A high-precision study of the hopfield model in the phase of broken replica symmetry

Using a multi-spin coding algorithm, the Hopfield model is studied for network sizes up toN=33,968. Thermodynamically stable states are found in a region where the replica-symmetric solution predicts none should exist, but where a one-step replica symmetry-breaking calculation predicts some should exist. Furthermore, the order parameter in this region is found to take on two distinct values, one of which is not predicted by any theory.     

Gibbs states of the Hopfield model with extensively many patterns

We consider the Hopfield model withM(N)=N patterns, whereN is the number of neurons. We show that if is sufficiently small and the temperature sufficiently low, then there exist disjoint Gibbs states for each of the stored patterns, almost surely with respect to the distribution of the random patterns. This solves a provlem left open in previous work. The key new ingredient is a self-averaging result on the free energy functional. This result has considerable additional interest and some consequences are discussed. A similar result for the free energy of the Sherrington-Kirkpatrick model is also given.     

Pair-correlated patterns in Hopfield model of neural networks

We study the retrieval properties of the Hopfield model of neural networks when the memorized patterns are statistically correlated in pairs. There is a finite correlationk between the memories of each pair, but memories of different pairs are uncorrelated. The analysis is restricted to the case of an arbitrary but finite number of memories in the thermodynamic limit. We find that there are two retrieval regimes: for 0<T<(1–k) the system recognizes the stored patterns and for (1–k)<T<(1+k) the system is able to recognize pairs, but it is not able to distinguish between its two patterns.     

Decay of the remanent magnetization in the asymmetric spin chain

The dynamics of the one-dimensional spin glass with asymmetric interactions between neighboring spins is considered. We confine ourselves to discrete couplings with values ±J. We show that the algebraic decay of the remanent magnetization of the infinite ±J-spin chain at zero temperature is only valid for symmetric couplings. Our analytical investigations as well as computer simulations show stretched exponential decay for any finite concentration of antisymmetric bonds. Thus, the asymmetric ±J-spin chain shows an asymmetry-induced phase transition at zero temperature.     

More relaxed condition for dynamics of discrete time delayed Hopfield neural networks

The dynamics of discrete time delayed Hopfield neural networks is investigated. By using a difference inequality combining with the linear matrix inequality, a sufficient condition ensuring global exponential stability of the unique equilibrium point of the networks is found. The result obtained holds not only for constant delay but also for time-varying delays.     

More relaxed condition for dynamics of discrete time delayed Hopfield neural networks

The dynamics of discrete time delayed Hopfield neural networks is investigated. By using a difference inequality combining with the linear matrix inequality, a sufficient condition ensuring global exponential stability of the unique equilibrium point of the networks is found. The result obtained holds not only for constant delay but also for time-varying delays.     

Rigorous results on the thermodynamics of the dilute Hopfield model

We study the Hopfield model of an autoassociative memory on a random graph onN vertices where the probability of two vertices being joined by a link isp(N). Assuming thatp(N) goes to zero more slowly thanO(1/N), we prove the following results: (1) If the number of stored patternsm(N) is small enough such thatm(N)/Np(N) 0, asN, then the free energy of this model converges, upon proper rescaling, to that of the standard Curie-Weiss model, for almost all choices of the random graph and the random patterns. (2) If in additionm(N) < ln N/ln 2, we prove that there exists, forT< 1, a Gibbs measure associated to each original pattern, whereas for higher temperatures the Gibbs measure is unique. The basic technical result in the proofs is a uniform bound on the difference between the Hamiltonian on a random graph and its mean value.     

Metastates in disordered mean-field models: Random field and hopfield models

We rigorously investigate the size dependence of disordered mean-field models with finite local spin space in terms of metastates. Thereby we provide an illustration of the framework of metastates for systems of randomly competing Gibbs measures. In particular we consider the thermodynamic limit of the empirical metastate , whereμ _n (η) is the Gibbs measure in the finite volume {1,…,n} and the frozen disorder variableη is fixed. We treat explicitly the Hopfield model with finitely many patterns and the Curie-Weiss random field Ising model. In both examples in the phase transition regime the empirical metastate is dispersed for largeN. Moreover, it does not converge for a.e.η, but rather in distribution, for whose limits we given explicit expressions. We also discuss another notion of metastates, due to Aizenman and Wehr.     

Slowing down in the three-dimensional three-state Potts glass with nearest neighbor exchange ± J: A Monte Carlo study

,Static and dynamic properties of the Potts model on the simple cubic lattice with nearest neighbor -interaction are obtained from Monte Carlo simulations in a temperature range where full thermal equilibrium still can be achieved (). For a lattice size L = 16, in this range finite size effects are still negligible, but the data for the spin glass susceptibility agree with previous extrapolations based on finite size scaling of very small lattices. While the static properties are compatible with a zero temperature transition, they certainly do not prove it. Unlike the Ising spin glass, the decay of the time-dependent order parameter is compatible with a simple Kohlrausch function, , while a power law prefactor cannot be distinguished. The Kohlrausch exponent y ( T ) decreases from at [0pt] to at [0pt] however. The relaxation time is compatible with the exponential divergence postulated by McMillan for spin glasses at their lower critical dimension, but the exponent that can be extracted still differs significantly from the theoretical value, . Thus the present results support the conclusion that the Potts spin glass in d = 3 dimensions differs qualitatively from the Ising spin glass. Received: 8 October 1997 / Accepted: 27 November 1997     

