content_image/416929.jpg content_image/416921.jpg content_image/416924.jpg content_image/416916.jpg
where t is the dimensionless time, C measures the coupling strength amongst the different units and A sin(2 πt/T ) is a periodic external signal with amplitude A and period T . The interaction matrix J$_{ij}$ reflects the presence of attractive and repulsive interactions between the units. More specifically, we adopt the following val- ues at random: 

The single-element case, N = 1, with added noise is the prototypical double-well potential system for which stochastic resonance was first considered. The case with- out repulsive interactions, p = 0, can still be described globally by a bistable potential (see next section) and, in the presence of noise, has been widely studied as a model case for stochastic resonance in extended systems [7]; it has also been considered, in the presence of a random field, as a prototypical example for the diversity-induced resonance effect [21]. For p > 0, the coexistence of at- tractive and repulsive interactions is characteristic of a wide class of spin-glass-type systems [30]. 

We will focus on the macroscopic variable S ( t ) = 1 N ∑ $_{i}$s$_{i}$ ( t ), and use as a measure of response the spec- tral power amplification factor [31], defined as the ratio of the output to input power at the corresponding driving frequency: 

where 〈· · · 〉 is a time average. R is roughly proportional to the amplitude of the oscillations of S ( t ). If R < 1, then the amplitude of the response is less than that of the signal, and vice versa for R > 1. 

It is convenient to analyze first the structure of the steady-state solutions for the system of equations (1) in the non-forced case, A = 0. The dynamics is relaxational ds$_{i}$ dt = − ∂V ∂s$_{i}$ [32], being 

<img src='content_image/416929.jpg'>

the Lyapunov potential. Therefore, the stable steady states are the configurations ( s 1 , . . . , s$_{N}$ ) which are abso- lute minima of V . If there are no repulsive links, p = 0, the Lyapunov potential has just two equivalent minima at s$_{i}$ = +1 or s$_{i}$ = − 1, ∀ i = 1 , . . . , N and, hence, the macroscopic variable will reach the stable asymptotic val- ues S = +1 or S = − 1, a typical situation of bistability. Whether S = +1 or S = − 1 is reached depends exclu- sively on the initial conditions. As p increases, the abso- lute minima depart from S = ± 1 and, furthermore, new 

<img src='content_image/416921.jpg'>

## III. SIGNAL AMPLIFICATION 

<img src='content_image/416924.jpg'>

<img src='content_image/416916.jpg'>

metastable minima of V appear. The dynamical equa- tions (1) may or may not get stuck in one of these min- ima, depending on initial conditions and the particular realization of the coupling constants J$_{ij}$ . We have used throughout the paper random initial conditions drawn from a uniform distribution in the ( − 1 , 1) interval, al- though we have observed the same type of phenomenol- ogy when using other random , but still symmetric, dis- tributions such as truncated Gaussian or the Johnson family of distributions. 

¿From our simulations we compute numerically the probability distribution P ( S ) of the final values of S reached during the dynamical evolution for different re- alizations of the coupling constants J$_{ij}$ and initial condi- tions. This is plotted in Fig. 1. We can observe a second- order phase transition as the average value 〈| S ( t ) |〉 van- ishes for p > p$_{c}$ ≈ 0 . 44. One can interpret these results in terms of an effective potential V eff ( S ) ≡ − ln P ( S ) which has two equivalent absolute minima at S = ± S 0 ( p ), where 1 > S 0 ( p ) > 0 for 0 < p < p$_{c}$ , and one absolute minimum at S = 0 for p ≥ p$_{c}$ . The effective potential V eff presents many relative minima for all values of p > 0, especially in the critical region p ≈ p$_{c}$ , a typical situation for the spin-glass models [30]. 

