Geometrical frustration and cluster spin glass with random graphs.

We develop a based on a sparse random graph to account for the interplay between geometric frustration and disorder in cluster magnetism. Our theory allows introduction of the cluster network connectivity as a controllable parameter. Two types of inner cluster geometry are considered: triangular and tetrahedral. The theory was developed for general, nonuniform intracluster interactions, but in the present paper the results presented correspond to uniform, antiferromagnetic (AF) intraclusters interaction J_{0}/J. The clusters are represented by nodes on a finite connectivity random graph, and the intercluster interactions are randomly Gaussian distributed. The graph realizations are treated in replica theory using the formalism of order parameter functions, which allows one to calculate the distribution of local fields and, as a consequence, the relevant observable. In the case of triangular cluster geometry, there is the onset of a classical spin liquid state at a temperature T^{*}/J and then, a cluster spin glass (CSG) phase at a temperature T_{/}J. The CSG ground state is robust even for very weak disorder or large negative J_{0}/J. These results does not depend on the network connectivity. Nevertheless, variations in the connectivity strongly affect the level of frustration f_{p}=-Θ_{CW}/T_{f} for large J_{0}/J. In contrast, for the nonfrustrated tetrahedral cluster geometry, the CSG ground state is suppressed for weak disorder or large negative J_{0}/J. The CSG boundary phase presents a reentrance which is dependent on the network connectivity.

Ising spin glass in a random network with a Gaussian random field.

We investigate thermodynamic phase transitions of the joint presence of spin glass (SG) and random field (RF) using a random graph model that allows us to deal with the quenched disorder. Therefore, the connectivity becomes a controllable parameter in our theory, allowing us to answer what the differences are between this description and the mean-field theory i.e., the fully connected theory. We have considered the random network random field Ising model where the spin exchange interaction as well as the RF are random variables following a Gaussian distribution. The results were found within the replica symmetric (RS) approximation, whose stability is obtained using the two-replica method. This also puts our work in the context of a broader discussion, which is the RS stability as a function of the connectivity. In particular, our results show that for small connectivity there is a region at zero temperature where the RS solution remains stable above a given value of the magnetic field no matter the strength of RF. Consequently, our results show important differences with the crossover between the RF and SG regimes predicted by the fully connected theory.

Multicritical points and topology-induced inverse transition in the random-field Blume-Capel model in a random network.

The interplay between quenched disorder provided by a random field (RF) and network connectivity in the Blume-Capel (BC) model is the subject of this paper. The replica method is used to average over the network randomness. It offers an alternative analytic route to both numerical simulations and standard mean field approaches. The results reveal a rich thermodynamic scenario with multicritical points that are strongly dependent on network connectivity. In addition, we also demonstrate that the RF has a deep effect on the inverse melting transition. This highly nontrivial type of phase transition has been proposed to exist in the BC model as a function of network topology. Our results confirm that the topological mechanism can lead to an inverse melting transition. Nevertheless, our results also show that as the RF becomes stronger, the paramagnetic phase is affected in such way that the topological mechanism for inverse melting is disabled.

Inverse melting and inverse freezing in a three-state spin-glass model with finite connectivity.

The phase diagrams of the three-state Ghatak-Sherrington spin-glass (or random Blume-Capel) model are obtained in mean-field theory with replica symmetry in order to study the effects of a ferromagnetic bias and finite random connectivity in which each spin is connected to a finite number of other spins. It is shown that inverse melting from a ferromagnetic to a low-temperature paramagnetic phase may appear for small but finite disorder and that inverse freezing appears for large disorder. There can also be a continuous inverse ferromagnetic to spin-glass transition.

Phase transitions in the three-state Ising spin-glass model with finite connectivity.

