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This is a rather bold attempt to bridge the gap between neuron structure and psychological data. We try to answer the question: Is there a relation between the neuronal connectivity in the human cortex (around 5,000) and the short-term memory capacity (7±2)? Our starting point is the Hopfield model (Hopfield 1982), presented in this volume by D.J. Amit. [ABSTRACT FROM AUTHOR]

In this paper some new techniques to analyze nonlinear neural networks are reviewed. A neural network is called nonlinear if the introduction of new data into the synaptic efficacies has to be performed through a non linear operation. The original Hopfield model is linear whereas, for instance, clipped synapses constitute a nonlinear model. We examine the statistical mechanics of a nonlinear neural network with finitely many patterns and arbitrary synaptic kernel, study the information retrieval, and show how the abundantly present spurious states which are a consequence of the nonlinearity can be eliminated. [ABSTRACT FROM AUTHOR]

We study the way in which the properties of spin glasses and associative memory networks are changed when the interactions between the units are not symmetrical. Our models are analog networks subject to thermal noise (Langevin models). In an approximation which becomes exact in the limit of large spin dimensionality, we find that spin glass phases are suppressed, even for arbitrarily small asymmetry. However, in the associative networks, memory states are not seriously degraded; their critical temperature is simply lowered from its value in the corresponding symmetric model. The effect of making the number of memories a finite fraction of the number of units in the system is also qualitatively the same as in the symmetric case. We suggest that asymmetric couplings may make retrieval of the desired memory states faster, since the system will not get trapped in spin glass states. [ABSTRACT FROM AUTHOR]

The Hopfield model with asymmetric bonds is studied. It is shown that some properties of a network with symmetric bonds hold for the complete asymmetric case, too. In particular there is a discontinuous transition when the network is overloaded with patterns. Finally a layered structure is considered. It is shown that it improves its computational abilities by iteratively adjusting its output. patterns. [ABSTRACT FROM AUTHOR]

Recent studies of the statistical mechanics of neural network models of associative memory are reviewed. The paper discusses models which have an energy function but depart from the simple Hebb rule. This includes networks with static synaptic noise, dilute networks and synapses that are nonlinear functions of the Hebb rule (e.g., clipped networks). The properties of networks that employ the projection method are reviewed. [ABSTRACT FROM AUTHOR]

In the spirit of the Colloquium, the connections between Spin-glasses, Optimization problems and Neural Networks are outlined. Special emphasis is given on recent work on phase transition in spin-glasses. A complete cooling recipe for "spin-glass-like" optimization problems is given. The application of numerical spin-glass and optimization techniques to neural network problems is shown. [ABSTRACT FROM AUTHOR]

We discuss the dependence of the ground state energy as the Monte Carlo (MC) cooling rate in a variety of spin glass and other optimization problems. Our work is motivated by recent interest in the concept of simulated annealing and the need to find efficient numerical optimization schemes. We find that the ground state energy for six spin-glass models and the optimal tour length for the N-city traveling salesman problem are a sensitive function of the cooling rate. For the one dimensional nearest neighbor Gaussian and two two dimensional spin-glass models, the nearest neighbor ± J and Gaussian models, we find E(r) = E0 + C1 rx, while for the three dimensional t J model, a two layer t J model, the infinite range model and the N-city traveling salesman problem, the dependence on r is slower, E(r) = Eo - C2 (Inr)-1.Here C1 and C2 are constants.We believe that this difference is related to the fact that finding the ground state energy for the latter four problems is an intractable problem, i.e., it is Np-complete. We assert that a logarithmic cooling rate dependence for finding the ground state is a necessary consequence of Np-completeness. Non Np-complete problems may depend on the cooling rate either as a power-law or logarithmically. However, we have found no example for the latter. In addition to MC simulated annealing, we also studied the ground states obtained upon iterative solution of the mean field equations. In our earlier work we observed that for the Gaussian distribution of exchange interactions, a slow cooling mean field procedure found lower energy T=0 states than did Monte Carlo simulations which were run for the same amount of computer time. However, work on the ± J model did not corroborate this conclusion. The detailed studies presented here clearly establish that the difference lies in the nature of the distribution of exchange interactions. It is the MC method and not the mean field approach which is sensitive to the distribution. [ABSTRACT FROM AUTHOR]

