Dynamics of spin systems with randomly asymmetric bonds: Langevin dynamics and a spherical model

Cocaine diminishes functional network robustness and destabilizes the energy landscape of neuronal activity in the medial prefrontal cortex

We present analysis of neuronal activity recordings from a subset of neurons in the medial prefrontal cortex of rats before and after the administration of cocaine. Using an underlying modern Hopfield model as a description for the neuronal network, combined with a machine learning approach, we compute the underlying functional connectivity of the neuronal network. We find that the functional connectivity changes after the administration of cocaine with both functional-excitatory and functional-inhibitory neurons being affected. Using conventional network analysis, we find that the diameter of the graph, or the shortest length between the two most distant nodes, increases with cocaine, suggesting that the neuronal network is less robust. We also find that the betweenness centrality scores for several of the functional-excitatory and functional-inhibitory neurons decrease significantly, while other scores remain essentially unchanged, to also suggest that the neuronal network is less robust. Finally, we study the distribution of neuronal activity and relate it to energy to find that cocaine drives the neuronal network towards destabilization in the energy landscape of neuronal activation. While this destabilization is presumably temporary given one administration of cocaine, perhaps this initial destabilization indicates a transition towards a new stable state with repeated cocaine administration. However, such analyses are useful more generally to understand how neuronal networks respond to perturbations.

Machine learning at the mesoscale: A computation-dissipation bottleneck

The cost of information processing in physical systems calls for a trade-off between performance and energetic expenditure. Here we formulate and study a computation-dissipation bottleneck in mesoscopic systems used as input-output devices. Using both real data sets and synthetic tasks, we show how nonequilibrium leads to enhanced performance. Our framework sheds light on a crucial compromise between information compression, input-output computation and dynamic irreversibility induced by nonreciprocal interactions.

A reservoir of timescales emerges in recurrent circuits with heterogeneous neural assemblies

The temporal activity of many physical and biological systems, from complex networks to neural circuits, exhibits fluctuations simultaneously varying over a large range of timescales. Long-tailed distributions of intrinsic timescales have been observed across neurons simultaneously recorded within the same cortical circuit. The mechanisms leading to this striking temporal heterogeneity are yet unknown. Here, we show that neural circuits, endowed with heterogeneous neural assemblies of different sizes, naturally generate multiple timescales of activity spanning several orders of magnitude. We develop an analytical theory using rate networks, supported by simulations of spiking networks with cell-type specific connectivity, to explain how neural timescales depend on assembly size and show that our model can naturally explain the long-tailed timescale distribution observed in the awake primate cortex. When driving recurrent networks of heterogeneous neural assemblies by a time-dependent broadband input, we found that large and small assemblies preferentially entrain slow and fast spectral components of the input, respectively. Our results suggest that heterogeneous assemblies can provide a biologically plausible mechanism for neural circuits to demix complex temporal input signals by transforming temporal into spatial neural codes via frequency-selective neural assemblies.

Generalized Lotka-Volterra Equations with Random, Nonreciprocal Interactions: The Typical Number of Equilibria

We compute the typical number of equilibria of the generalized Lotka-Volterra equations describing species-rich ecosystems with random, nonreciprocal interactions using the replicated Kac-Rice method. We characterize the multiple-equilibria phase by determining the average abundance and similarity between equilibria as a function of their diversity (i.e., of the number of coexisting species) and of the variability of the interactions. We show that linearly unstable equilibria are dominant, and that the typical number of equilibria differs with respect to the average number.

Nonequilibrium thermodynamics of the asymmetric Sherrington-Kirkpatrick model

Most natural systems operate far from equilibrium, displaying time-asymmetric, irreversible dynamics characterized by a positive entropy production while exchanging energy and matter with the environment. Although stochastic thermodynamics underpins the irreversible dynamics of small systems, the nonequilibrium thermodynamics of larger, more complex systems remains unexplored. Here, we investigate the asymmetric Sherrington-Kirkpatrick model with synchronous and asynchronous updates as a prototypical example of large-scale nonequilibrium processes. Using a path integral method, we calculate a generating functional over trajectories, obtaining exact solutions of the order parameters, path entropy, and steady-state entropy production of infinitely large networks. Entropy production peaks at critical order-disorder phase transitions, but is significantly larger for quasi-deterministic disordered dynamics. Consequently, entropy production can increase under distinct scenarios, requiring multiple thermodynamic quantities to describe the system accurately. These results contribute to developing an exact analytical theory of the nonequilibrium thermodynamics of large-scale physical and biological systems and their phase transitions.