The statistical mechanics of a two-state Ising spin-glass model with finite random connectivity, in which each site is connected to a finite number of other sites, is extended in this work within the replica technique to study the phase transitions in the three-state Ghatak-Sherrington (or random Blume-Capel) model of a spin glass with a crystal-field term. The replica symmetry ansatz for the order function is expressed in terms of a two-dimensional effective-field distribution, which is determined numerically by means of a population dynamics procedure. Phase diagrams are obtained exhibiting phase boundaries that have a reentrance with both a continuous and a genuine first-order transition with a discontinuity in the entropy. This may be seen as "inverse freezing," which has been studied extensively lately, as a process either with or without exchange of latent heat.

Multistability in networks of Hindmarsh-Rose neurons.

We investigate the dynamical states of a two-dimensional network of Hindmarsh-Rose spiking neurons, in the vicinity of the current threshold where the single neuron becomes active. Each neuron is electrically coupled with neurons in its close neighborhood. The existence of multistable synchronization states is established and discussed. We also show that, provided adequate initial conditions, the collective behavior is able to keep the network in activity, even for current values far below the activity threshold of the single neuron. A phase diagram of the different network states is presented for a large interval of the coupling-current parameter space.

Periodicity and chaos in electrically coupled Hindmarsh-Rose neurons.

The Hindmarsh-Rose (HR) system of equations is a model that captures the essential of the spiking activity of biological neurons. In this work we present an exploratory numerical study of the time activities of two HR neurons interacting through electrical synapses. The knowledge of this simple system is a first step towards the understanding of the cooperative behavior of large neural assemblies. Several periodic and chaotic attractors where identified, as the coupling strength is increased from zero until the perfect synchronization regime. In addition to the known phase locking synchronization at weak coupling, electrical synapses also allow for both in-phase and antiphase synchronization from moderate to strong coupling. A regime where the system changes apparently randomly between in-phase and antiphase locking evolves to a bistability regime, where both in-phase and antiphase periodic attractors are locally stable. At the strong coupling regime in-phase chaotic evolution dominates, but windows with complex periodic behavior are also present.

Retrieval behavior and thermodynamic properties of symmetrically diluted Q-Ising neural networks.

The retrieval behavior and thermodynamic properties of symmetrically diluted Q-Ising neural networks are derived and studied in replica-symmetric mean-field theory generalizing earlier works on either the fully connected or the symmetrical extremely diluted network. Capacity-gain parameter phase diagrams are obtained for the Q=3, Q=4, and Q=infinity state networks with uniformly distributed patterns of low activity in order to search for the effects of a gradual dilution of the synapses. It is shown that enlarged regions of continuous changeover into a region of optimal performance are obtained for finite stochastic noise and small but finite connectivity. The de Almeida-Thouless lines of stability are obtained for arbitrary connectivity, and the resulting phase diagrams are used to draw conclusions on the behavior of symmetrically diluted networks with other pattern distributions of either high or low activity.

Coupling of low-frequency modes with the complex Ginzburg-Landau equation: Generalized Zakharov equations.

In this paper we introduce and examine a generalization of the complex Ginzburg-Landau equation (CGLE) where the self-interaction contained in the cubic term is replaced by a coupling involving the original field and a low-frequency one. New instabilities arise and a radically new asymptotic dynamical behavior emerges displaying defect turbulence over wide regions of the parameter space.

Categorization in fully connected multistate neural network models.

The categorization ability of fully connected neural network models, with either discrete or continuous Q-state units, is studied in this work in replica symmetric mean-field theory. Hierarchically correlated multistate patterns in a two level structure of ancestors and descendents (examples) are embedded in the network and the categorization task consists in recognizing the ancestors when the network is trained exclusively with their descendents. Explicit results for the dependence of the equilibrium properties of a Q=3-state model and a Q=infinity-state model are obtained in the form of phase diagrams and categorization curves. A strong improvement of the categorization ability is found when the network is trained with examples of low activity. The categorization ability is found to be robust to finite threshold and synaptic noise. The Almeida-Thouless lines that limit the validity of the replica-symmetric results, are also obtained.