In this paper we describe recent developments in the theory and algorithm design of combinatorial optimization that are related to questions concerning ground states of spin glasses. In particular, we outline an approach, based on polyhedral combinatorics, that has led to the implementation of a cutting plane method for calculating exact ground states of spin glasses in the Ising model. With this method exact ground states for planar grids of size up to 40 x 40 with periodic boundary conditions and exterior magnetic field can be determined in reasonable running times. [ABSTRACT FROM AUTHOR]

Simulated annealing is a combinatorial optimization method based on randomization techniques. The method originates from the analogy between the annealing of solids, as described by the theory of statistical physics, and the optimization of large combinatorial problems. Here we review the basic theory of simulated annealing and recite a number of applications of the method. The theoretical review includes concepts of the theory of homogeneous and inhomogeneous Markov chains, an analysis of the asymptotic convergence of the algorithm, and a discussion of the finite-time behaviour. Furthermore, a section is devoted to the relation between statistical physics and the optimization of combinatorial problems. General aspects related to the application of the algorithm are briefly discussed. The list of applications includes combinatorial optimization problems related to VLSI design, image processing, code design and neural networks. [ABSTRACT FROM AUTHOR]

The phenomenon of slow relaxation is discussed with emphasis on finding a general explanation of its ubiquity. After a review of the phenomenon, possible causes are considered both at the microscopic level and from a configuration space viewpoint. Quenched randomness and strongly constrained dynamics appear most important microscopically. In configuration space the central requirement seems to be the existence of many valleys connected sparsely in a hierarchical way. Recent toy models are shown to fit into this overall picture. [ABSTRACT FROM AUTHOR]

In this review we discuss the diffusion in ultrametric spaces and analyze the long time behaviour of characteristic observables, e.g. the autocorrelation function. We consider both models with stochastic and nonstochastic parameters (branching ratios, transition rates). Furthermore, results for random energy and random free energy models are given. [ABSTRACT FROM AUTHOR]

This is in no case a review on ultrametricity (a recent one can be found in [2]), but a brief compilation of some comments about "ultrametricity today". This contribution will be divided into three parts : an introduction which contains basic definitions and remarks about ultrametricity in other fields than spin glasses, a section on ultrametricity in the SK model, and a section on numerical tests of ultrametricity in spin glasses and optimization problems. [ABSTRACT FROM AUTHOR]

We survey the progress in numerical studies of spin glasses since the first Heidelberg Colloquium, and discuss the resulting advances in our understanding of the field. A clear case can be made for the importance of using multiple methods of attack for the same problem. Combination methods, e.g. finite size scaling and Monte Carlo simulations, which we cover in some detail, are found to be particularly useful. Some remaining outstanding issues are raised, and directions for future work suggested. [ABSTRACT FROM AUTHOR]

This talk reviews phenomenological theories and Monte Carlo simulations on the dynamics of ordering processes, and the associated statistical fluctuations, both in pure systems and in systems with quenched disorder. Particular attention is paid to understand size effects and the approach to the thermodynamic limit: we discuss time-scales on which fluctuations around an equilibrium state decay, the time-scale for growth of a domain size comparable to the system volume, and the "ergodic time" on which a system dynamically averages over its various ordered configurations. [ABSTRACT FROM AUTHOR]

The ordered phase of short-range spin glasses is described in terms of the scaling behaviour associated with a zero-temperature fixed point. The main ingredient of the theory is the exponent y which describes the growth with length scale L of the characteristic coupling at zero temperature, J(L) − JLY. The exponent y is estimated numerically for dimensions d=2,3. For Ising spin glasses we find y − -0.3 for d=2 and y − 0.2 for d=3, implying scaling to weak (strong) coupling for d=2(3), i.e. the "lower critical dimension" dℓ satisfies 2dℓ it determines the large scale properties of the ordered phase, such as the long-distance behaviour of connected correlation functions, G(r) ∝ r−y, and the singular response to a weak magnetic field, msing ∝ hd/(d-2y), The decay of the connected correlation functions at large distances implies that the pure-state overlap distribution function P(q) is trivial, in contrast to the Sherrington-Kirkpatrick model. The dynamics of the system are also discussed, as is the extension to vector spin models. [ABSTRACT FROM AUTHOR]