Power law decay of stored pattern stability in sparse Hopfield neural networks

Hopfield neural networks on scale-free networks display the power law relation between the stability of patterns and the number of patterns.The stability is measured by the overlap between the output state and the stored pattern which is presented to a neural network.In simulations the overlap declines to a constant by a power law decay.Here we provide the explanation for the power law behavior through the signal-to-noise ratio analysis.We show that on sparse networks storing a plenty of patterns the stability of stored patterns can be approached by a power law function with the exponent-0.5.There is a difference between analytic and simulation results that the analytic results of overlap decay to 0.The difference exists because the signal and noise term of nodes diverge from the mean-field approach in the sparse finite size networks.     

Some new results on the two-dimensional kinetic Ising model in the phase coexistence region

We consider a Glauber dynamics reversible with respect to the two-dimensional Ising model in a finite square of sideL with open boundary conditions, in the absence of an external field and at large inverse temperature . We prove that the gap in the spectrum of the generator restricted to the invariant subspace of functions which are even under global spin flip is much larger than the true gap. As a consequence we are able to show that there exists a new time scalet _even, much smaller than the global relaxation timet _rel, such that, with large probability, any initial configuration first relaxes to one of the two phases in a time scale of ordert _even and only after a time scale of the order oft _rel does it reach the final equilibrium by jumping, via a large deviation, to the opposite phase. It also follows that, with large probability, the time spent by the system during the first jump from one phase to the opposite one is much shorter than the relaxation time.     

Slowing down of the wave packets in quantum graphs

Necklace-type devices are studied, which are optical lines with periodically located segments replaced by two arches of different lengths. It is shown that one can choose parameters of the necklace in such a way that the dispersion relation for the whole line will be very flat in the neighbourhood of the prescribed frequency (although the corresponding band is not narrow). It is also shown that a finite piece of such a line creates a slowdown of wave packets without any significant loss of the energy due to the reflection.     

Slowing down of the wave packets in quantum graphs

Necklace-type devices are studied, which are optical lines with periodically located segments replaced by two arches of different lengths. It is shown that one can choose parameters of the necklace in such a way that the dispersion relation for the whole line will be very flat in the neighbourhood of the prescribed frequency (although the corresponding band is not narrow). It is also shown that a finite piece of such a line creates a slowdown of wave packets without any significant loss of the energy due to the reflection.     

A mean-field,effective medium theory of random binary alloys III. The ising model with competing interactions

The Ising model with competing interactions is studied in a mean field effective medium approach. The phase diagram of such model alloys is studied. We conclude that for all ratios of the competing interaction moments, a spin glass phase always exists at low temperatures for certain concentration regimes.     

A lower bound for the memory capacity in the Potts-Hopfield model

We consider Potts-Hopfield networks of sizeN. We prove the result: _c >0 such that for all 0<< _c we can find, >0 in such a way that, whenN, we can store N patterns, all of them being sorrounded by -energy barriers at distance.     

Metastates in the Hopfield Model in the Replica Symmetric Regime

We study the finite dimensional marginals of the Gibbs measure in the Hopfield model at low temperature when the number of patterns, M, is proportional to the volume with a sufficiently small proportionality constant > 0. It is shown that even when a single pattern is selected (by a magnetic field or by conditioning), the marginals do not converge almost surely, but only in law. The corresponding limiting law is constructed explicitly. We fit our result in the recently proposed language of metastates which we discuss some length. As a byproduct, in a certain regime of the parameters and (the inverse temperature), we also give a simple proof of Talagrands recent result that the replica symmetric solution found by Amit, Gutfreund, and Sompolinsky can be rigorously justified.     

Interpretation of protein structure and dynamics from the statistics of the open and closed times measured in a single ion-channel protein

Ion channels are proteins in the lipid cell membrane. They spontaneously fluctuate between conformational shapes that are open or closed to the passage of ions. The ionic currents through an individual channel can be resolved by the patch clamp technique. Thus, the time sequence of open and closed conformational states can be measured in one channel molecule. The probability density function of the dwell times in the open and closed states displays scaling functions that may arise from: (1) a large number of conformational substates having a continuous distribution of activation energy barriers, (2) time-dependent changes in the energy barriers between states, or (3) local interactions that constrain local structures which interact hierarchically to form global structure.     

Fluctuation of the free energy in the Hopfield model

We prove that in the ergodic region [T>J ²(1 + r)] the deviation of the total free energy of the Hopfield neural network converges in distribution asN to a (shifted) Gaussian variable. Moreover, the free energy per site converges in probability to lim(1/N)ln _N .     

1/f noise and resistivity of spin glasses

Electrical resistivityρ(T) of spin glasses within the framework of Mookerjee and Chowdhury’s percolation model where there is a distribution of relaxation times (drt) is calculated.ρ(T) thus calculated is in better qualitative agreement with experimental results than that in the single relaxation time model.