Instability of frozen-in states in synchronous Hebbian neural networks

The full dynamics of a synchronous recurrent neural network model with Ising binary units and a Hebbian learning rule with a finite self-interaction is studied in order to determine the stability to synaptic and stochastic noise of frozen-in states that appear in the absence of both kinds of noise. Both, the numerical simulation procedure of Eissfeller and Opper and a new alternative procedure that allows to follow the dynamics over larger time scales have been used in this work. It is shown that synaptic noise destabilizes the frozen-in states and yields either retrieval or paramagnetic states for not too large stochastic noise. The indications are that the same results may follow in the absence of synaptic noise, for low stochastic noise.Comment: 14 pages and 4 figures; accepted for publication in J. Phys. A: Math. Ge

Revisiting the effect of external fields in Axelrod's model of social dynamics

The study of the effects of spatially uniform fields on the steady-state properties of Axelrod's model has yielded plenty of controversial results. Here we re-examine the impact of this type of field for a selection of parameters such that the field-free steady state of the model is heterogeneous or multicultural. Analyses of both one and two-dimensional versions of Axelrod's model indicate that, contrary to previous claims in the literature, the steady state remains heterogeneous regardless of the value of the field strength. Turning on the field leads to a discontinuous decrease on the number of cultural domains, which we argue is due to the instability of zero-field heterogeneous absorbing configurations. We find, however, that spatially nonuniform fields that implement a consensus rule among the neighborhood of the agents enforces homogenization. Although the overall effects of the fields are essentially the same irrespective of the dimensionality of the model, we argue that the dimensionality has a significant impact on the stability of the field-free homogeneous steady state

A Population Genetic Approach to the Quasispecies Model

A population genetics formulation of Eigen's molecular quasispecies model is proposed and several simple replication landscapes are investigated analytically. Our results show a remarcable similarity to those obtained with the original kinetics formulation of the quasispecies model. However, due to the simplicity of our approach, the space of the parameters that define the model can be explored. In particular, for the simgle-sharp-peak landscape our analysis yelds some interesting predictions such as the existence of a maximum peak height and a mini- mum molecule length for the onset of the error threshold transition.Comment: 16 pages, 4 Postscript figures. Submited to Phy. Rev.

Complementarity and diversity in a soluble model ecosystem

Complementarity among species with different traits is one of the basic processes affecting biodiversity, defined as the number of species in the ecosystem. We present here a soluble model ecosystem in which the species are characterized by binary traits and their pairwise interactions follow a complementarity principle. Manipulation of the species composition, and so the study of its effects on the species diversity is achieved through the introduction of a bias parameter favoring one of the traits. Using statistical mechanics tools we find explicit expressions for the allowed values of the equilibrium species concentrations in terms of the control parameters of the model

Error threshold in the evolution of diploid organisms

The effects of error propagation in the reproduction of diploid organisms are studied within the populational genetics framework of the quasispecies model. The dependence of the error threshold on the dominance parameter is fully investigated. In particular, it is shown that dominance can protect the wild-type alleles from the error catastrophe. The analysis is restricted to a diploid analogue of the single-peaked landscape.Comment: 9 pages, 4 Postscript figures. Submitted to J. Phy. A: Mat. and Ge

Synchronous versus sequential updating in the three-state Ising neural network with variable dilution

The three-state Ising neural network with synchronous updating and variable dilution is discussed starting from the appropriate Hamiltonians. The thermodynamic and retrieval properties are examined using replica mean-field theory. Capacity-temperature phase diagrams are derived for several values of the pattern activity and different gradations of dilution, and the information content is calculated. The results are compared with those for sequential updating. The effect of self-coupling is established. Also the dynamics is studied using the generating function technique for both synchronous and sequential updating. Typical flow diagrams for the overlap order parameter are presented. The differences with the signal-to-noise approach are outlined.Comment: 21 pages Latex, 12 eps figures and 1 ps figur

Landscape statistics of the p-spin Ising model

The statistical properties of the local optima (metastable states) of the infinite range Ising spin glass with p-spin interactions in the presence of an external magnetic field h are investigated analytically. The average number of optima as well as the typical overlap between pairs of identical optima are calculated for general p. Similarly to the thermodynamic order parameter, for p>2 and small h the typical overlap q_t is a discontinuous function of the energy. The size of the jump in q_t increases with p and decreases with h, vanishing at finite values of the magnetic field.Comment: 12 pages,te

Bi-stability of mixed states in neural network storing hierarchical patterns

We discuss the properties of equilibrium states in an autoassociative memory model storing hierarchically correlated patterns (hereafter, hierarchical patterns). We will show that symmetric mixed states (hereafter, mixed states) are bi-stable on the associative memory model storing the hierarchical patterns in a region of the ferromagnetic phase. This means that the first-order transition occurs in this ferromagnetic phase. We treat these contents with a statistical mechanical method (SCSNA) and by computer simulation. Finally, we discuss a physiological implication of this model. Sugase et al. analyzed the time-course of the information carried by the firing of face-responsive neurons in the inferior temporal cortex. We also discuss the relation between the theoretical results and the physiological experiments of Sugase et al.Comment: 18 pages, 6 figure

Finite-size scaling of the quasiespecies model

We use finite-size scaling to investigate the critical behavior of the quasiespecies model of molecular evolution in the single-sharp-peak replication landscape. This model exhibits a sharp threshold phenomenon at Q=Q_c=1/a, where Q is the probability of exact replication of a molecule of length L and a is the selective advantage of the master string. We investigate the sharpness of the threshold and find that its characteristic persist across a range of Q of order L^(-1) about Q_c. Furthermore, using the data collapsing method we show that the normalized mean Hamming distance between the master string and the entire population, as well as the properly scaled fluctuations around this mean value, follow universal forms in the critical region.Comment: 8 pages,tex. Submitted to Physical Review