Experimental evidence for non-exponential decay of the polarization is presented for KTaO3:Na. These data are viewed in the light of analogous findings in spin glasses. They are in support of the decay law P t tt-ga found to be valid for a large number of dielectrics since its discovery by Curie in 1889. [ABSTRACT FROM AUTHOR]

The thermoremanent magnetization of the insulating spin-glass Eu(0.4)Sr(0.6)S has been studied in the range 1.01 Tc down to 0.88 Tc for various initial fields, over 3 decades of time (0.30 - 300 s) by means of Faraday rotation. Within our accuracy, the decay of the TRM is well characterized by a stretched exponential law $$\sigma = \sigma _{\rm O} \exp \{ - (t/\tau )^{1 - n} \}$$ We have studied the variation of the parameters n and τ as a function of the temperature and field. Besides the well known acceleration of the decay with increasing initially applied fields, our results in the weakest field H = 0.2 Oe exhibit a clear-cut anomaly right at the temperature Tc(H) where the transition is expected to occur in the field H = 0.2 Oe, as determined from our previous dynamical measurements. It suggests a characteristic field scale for a spin-glass which defines the field range over which linear response theory may be applicable. [ABSTRACT FROM AUTHOR]

The idea in these experiments is to infer the nature of the spin-glass transition by making dynamic measurements and then extrapolating the time dependence to infinitely long measuring times. We report that such measurements of the ac susceptibility, χ(T, ω) = χ' + iχ• of the insulating spin glass Eu0.40Sr0.60S reveal properties in the zero-frequency limit that indicate the system undergoes a phas transition into the spin-glass state. [ABSTRACT FROM AUTHOR]

Recent measurements of the specific heat and of the thermal conductivity of Eux Sr1−xS in high magnetic fields are reviewed. In the spin-glass and ferromagnetic concentration ranges the specific heat can be quantitavively accounted for in terms of magnon-like excitations, as shown by numerical calculations without any fit parameter. In particular, the exponential decrease of the magnetic specific heat with decreasing temperature in high fields, indicating a gap in the magnetic excitation spectrum, is reproduced. In the spin-glass range, heat transport is strongly facilitated by an external magnetic field, while below the percolation threshold xp = 0.13 no field effect on the thermal conductivity is seen. [ABSTRACT FROM AUTHOR]

Utilizing various experimental techniques the dynamics in spin glasses can be studied over more than 17 decades in time. A concordant experimental quantity is the relaxation rate, which in the linear response regime probes the time dependence of the spin-spin time correlation function. Some fundamental experimental observations of the zero-field equilibrium relaxation as well as the influence of aging on the observed relaxation are discussed. It is inferred that the relaxation in spin glasses exhibits a qualitatively universal behaviour. [ABSTRACT FROM AUTHOR]

This review of the spin-glass problem will begin by considering the basic ingredients of a spin glass. Next I shall summarize the fundamental experimental properties and the real systems which are being studied. A bit of theory will be presented to show how idealized the models are when compared to the real materials. Then I shall discuss the dynamics of the spin-glass phase transition and relaxations within the frozen state. Finally, three recent experiments will be considered which contrast the spin-glass behavior with different types of phase transitions, to wit random-field Ising model, antiferromagnet, and spin-density wave. [ABSTRACT FROM AUTHOR]

Confronted with the contradiction between mounting evidence for a phase transition in three dimension with only two states (in zero field) and the necessity for a symmetry breaking ä la PARISI when treating an Ising spin glass with a gaussian bond distribution, we look, as an escape, for possible stable replica symmetric solutions, in the tree approximation, associated with non-gaussian distributions. [ABSTRACT FROM AUTHOR]