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: 2 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: J$_{ij}$ = J$_{ji}$ = { − 1 , with probability p, 1 , with probability 1 − p. (2) FIG. 1: We plot in a gray scale the stationary probability distribution P ( S ), in the absence of external signal A = 0, coming from numerical simulations of Eqs. (1). For better viewing, the distribution has been rescaled by its maximum value at each p . The data show that at p < p c ≈ 0 . 44 the sys- tem presents two equivalent absolute maxima for P ( s ), while there is only one absolute maximum for p > p c . We note, however, that there are many relative maxima for all values of p , specially around the region p ≈ p c . Other parameter values are: N = 200, C = 8. The probability has been com- puted after averaging over 1000 realizations of the couplings J ij and initial conditions drawn from an uniform distribu- tion in the interval ( − 1 , 1). For the numerical integration we used a fourth-order Runge-Kutta method with a time step Δ t = 0 . 1. 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. R = 4 A − 2 ∣ 〈 e − i 2 $^{πt/T}$S ( t ) 〉 ∣ (3) 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]. 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 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 V ( s 1 , . . . , s$_{N}$ ) = N ∑ i =1   − s 2 i 2 + s 4 i 4 + C 4 N N ∑ j =1 J$_{ij}$ ( s$_{i}$ − s$_{j}$ ) 2   (4) III. SIGNAL AMPLIFICATION