Aging by Near-Extinctions in Many-Variable Interacting Populations

Models of many-species ecosystems, such as the Lotka-Volterra and replicator equations, suggest that these systems generically exhibit near-extinction processes, where population sizes go very close to zero for some time before rebounding, accompanied by a slowdown of the dynamics (aging). Here, we investigate the connection between near-extinction and aging by introducing an exactly solvable many-variable model, where the time derivative of each population size vanishes at both zero and some finite maximal size. We show that aging emerges generically when random interactions are taken between populations. Population sizes remain exponentially close (in time) to the absorbing values for extended periods of time, with rapid transitions between these two values. The mechanism for aging is different from the one at play in usual glassy systems: At long times, the system evolves in the vicinity of unstable fixed points rather than marginal ones.

Path integral approach to universal dynamics of reservoir computers

In this work, we give a characterization of the reservoir computer (RC) by the network structure, especially the probability distribution of random coupling constants. First, based on the path integral method, we clarify the universal behavior of the random network dynamics in the thermodynamic limit, which depends only on the asymptotic behavior of the second cumulant generating functions of the network coupling constants. This result enables us to classify the random networks into several universality classes, according to the distribution function of coupling constants chosen for the networks. Interestingly, it is revealed that such a classification has a close relationship with the distribution of eigenvalues of the random coupling matrix. We also comment on the relation between our theory and some practical choices of random connectivity in the RC. Subsequently, we investigate the relationship between the RC's computational power and the network parameters for several universality classes. We perform several numerical simulations to evaluate the phase diagrams of the steady reservoir states, common-signal-induced synchronization, and the computational power in the chaotic time series inference tasks. As a result, we clarify the close relationship between these quantities, especially a remarkable computational performance near the phase transitions, which is realized even near a nonchaotic transition boundary. These results may provide us with a new perspective on the designing principle for the RC.

Diffusions interacting through a random matrix: universality via stochastic Taylor expansion

Consider $$(X_{i}(t))$$ ( X i ( t ) ) solving a system of N stochastic differential equations interacting through a random matrix $${\mathbf {J}} = (J_{ij})$$ J = ( J ij ) with independent (not necessarily identically distributed) random coefficients. We show that the trajectories of averaged observables of $$(X_i(t))$$ ( X i ( t ) ) , initialized from some $$\mu $$ μ independent of $${\mathbf {J}}$$ J , are universal, i.e., only depend on the choice of the distribution $$\mathbf {J}$$ J through its first and second moments (assuming e.g., sub-exponential tails). We take a general combinatorial approach to proving universality for dynamical systems with random coefficients, combining a stochastic Taylor expansion with a moment matching-type argument. Concrete settings for which our results imply universality include aging in the spherical SK spin glass, and Langevin dynamics and gradient flows for symmetric and asymmetric Hopfield networks.

Dynamics and orientation selectivity in a cortical model of rodent V1 with excess bidirectional connections

Recent experiments have revealed fine structure in cortical microcircuitry. In particular, bidirectional connections are more prevalent than expected by chance. Whether this fine structure affects cortical dynamics and function has not yet been studied. Here we investigate the effects of excess bidirectionality in a strongly recurrent network model of rodent V1. We show that reciprocal connections have only a very weak effect on orientation selectivity. We find that excess reciprocity between inhibitory neurons slows down the dynamics and strongly increases the Fano factor, while for reciprocal connections between excitatory and inhibitory neurons it has the opposite effect. In contrast, excess bidirectionality within the excitatory population has a minor effect on the neuronal dynamics. These results can be explained by an effective delayed neuronal self-coupling which stems from the fine structure. Our work suggests that excess bidirectionality between inhibitory neurons decreases the efficiency of feature encoding in cortex by reducing the signal to noise ratio. On the other hand it implies that the experimentally observed strong reciprocity between excitatory and inhibitory neurons improves the feature encoding.

