A recently introduced lattice model, describing an extended system which exhibits a reentrant (symmetry-breaking, second-order) noise-induced nonequilibrium phase transition, is studied under the assumption that the multiplicative noise leading to the transition is colored. Within an effective Markovian approximation and a mean-field scheme it is found that when the self-correlation time $\tau$ of the noise is different from zero, the transition is also reentrant with respect to the spatial coupling D. In other words, at variance with what one expects for equilibrium phase transitions, a large enough value of D favors disorder. Moreover, except for a small region in the parameter subspace determined by the noise intensity $\sigma$ and D, an increase in $\tau$ usually prevents the formation of an ordered state. These effects are supported by numerical simulations.

We obtain an analytically exact nonequilibrium potential for the space-independent FitzHugh-Nagumo model, valid in the excitable and bistable regimes (in the limit of small noise). This potential allows us to characterize the nature of the barrier in the excitable regime and, after subjecting the system to a modulated weak signal that rocks the potential inducing the phenomenon of stochastic resonance, to calculate the signal-to-noise ratio in the bistable regime.

In order to perform numerical simulations of the Kardar-Parisi-Zhang (KPZ) equation, in any dimensionality, a spatial discretization scheme must be prescribed. The known fact that the KPZ equation can be obtained as a result of a Hopf-Cole transformation applied to a diffusion equation (with multiplicative noise) is shown here to strongly restrict the arbitrariness in the choice of spatial discretization schemes. On one hand, the discretization prescriptions for the Laplacian and the nonlinear (KPZ) term cannot be independently chosen. On the other hand, since the discretization is an operation performed on space and the Hopf-Cole transformation is local both in space and time, the former should be the same regardless of the field to which it is applied. It is shown that whereas some discretization schemes pass both consistency tests, known examples in the literature do not. The requirement of consistency for the discretization of Lyapunov functionals is argued to be a natural and safe starting point in choosing spatial discretization schemes. We also analyze the relation between real-space and pseudospectral discrete representations. In addition we discuss the relevance of the Galilean-invariance violation in these consistent discretization schemes and the alleged conflict of standard discretization with the fluctuation-dissipation theorem, peculiar of one dimension.

Recent massive numerical simulations have shown that the response of a “stochastic resonator” is enhanced as a consequence of spatial coupling. Similar results have been analytically obtained in a reaction-diffusion model, using nonequilibrium potential techniques. We now consider a field-dependent diffusivity and show that the selectivity of the coupling is more efficient for achieving stochastic-resonance enhancement than its overall value in the constant-diffusivity case.

As a mock-up of synaptic transmission between neurons, we revisit a problem that has recently risen the interest of several authors: the propagation of a low-frequency periodic signal through a chain of one-way coupled bistable oscillators, subject to uncorrelated additive noise. On a numerical study performed in the optimal range of noise intensity for which essentially undamped transmission along the chain has been reported, we focus on the outcome of feeding with noise all the nodes but the central one. A (moderate) critical value of the coupling between oscillators is found such that whereas below it the propagation can be considered to be interrupted at the “dead neuron,” it is reestablished above it. Thus, one of the distinguishing features of synaptic transmission, namely, a fault-tolerant behavior that enhances reliability at the expense of efficiency, arises here as an emergent property of the system.