Elementary solution of Classical Spin-Glass models

Hemmen, J. Leo van; Grensing, D.; Huber, A.; Kühn, Reiner

doi:10.1007/bf01308399

Cited by 36 publications

(11 citation statements)

References 3 publications

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“…¿From this we want to derive a flow equation for the overlap order parameters. Because of the multi-state character of the model (due to the four-spin interaction term) the summation over i has to be carried out by generalizing the method of submagnetizations or suboverlaps connected with partitions of the network with respect to the built-in patterns [13]- [14]. We note that for Hopfield networks we do not need such a partitioning [15].…”

Section: Dynamicsmentioning

confidence: 99%

Statics and dynamics of an Ashkin-Teller neural network with low loading

Bollé¹,

Kozłowski²

1998

J. Phys. A: Math. Gen.

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An Ashkin-Teller neural network, allowing for two types of neurons is considered in the case of low loading as a function of the strength of the respective couplings between these neurons. The storage and retrieval of embedded patterns built from the two types of neurons, with different degrees of (in)dependence is studied. In particular, thermodynamic properties including the existence and stability of Mattis states are discussed. Furthermore, the dynamic behaviour is examined by deriving flow equations for the macroscopic overlap. It is found that for linked patterns the model shows better retrieval properties than a corresponding Hopfield model.One of the best known physical models for neural networks is the Hopfield model [1]. In theoretical investigations of network properties, e.g., the retrieval of learned patterns, it plays a similar role as the Ising model does in the theory of magnetism. Extensions of this model to multi-state neurons have received a lot of attention recently (see, e.g., [2] -[5] and the references cited therein). Thereby the ability to store and retrieve so-called grey-toned and coloured patterns has been investigated.In this work we consider another extension of the Hopfield model to allow for multi-functional neurons. The specific model we have in mind is the neural network version of the Ashkin-Teller spin-glass ([6]-[9]). Indeed, on the one hand the Ashkin-Teller model has two different kinds of neurons (spins) at each site interacting with each other. This allows us to interprete this model as a neural network with two types of neurons having different functions. On the other hand, this Ashkin-Teller neural network (ATNN) can be considered as a model consisting out of two interacting Hopfield models.We expect the behaviour of the ATNN to be different from the one of the Hopfield model in a non trivial way. One of the things we want to find out, e.g., is whether this (four-neuron) interaction between the two types of neurons can improve the retrieval process for embedded patterns built from these two types of neurons. We will see, indeed, that for a particular choice of this interaction term the retrieval quality of the embedded patterns is very high in comparison with a corresponding Hopfield model. Therefore, independent of the possible biological relevance of this model, if any, such a study is interesting from the pure physical point of view.In this work we consider both the thermodynamic and dynamic properties of this model in the case of loading of a finite number of patterns.The rest of this paper is organized as follows. In section 2 the ATNN model is introduced. Section 3 discusses the methods used for analyzing both the equilibrium properties and the dynamics of the model. In particular, fixed-point equations as well as flow equations for the relevant macroscopic overlap order parameters are derived. In section 4 numerical solutions of these equations are discussed for a representative set of network parameters. The retrieval properties of embedded patterns with different...

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Section: Dynamicsmentioning

confidence: 99%

Statics and dynamics of an Ashkin-Teller neural network with low loading

Bollé¹,

Kozłowski²

1998

J. Phys. A: Math. Gen.

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“…To this end we introduce sublattices (van Hemmen et al 1986van Hemmen and K~hn 1986) Z(x, O ax, /~, e) = (i I ~ = x, A~ x = D ~x, h~h(t) = --r/, hr~f(t) = --O}, (10) that gather neurons with the same storage vector (r 1 <~/~ ~< q), with an identical axonal delay A ax, the same (momentary) inhibition strength h~h(t) and refractory field h~f(t) into a common class. If p(x, D ~x, ~/, Q, t) is the portion of neurons that belong at time t to the sublattice L(x, D "x, ~/, Q), then the overlap can be written (Herz et al 1988(Herz et al , 1989 ,…”

Section: Equation Of Motionmentioning

confidence: 99%

A biologically motivated and analytically soluble model of collective oscillations in the cortex

1993

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Almtraet. A model of an associative network of spikingneurons with stationary states, globally locked oscillations, and weakly locked oscillatory states is presented and analyzed. The network is close to biology in the following sense. First, the neurons spike and our model includes an absolute refractory period after each spike. Second, we consider a distribution of axonal delay times. Finally, we describe synaptic signal transmission by excitatory and inhibitory potentials (EPSP and IPSP) with a realistic shape, that is, through a response kernel. During retrieval of a pattern, all active neurons exhibit periodic spike bursts which may or may not be synchronized ('locked') into a coherent oscillation. We derive an analytical condition of locking and calculate the period of collective activity during oscillatory retrieval. In a stationary retrieval state, the overlap assumes a constant value proportional to the mean firing rate of the neurons. It is argued that in a biological network an intermediate scenario of "weak locking' is most likely.

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“…To do so, we shall perform an expansion of W s in powers of 1/ √ N and keep only the non vanishing terms in the thermodynamical limit. This expansion is made possible by the mean-field nature of the Hopfield Hamiltonian [9]. We restrict to quenched patterns that differ from each other on a macroscopic number of sites, so each block G τ contains a number N c τ of sites of the order of N ; note that if the patterns are randomly drawn from an unbiased distribution c τ = 2 −p in the limit of large sizes.…”

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confidence: 99%

Relationship between long timescales and the static free-energy in the Hopfield model

Biroli

Monasson

1998

J. Phys. A: Math. Gen.

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The Glauber dynamics of the Hopfield model at low storage level is considered. We analytically derive the spectrum of relaxation times for large system sizes. The longest time scales are gathered in families, each family being in one to one correspondence with a stationary (not necessarily stable) point of the static mean-field free-energy. Inside a family, the time scales are given by the reciprocals (of the absolute values) of the eigenvalues of the free-energy Hessian matrix. * Laboratoire de Physique Théorique de l'ENS; preprint number 98/04

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Elementary solution of Classical Spin-Glass models

Cited by 36 publications

References 3 publications

Statics and dynamics of an Ashkin-Teller neural network with low loading

Statics and dynamics of an Ashkin-Teller neural network with low loading

A biologically motivated and analytically soluble model of collective oscillations in the cortex

Relationship between long timescales and the static free-energy in the Hopfield model

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