Correlations between synapses in pairs of neurons slow down dynamics in randomly connected neural networks

Networks of randomly connected neurons are among the most popular models in theoretical neuroscience. The connectivity between neurons in the cortex is however not fully random, the simplest and most prominent deviation from randomness found in experimental data being the overrepresentation of bidirectional connections among pyramidal cells. Using numerical and analytical methods, we investigate the effects of partially symmetric connectivity on the dynamics in networks of rate units. We consider the two dynamical regimes exhibited by random neural networks: the weak-coupling regime, where the firing activity decays to a single fixed point unless the network is stimulated, and the strong-coupling or chaotic regime, characterized by internally generated fluctuating firing rates. In the weak-coupling regime, we compute analytically, for an arbitrary degree of symmetry, the autocorrelation of network activity in the presence of external noise. In the chaotic regime, we perform simulations to determine the timescale of the intrinsic fluctuations. In both cases, symmetry increases the characteristic asymptotic decay time of the autocorrelation function and therefore slows down the dynamics in the network.

Analysis of an Attractor Neural Network's Response to Conflicting External Inputs

The theory of attractor neural networks has been influential in our understanding of the neural processes underlying spatial, declarative, and episodic memory. Many theoretical studies focus on the inherent properties of an attractor, such as its structure and capacity. Relatively little is known about how an attractor neural network responds to external inputs, which often carry conflicting information about a stimulus. In this paper we analyze the behavior of an attractor neural network driven by two conflicting external inputs. Our focus is on analyzing the emergent properties of the megamap model, a quasi-continuous attractor network in which place cells are flexibly recombined to represent a large spatial environment. In this model, the system shows a sharp transition from the winner-take-all mode, which is characteristic of standard continuous attractor neural networks, to a combinatorial mode in which the equilibrium activity pattern combines embedded attractor states in response to conflicting external inputs. We derive a numerical test for determining the operational mode of the system a priori. We then derive a linear transformation from the full megamap model with thousands of neurons to a reduced 2-unit model that has similar qualitative behavior. Our analysis of the reduced model and explicit expressions relating the parameters of the reduced model to the megamap elucidate the conditions under which the combinatorial mode emerges and the dynamics in each mode given the relative strength of the attractor network and the relative strength of the two conflicting inputs. Although we focus on a particular attractor network model, we describe a set of conditions under which our analysis can be applied to more general attractor neural networks.

Pattern Storage, Bifurcations, and Groupwise Correlation Structure of an Exactly Solvable Asymmetric Neural Network Model

Despite their biological plausibility, neural network models with asymmetric weights are rarely solved analytically, and closed-form solutions are available only in some limiting cases or in some mean-field approximations. We found exact analytical solutions of an asymmetric spin model of neural networks with arbitrary size without resorting to any approximation, and we comprehensively studied its dynamical and statistical properties. The network had discrete time evolution equations and binary firing rates, and it could be driven by noise with any distribution. We found analytical expressions of the conditional and stationary joint probability distributions of the membrane potentials and the firing rates. By manipulating the conditional probability distribution of the firing rates, we extend to stochastic networks the associating learning rule previously introduced by Personnaz and coworkers. The new learning rule allowed the safe storage, under the presence of noise, of point and cyclic attractors, with useful implications for content-addressable memories. Furthermore, we studied the bifurcation structure of the network dynamics in the zero-noise limit. We analytically derived examples of the codimension 1 and codimension 2 bifurcation diagrams of the network, which describe how the neuronal dynamics changes with the external stimuli. This showed that the network may undergo transitions among multistable regimes, oscillatory behavior elicited by asymmetric synaptic connections, and various forms of spontaneous symmetry breaking. We also calculated analytically groupwise correlations of neural activity in the network in the stationary regime. This revealed neuronal regimes where, statistically, the membrane potentials and the firing rates are either synchronous or asynchronous. Our results are valid for networks with any number of neurons, although our equations can be realistically solved only for small networks. For completeness, we also derived the network equations in the thermodynamic limit of infinite network size and we analytically studied their local bifurcations. All the analytical results were extensively validated by numerical simulations.

Inferring hidden states in Langevin dynamics on large networks: Average case performance

We present average performance results for dynamical inference problems in large networks, where a set of nodes is hidden while the time trajectories of the others are observed. Examples of this scenario can occur in signal transduction and gene regulation networks. We focus on the linear stochastic dynamics of continuous variables interacting via random Gaussian couplings of generic symmetry. We analyze the inference error, given by the variance of the posterior distribution over hidden paths, in the thermodynamic limit and as a function of the system parameters and the ratio α between the number of hidden and observed nodes. By applying Kalman filter recursions we find that the posterior dynamics is governed by an "effective" drift that incorporates the effect of the observations. We present two approaches for characterizing the posterior variance that allow us to tackle, respectively, equilibrium and nonequilibrium dynamics. The first appeals to Random Matrix Theory and reveals average spectral properties of the inference error and typical posterior relaxation times; the second is based on dynamical functionals and yields the inference error as the solution of an algebraic equation.

