paths propagating along the edges of the graph. These equations are analogous to the cavity equations for equilibrium models and are exact on a tree. On graphs with exclusively directed edges we find an exact expression for the stationary distribution of the spins. We present the phase diagrams for an Ising model on an asymmetric Bethe lattice and for a neural network with Hebbian interactions on an asymmetric scale-free graph. For graphs with a nonzero fraction of symmetric edges the equations can be solved for a finite number of time steps. Theoretical predictions are confirmed by simulation results. Using a heuristic method, the cavity equations are extended to a set of equations that determine the marginals of the stationary distribution of Ising models on graphs with a nonzero fraction of symmetric edges. The results of this method are discussed and compared with simulations

Parallel dynamics of continuous Hopfield model revisited

We have applied the generating functional analysis (GFA) to the continuous Hopfield model. We have also confirmed that the GFA predictions in some typical cases exhibit good consistency with computer simulation results. When a retarded self-interaction term is omitted, the GFA result becomes identical to that obtained using the statistical neurodynamics as well as the case of the sequential binary Hopfield model.Comment: 4 pages, 2 figure

Dynamical replica theoretic analysis of CDMA detection dynamics

We investigate the detection dynamics of the Gibbs sampler for code-division multiple access (CDMA) multiuser detection. Our approach is based upon dynamical replica theory which allows an analytic approximation to the dynamics. We use this tool to investigate the basins of attraction when phase coexistence occurs and examine its efficacy via comparison with Monte Carlo simulations.Comment: 18 pages, 2 figure

Bi-stability of mixed states in neural network storing hierarchical patterns

We discuss the properties of equilibrium states in an autoassociative memory model storing hierarchically correlated patterns (hereafter, hierarchical patterns). We will show that symmetric mixed states (hereafter, mixed states) are bi-stable on the associative memory model storing the hierarchical patterns in a region of the ferromagnetic phase. This means that the first-order transition occurs in this ferromagnetic phase. We treat these contents with a statistical mechanical method (SCSNA) and by computer simulation. Finally, we discuss a physiological implication of this model. Sugase et al. analyzed the time-course of the information carried by the firing of face-responsive neurons in the inferior temporal cortex. We also discuss the relation between the theoretical results and the physiological experiments of Sugase et al.Comment: 18 pages, 6 figure