Spectra of sparse regular graphs with loops.

We derive exact equations that determine the spectra of undirected and directed sparsely connected regular graphs containing loops of arbitrary lengths. The implications of our results for the structural and dynamical properties of network models are discussed by showing how loops influence the size of the spectral gap and the propensity for synchronization. Analytical formulas for the spectrum are obtained for specific lengths of the loops.

Localization transition in symmetric random matrices.

We study the behavior of the inverse participation ratio and the localization transition in infinitely large random matrices through the cavity method. Results are shown for two ensembles of random matrices: Laplacian matrices on sparse random graphs and fully connected Lévy matrices. We derive a critical line separating localized from extended states in the case of Lévy matrices. Comparison between theoretical results and diagonalization of finite random matrices is shown.

Small-world hypergraphs on a bond-disordered Bethe lattice.

We study the thermodynamic properties of spin systems with bond-disorder on small-world hypergraphs, obtained by superimposing a one-dimensional Ising chain onto a random Bethe graph with p-spin interactions. Using transfer-matrix techniques, we derive fixed-point equations describing the relevant order parameters and the free energy, both in the replica symmetric and one step replica symmetry breaking approximation. We determine the static and dynamic ferromagnetic transition and the spinglass transition within replica symmetry for all temperatures, and demonstrate corrections to these results when one step replica symmetry breaking is taken into account. The results obtained are in agreement with Monte-Carlo simulations.

Adaptive thresholds for neural networks with synaptic noise.

The inclusion of a macroscopic adaptive threshold is studied for the retrieval dynamics of both layered feedforward and fully connected neural network models with synaptic noise. These two types of architectures require a different method to be solved numerically. In both cases it is shown that, if the threshold is chosen appropriately as a function of the cross-talk noise and of the activity of the stored patterns, adapting itself automatically in the course of the recall process, an autonomous functioning of the network is guaranteed. This self-control mechanism considerably improves the quality of retrieval, in particular the storage capacity, the basins of attraction and the mutual information content.

Metastable configurations of small-world networks.

We calculate the number of metastable configurations of Ising small-world networks that are constructed upon superimposing sparse Poisson random graphs onto a one-dimensional chain. Our solution is based on replicated transfer-matrix techniques. We examine the denegeracy of the ground state and find a jump in the entropy of metastable configurations exactly at the crossover between the small-world and the Poisson random graph structures. We also examine the difference in entropy between metastable and all possible configurations, for both ferromagnetic and bond-disordered long-range couplings.

Thermodynamics of spin systems on small-world hypergraphs.

We study the thermodynamic properties of spin systems on small-world hypergraphs, obtained by superimposing sparse Poisson random graphs with p -spin interactions onto a one-dimensional Ising chain with nearest-neighbor interactions. We use replica-symmetric transfer-matrix techniques to derive a set of fixed-point equations describing the relevant order parameters and free energy, and solve them employing population dynamics. In the special case where the number of connections per site is of the order of the system size, we are able to solve the model analytically. In the more general case where the number of connections is finite, we determine the static and dynamic ferromagnetic-paramagnetic transitions using population dynamics. The results are tested against Monte-Carlo simulations.

Optimal capacity of the Blume-Emery-Griffiths perceptron.

A Blume-Emery-Griffiths perceptron model is introduced and its optimal capacity is calculated within the replica-symmetric Gardner approach, as a function of the pattern activity and the embedding stability parameter. The stability of the replica-symmetric approximation is studied via the analog of the de Almeida-Thouless line. A comparison is made with other three-state perceptrons.

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

A study of the time evolution and a stability analysis of the phases in the extremely diluted Blume-Emery-Griffiths neural network model are shown to yield new phase diagrams in which fluctuation retrieval may drive pattern retrieval. It is shown that saddle-point solutions associated with fluctuation overlaps slow down the flow of the network states towards the retrieval fixed points. A comparison of the performance with other three-state networks is also presented.