Nonequilibrium landscape theory of neural networks

The brain map project aims to map out the neuron connections of the human brain. Even with all of the wirings mapped out, the global and physical understandings of the function and behavior are still challenging. Hopfield quantified the learning and memory process of symmetrically connected neural networks globally through equilibrium energy. The energy basins of attractions represent memories, and the memory retrieval dynamics is determined by the energy gradient. However, the realistic neural networks are asymmetrically connected, and oscillations cannot emerge from symmetric neural networks. Here, we developed a nonequilibrium landscape-flux theory for realistic asymmetrically connected neural networks. We uncovered the underlying potential landscape and the associated Lyapunov function for quantifying the global stability and function. We found the dynamics and oscillations in human brains responsible for cognitive processes and physiological rhythm regulations are determined not only by the landscape gradient but also by the flux. We found that the flux is closely related to the degrees of the asymmetric connections in neural networks and is the origin of the neural oscillations. The neural oscillation landscape shows a closed-ring attractor topology. The landscape gradient attracts the network down to the ring. The flux is responsible for coherent oscillations on the ring. We suggest the flux may provide the driving force for associations among memories. We applied our theory to rapid-eye movement sleep cycle. We identified the key regulation factors for function through global sensitivity analysis of landscape topography against wirings, which are in good agreements with experiments.

Dynamic mean-field and cavity methods for diluted Ising systems

We compare dynamic mean-field and dynamic cavity methods to describe the stationary states of dilute kinetic Ising models. We compute dynamic mean-field theory by expanding in interaction strength to third order, and we compare to the exact dynamic mean-field theory for fully asymmetric networks. We show that in diluted networks, the dynamic cavity method generally predicts magnetizations of individual spins better than both first-order ("naive") and second-order ("TAP") dynamic mean-field theory.

Lateral thinking, from the Hopfield model to cortical dynamics

Self-organizing attractor networks may comprise the building blocks for cortical dynamics, providing the basic operations of categorization, including analog-to-digital conversion, association and auto-association, which are then expressed as components of distinct cognitive functions depending on the contents of the neural codes in each region. To assess the viability of this scenario, we first review how a local cortical patch may be modeled as an attractor network, in which memory representations are not artificially stored as prescribed binary patterns of activity as in the Hopfield model, but self-organize as continuously graded patterns induced by afferent input. Recordings in macaques indicate that such cortical attractor networks may express retrieval dynamics over cognitively plausible rapid time scales, shorter than those dominated by neuronal fatigue. A cortical network encompassing many local attractor networks, and incorporating a realistic description of adaptation dynamics, may be captured by a Potts model. This network model has the capacity to engage long-range associations into sustained iterative attractor dynamics at a cortical scale, in what may be regarded as a mathematical model of spontaneous lateral thought. This article is part of a Special Issue entitled: Neural Coding.

Correlation between eigenvalue spectra and dynamics of neural networks

This letter presents a study of the correlation between the eigenvalue spectra of synaptic matrices and the dynamical properties of asymmetric neural networks with associative memories. For this type of neural network, it was found that there are essentially two different dynamical phases: the chaos phase, with almost all trajectories converging to a single chaotic attractor, and the memory phase, with almost all trajectories being attracted toward fixed-point attractors acting as memories. We found that if a neural network is designed in the chaos phase, the eigenvalue spectrum of its synaptic matrix behaves like that of a random matrix (i.e., all eigenvalues lie uniformly distributed within a circle in the complex plan), and if it is designed in the memory phase, the eigenvalue spectrum will split into two parts: one part corresponds to a random background, the other part equal in number to the memory attractors. The mechanism for these phenomena is discussed in this letter.

Learning of spatiotemporal patterns in Ising-spin neural networks: analysis of storage capacity by path integral methods

We encode periodic spatiotemporal patterns in Ising-spin neural networks, using the simple learning rule inspired by the spike-timing-dependent synaptic plasticity. It is then found that periodically oscillating spin neurons successfully reproduce phase differences of the encoded periodic patterns. The storage capacity of this associative memory neural network is enhanced with an adequate level of asymmetry in synapse connections. To understand the properties of these nonequilibrium retrieval states of the neural network, we carry out an analysis based on a path integral method. The relation of a dynamic crosstalk term to time-persistent oscillation of a correlation function well explains the enhancement of the storage capacity in spite of our approximation on nonpersistent terms. We investigate the accuracy of this approximation further by detailed comparison with numerical simulations.

