1900 2000 2100 2200 2300 2400 t -1 0 1 S 1900 2000 2100 2200 2300 2400 t 0 2 4 6 8 Number of states p=0.44, A=0.2 p=0.0, A=0.4 p=0.44, A=0.2 p=0.0, A=0.4 0 5 10 15 20 C 0 0.02 0.04 M p=0.44 p=0.45 0 5 10 15 20 C 5 10 15 20 25 30 35 R FIG. 5: We amplify some representative trajectories (upper panel), and count the number of states through which the system moves in each trajectory (lower panel). Values of T , N and C as in Fig. 2. FIG. 6: Coupling-induced resonance. Main plot: the response R shows a maximum as a function of coupling constant C . As shown in the insert, the same maximum appears as a function of the localization measure M , see section IV. Values of N , T and A as in figure 3 and K = 0 . 2 in the insert. tories are displayed in Figs. 6 and 7, respectively. In the weak coupling limit, the units behave basically as inde- pendent from each other and, as the signal amplitude A is subthreshold for a single variable, the overall response is small. In the large coupling limit, the interaction term is too big to allow an unit that could first follow the signal to depart from the influence of its neighbors. The resonance with the number of units N and some representative trajectories is presented in Figs. 8 and 9. Since fluctuations in the number of repulsive links de- crease with N , a larger system requires a greater fraction of repulsive links to achieve the same level of disorder than a smaller system. As a consequence, the response riodic forcing can be seen, approximately, as a periodic modulation to the potential V eff − S · A sin(2 πt/T ). As discussed above, the effect of the repulsive links is such that V eff ( S ) changes from bistable at p = 0 to having many metastable minima at p ≈ p$_{c}$ and a single absolute minimum for p > p$_{c}$ . Hence, the deep potential barrier separating the S = ± 1 solutions for p = 0 lowers under the effect of the repulsive links. As a consequence, the modulation induced by the periodic forcing is now large enough, and the global variable is then able to oscillate from the minimum + S 0 ( p ) to − S 0 ( p ) and vice versa. As p approaches p$_{c}$ a more complicated scenario appears. In this region the effective potential presents already a rich structure with many metastable minima in the non- forced case. Those minima can be mod ified or even disappear by the effect of the periodic modulation. It is particularly illustrative to compare the responses to a suprathreshold signal of amplitude A = 0 . 4 in the case p = 0, and to a signal of amplitude A = 0 . 2 (which would be subthreshold in the case p = 0) at the optimal fraction of repulsive links p = p$_{c}$ . In both cases, the amplitude of the oscillations is approximately the same, as the system makes large excursions from S ≈ − 1 to S ≈ +1 and vice versa. However, the shape of the oscillations is rather different, as shown in the upper panel of Fig 5. In the p = 0 case, the transition from one minimum to the other is rather fast (vertical portion of the dashed line), while in the case p = p$_{c}$ , the transition is slower as the system seems to be spending more time in intermediate states. To determine what those differences reveal about the underlying effective potential, we have used a method [33] that allows us to detect the number of states a system visits from an analysis of its time series. A typical ex- ample is shown in the lower panel of Fig. 5. We only detect two states in the global variable S when p = 0, corresponding, as expected, to the modulated bistable potential. By contrast, the slight irregularities in the tra- jectory for p = 0 . 44, hardly visible by eye, correspond to several very shallow potential wells. The system evolves through many states at the optimal probability of repul- sive links, as shown in the lower panel of Fig 5. This image explains why signals of every amplitude and pe- riod can be amplified for p ≈ p$_{c}$ . In this case, the system can access the many intermediate states, covering a dis- tance proportional to T and A , in case of very fast or very weak signals. The previous results show that the disorder induced by an intermediate level of repulsive links is an essential ingredient to get an optimal response to the external forc- ing. This can be explained as, in the absence of forcing, the metastable states correspond to a wide distribution for the values of s$_{i}$ ’s. When the forcing is turned on, some units will be responsive to the signal, and then they will pull others which are positively coupled to them. This basic mechanism is further highlighted by the observation of a resonance behavior with both the coupling constant C and the number of units N . The resonance with C and some representative trajec- 4