Gap structure of the local field in symmetric Q Ising neural networks.

The time evolution of the local field in symmetric Q-Ising neural networks is studied for arbitrary Q. In particular, the structure of the noise and the appearance of gaps in the probability distribution are discussed. Results are presented for several values of Q and compared with numerical simulations.

Equivalence of the Ashkin-Teller and the four-state Potts-glass models of neural networks.

We show that for a particular choice of the coupling parameters the Ashkin-Teller spin-glass neural network model with the Hebb learning rule and one condensed pattern yields the same thermodynamic properties as the four-state anisotropic Potts-glass neural network model. This equivalence is not seen at the level of the Hamiltonians.

Optimal colored perceptrons.

Ashkin-Teller type perceptron models are introduced. Their maximal capacity per number of couplings is calculated within a first-step replica-symmetry-breaking Gardner approach. The results are compared with extensive numerical simulations using several algorithms.

Correlated patterns in nonmonotonic graded-response perceptrons

The optimal capacity of graded-response perceptrons storing biased and spatially correlated patterns with nonmonotonic input-output relations is studied. It is shown that only the structure of the output patterns is important for the overall performance of the perceptrons.

Mutual information of sparsely coded associative memory with self-control and ternary neurons.

The influence of a macroscopic time-dependent threshold on the retrieval dynamics of attractor associative memory models with ternary neurons ¿-1, 0, +1¿ is examined. If the threshold is chosen appropriately as a function of the cross-talk noise and of the activity of the memorized patterns in the model, adapting itself in the course of the time evolution, it guarantees an autonomous functioning of the model. Especially in the limit of sparse coding, it is found that this self-control mechanism considerably improves the quality of the fixed-point retrieval dynamics, in particular the storage capacity, the basins of attraction and the information content. The mutual information is shown to be the relevant parameter to study the retrieval quality of such sparsely coded models. Numerical results confirm these observations.

H19 overexpression in breast adenocarcinoma stromal cells is associated with tumor values and steroid receptor status but independent of p53 and Ki-67 expression.

In a previous study we described the expression of the H19 gene by in situ hybridization (ISH) in normal breast and in benign or malignant breast tumors (Dugimont T, Curgy JJ, Wernert N, Delobelle A, Raes MB, Joubel A, Stehelin D, Coll J: Biol Cell 1995, 85:117-124). In the present work, 1) we extend the previous one to a statistically useful number of adenocarcinomas, including 10 subclasses, 2) we provide information on the precise ISH localization of the H19 RNA by using, on serial tissue sections, antibodies delineating specifically the stromal or the epithelial component of the breast, and 3) we consider relationships between the H19 gene expression and various clinicopathological information as tumor values (T0 to T4), grades, steroid receptors, lymph node status, and molecular features as the p53 gene product and the Ki-67/MIB1 protein, which is specific to proliferating cells. Data indicate that 1) in 72.5% of studied breast adenocarcinomas an overall H19 gene expression is increased when compared with healthy tissues, 2) the H19 gene is generally overexpressed in stromal cells (92.2%) and rarely in epithelial cells (2.9% only), 3) an up-regulation of the H19 gene is significantly correlated with the tumor values and the presence of both estrogen and progesterone receptors, and 4) at the cellular level, the H19 gene demonstrates an independent expression versus accumulation of both the p53 protein and the Ki-67/MIB-1 cell-cycle marker.

Optimal nonlinear training in the multi-class proximity problem.

Using a signal-to-noise analysis, the effects of nonlinear modulation of the Hebbian learning rule in the multi-class proximity problem are investigated. Both random classification and classification provided by a Gaussian and a binary teacher are treated. Analytic expressions are derived for the learning and generalization rates around an old and a new prototype. For the proximity problem with binary inputs but Q'-state outputs, it is shown that the optimal modulation is a combination of a hyperbolic tangent and a linear function. As an illustration, numerical results are presented for the two-class and the Q' = 3 multi-class problem.