A constructive mean-field analysis of multi-population neural networks with random synaptic weights and stochastic inputs

We deal with the problem of bridging the gap between two scales in neuronal modeling. At the first (microscopic) scale, neurons are considered individually and their behavior described by stochastic differential equations that govern the time variations of their membrane potentials. They are coupled by synaptic connections acting on their resulting activity, a nonlinear function of their membrane potential. At the second (mesoscopic) scale, interacting populations of neurons are described individually by similar equations. The equations describing the dynamical and the stationary mean-field behaviors are considered as functional equations on a set of stochastic processes. Using this new point of view allows us to prove that these equations are well-posed on any finite time interval and to provide a constructive method for effectively computing their unique solution. This method is proved to converge to the unique solution and we characterize its complexity and convergence rate. We also provide partial results for the stationary problem on infinite time intervals. These results shed some new light on such neural mass models as the one of Jansen and Rit (1995): their dynamics appears as a coarse approximation of the much richer dynamics that emerges from our analysis. Our numerical experiments confirm that the framework we propose and the numerical methods we derive from it provide a new and powerful tool for the exploration of neural behaviors at different scales.

Behavior of heuristics on large and hard satisfiability problems

We study the behavior of a heuristic for solving random satisfiability problems by stochastic local search near the satisfiability threshold. The heuristic for average satisfiability (ASAT), is similar to the Focused Metropolis Search heuristic, and shares the property of being focused, i.e., only variables in unsatisfied clauses are updated in each step. It is significantly simpler than the benchmark WALKSAT heuristic. We show that ASAT solves instances as large as N=10(6) in linear time, on average, up to a ratio of 4.21 clauses per variable in random three-satisfiability. For K higher than 3, ASAT appears to solve instances of K -satisfiability up to the Montanari-Ricci-Tersenghi-Parisi full replica symmetry breaking (FSRB) threshold denoted alpha(s)(K) in linear time.

Shear-thickening and entropy-driven reentrance

We discuss a generic mechanism for shear thickening analogous to entropy-driven phase reentrance. We implement it in the context of nonrelaxational mean-field glassy systems: although very simple, the microscopic models we study present a dynamical phase diagram with second- and first-order stirring-induced jamming transitions leading to intermittency, metastability, and phase coexistence as seen in some experiments. The jammed state is fragile with respect to change in the stirring direction. Our approach provides a direct derivation of a mode-coupling theory of shear thickening.

Path-integral approach to dynamics in a sparse random network

We study the dynamics involved in a sparse random network model. We extend the standard mean-field approximation for the dynamics of a random network by employing the path-integral approach. The result indicates that the distribution of the variable is essentially identical to that obtained from globally coupled oscillators with random Gaussian interaction. We present the results of a numerical simulation of the Kuramoto transition in a random network, which is found to be consistent with this analysis.

Aging under shear: structural relaxation of a non-Newtonian fluid

The influence of an applied shear field on the dynamics of an aging colloidal suspension has been investigated by the dynamic light-scattering determination of the density autocorrelation function. Though a stationary state is never observed, the slow dynamics crosses between two different nonequilibrium regimes as soon as the structural relaxation time tau(s) approaches the inverse shear rate gamma (-1) . In the shear-dominated regime (at high gamma values) the structural relaxation time is found to be strongly sensitive to the shear rate ( tau(s) approximately gamma (-alpha) , with alpha approximately 1 ) while aging proceeds at a very slow rate. The effect of the shear on the detailed shape of the density autocorrelation function is quantitatively described, assuming that the structural relaxation process arises from the heterogeneous superposition of many relaxing units, each one independently coupled to shear with a parallel composition rule for time scales: 1/tau-->1/tau+A gamma .

Role of the interaction matrix in mean-field spin glass models

Mean-field models of two-spin Ising spin glasses with interaction matrices taken from ensembles that are invariant under O(N) transformations are studied. A general study shows that the nature of the spin glass transition can be deduced from the eigenvalue spectrum of the interaction matrix. A simple replica approach is derived to carry out the average over the O(N) disorder. The analytic results are confirmed by the extensive Monte Carlo simulations for large system sizes and by the exact enumeration for small system sizes.

