On the basis of the general form for the energy needed to adapt the connection strengths w ij of a network in which learning takes place, a local learning rule is found for the changes ∆w ij . This biologically realizable learning rule turns out to comply with Hebb's neurophysiological postulate, but is not of the form of any of the learning rules proposed in the literature.The learning rule possesses the property that the energy needed in each learning step is minimal, and is, as such, evolutionary attractive. Moreover, the pre-and post-synaptic neurons are found to influence the synaptic changes differently, resulting in a asymmetric connection matrix w ij , a fact which is in agreement with biological observation.It is shown that, if a finite set of the same patterns is presented over and over again to the network, the weights of the synapses converge to finite values.Furthermore, it is proved that the final values found in this biologically realizable limit are the same as those found via a mathematical approach to the problem of finding the weights of a partially connected neural network that can store a collection of patterns. The mathematical solution is obtained via a modified version of the so-called method of the pseudo-inverse, and has the inverse of a reduced correlation matrix, rather than the usual correlation matrix, as its basic ingredient. Thus, a biological network might realize the final results of the mathematician by the energetically economic rule for the adaption of the synapses found in this article.

We examine the problem of damage spreading in the off-equilibrium mode coupling equations. The study is done for the spherical p-spin model introduced by Crisanti, Horner and Sommers. For p > 2 we show the existence of a temperature transition T0 well above any relevant thermodynamic transition temperature. Above T0 the asymptotic damage decays to zero while below T0 it decays to a finite value independent of the initial damage. This transition is stable in the presence of asymmetry in the interactions. We discuss the physical origin of this peculiar phase transition which occurs as a consequence of the non-linear coupling between the damage and the two-time correlation functions.The theoretical understanding of the dynamical behavior of glasses is a long outstanding problem in statistical physics which has recently revealed new aspects related to the underlying mechanism responsible of the glass transition [1,2]. While there are still some obscure points in the theory (i.e. the inclusion of finite time activated process beyond the mean-field limit) a scenario has emerged which unifies the dynamical approach (modecoupling theory) with the thermodynamic Adam-Gibbs-Di Marzio approach. The scenario for the dynamical behavior of glasses can be summarized in three different temperatures which separate three different regimes. In the high-temperature regime T > T d the system behaves as a liquid and is very well described by the modecoupling equations of Götze in the equilibrium regime [3]. A crossover takes place at T d where there is a dynamical singularity and the correlation functions do not decay to zero in the infinite time limit (ergodicity breaking). This dynamical singularity is a genuine mean-field effect which turns out to be a crossover temperature when activated processes are taken into account. Below T d the relaxation time (or viscosity) starts to grow dramatically fast and seems to diverge at T s where the configurational entropy vanishes and a thermodynamic phase transition takes place. The glass transition T g (as defined where the viscosity is 10 13 Poise) lies between T s and T d and depends on the cooling rate. Hence T g does not correspond to a true dynamical singularity. Furthermore, there is a first order phase transition T M where the liquid (if cooled sufficiently slow) crystallizes.The essentials of this scenario have been corroborated in the context of mean-field spin glasses, and in particular in those models with a one step replica symmetry breaking transition [4]. While the first-order transition temperature T M is absent in spin glasses (disorder prevents the existence of a crystal state) the other two transitions (T s , T d ) have been clearly identified.The purpose of this paper is the study of the dam-age spreading in models for structural glasses. Damage spreading is the study of the time propagation of a perturbation or damage in the initial condition of a system. The propagation of the initial damage is a dynamical effect which has deserved considerably attention in the past (speci...

Abstract. A recurrent neural network with noisy input is studied analytically, on the basis of a Discrete Time Master Equation. The latter is derived from a biologically realizable learning rule for the weights of the connections. In a numerical study it is found that the fixed points of the dynamics of the net are time dependent, implying that the representation in the brain of a fixed piece of information (e.g., a word to be recognized) is not fixed in time.

We consider damage spreading transitions in the framework of mode-coupling theory. This theory describes relaxation processes in glasses in the mean-field approximation which are known to be characterized by the presence of an exponentially large number of metastable states. For systems evolving under identical but arbitrarily correlated noises, we demonstrate that there exists a critical temperature T 0 which separates two different dynamical regimes depending on whether damage spreads or not in the asymptotic long-time limit. This transition exists for generic noise correlations such that the zero damage solution is stable at high temperatures, being minimal for maximal noise correlations. Although this dynamical transition depends on the type of noise correlations, we show that the asymptotic damage has the good properties of a dynamical order parameter, such as ͑i͒ independence of the initial damage; ͑ii͒ independence of the class of initial condition; and ͑iii͒ stability of the transition in the presence of asymmetric interactions which violate detailed balance. For maximally correlated noises we suggest that damage spreading occurs due to the presence of a divergent number of saddle points ͑as well as metastable states͒ in the thermodynamic limit consequence of the ruggedness of the free-energy landscape which characterizes the glassy state. These results are then compared to extensive numerical simulations of a mean-field glass model ͑the Bernasconi model͒ with Monte Carlo heatbath dynamics. The freedom of choosing arbitrary noise correlations for Langevin dynamics makes damage spreading an interesting tool to probe the ruggedness of the configurational landscape.