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.

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.

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 $f_p=-{\mathrm{\Theta }}_{CW}/T_f$ 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.

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.

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.