Equivalence of driven and aging fluctuation-dissipation relations in the trap model

We study the nonequilibrium version of the fluctuation-dissipation (FD) relation in the glass phase of a trap model that is driven into a nonequilibrium steady state by external "shear." This extends our recent study of aging FD relations in the same model, where we found limiting, observable independent FD relations for "neutral" observables that are uncorrelated with the system's average energy. In this work, for such neutral observables, we find the FD relation for a stationary weakly driven system to be the same, to within small corrections, as for an infinitely aged system. We analyze the robustness of this correspondence with respect to non-neutrality of the observable, and with respect to changes in the driving mechanism.

Interrupted coarsening in a driven kinetically constrained Ising chain

We introduce a driven version of the one-dimensional kinetically constrained spin chain [J. Jackle and S. Eisinger, Zeitschr. Phys. B 84, 115 (1991)]. In its original undriven version, this model shows anomalous coarsening following a quench to a low temperature, with an equilibration time that diverges as approximately exp(1/T(2)) for T-->0. We show that driving of constant rate gamma; interrupts coarsening and stabilizes the chain in a state analogous to that of a coarsening chain of age 1/gamma;. We present an analytical theory for this steady state, and demonstrate it to be in excellent agreement with our simulation results.

Dynamic ultrametricity in spin glasses

We investigate the dynamics of spin glasses from the "rheological" point of view, in which aging is suppressed by the action of small, nonconservative forces. The different features can be expressed in terms of the scaling of relaxation times with the magnitude of the driving force, which plays the role of the critical parameter. Stated in these terms, ultrametricity loses much of its mystery and can be checked rather easily. This approach also seems a natural starting point to investigate what would be the real-space structures underlying the hierarchy of time scales. We study in detail the appearance of this many-scale behavior in a mean-field model, in which dynamic ultrametricity is clearly present. A similar analysis is performed on numerical results obtained for a three-dimensional spin glass: In that case, our results are compatible with either that dynamic ultrametricity is absent or that it develops so slowly that even in experimental time-windows it is still hardly observable.

A two-time-scale, two-temperature scenario for nonlinear rheology

We investigate a general scenario for "glassy" or "jammed" systems driven by an external, nonconservative force, analogous to a shear force in a fluid. In this scenario, the drive results in the suppression of the usual aging process, and the correlation and response functions become time translation invariant. The relaxation time and the response functions are then dependent on the intensity of the drive and on temperature. We investigate this dependence within the framework of a dynamical closure approximation that becomes exact for disordered, fully connected models. The relaxation time is shown to be a decreasing function of the drive ("shear thinning" effect). The correlation functions below the glass transition temperature (Tc) display a two-time-scale relaxation pattern, similar to that observed at equilibrium slightly above Tc. We also study the violation of the fluctuation-dissipation relationship in the driven system. This violation is very reminiscent of the one that takes place in a system aging below Tc at zero drive. It involves, in particular the appearance of a two-temperature regime, in the sense of an effective fluctuation-dissipation temperature [L. F. Cugliandolo, J. Kurchan, and L. Peliti, Phys. Rev. E 55, 3898 (1997)]. Although our results are, in principle, limited to the closure relations that hold for mean-field models, we argue that a number of the salient features are not inherent to the approximation scheme, and may be tested in experiments and simulations.

Retrieval properties of a Hopfield model with random asymmetric interactions

The process of pattern retrieval in a Hopfield model in which a random antisymmetric component is added to the otherwise symmetric synaptic matrix is studied by computer simulations. The introduction of the anti-symmetric component is found to increase the fraction of random inputs that converge to the memory states. However, the size of the basin of attraction of a memory state does not show any significant change when asymmetry is introduced in the synaptic matrix. We show that this is due to the fact that the spurious fixed points, which are destabilized by the introduction of asymmetry, have very small basins of attraction. The convergence time to spurious fixed-point attractors increases faster than that for the memory states as the asymmetry parameter is increased. The possibility of convergence to spurious fixed points is greatly reduced if a suitable upper limit is set for the convergence time. This prescription works better if the synaptic matrix has an antisymmetric component.

Damage spreading transition in glasses: a probe for the ruggedness of the configurational landscape

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 T0 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 (i) independence of the initial damage; (ii) independence of the class of initial condition; and (iii) 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 states) 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 model) with Monte Carlo heat-bath 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.