7 Having the optimal disorder, the system becomes more se nsitive to external signals of every kind. Furthermore, we have shown that varying the number of elements or th e coupling strength in an ensemble of coupled bistable elements can improve the sensitivity to an external forc- in g. These various ways to increase sensitivity make the ph enomenon less dependent on a fine tuning of the pro- po rtion of repulsive links, which can be a positive feature in practical applications. Apparently, when the system siz e becomes very large, it is difficult to get a resonance eff ect, unless we increase the coupling strength by many t imes. Arguably, thi s difficulty can be overcome by other ty pes of network settings [36]. Finally, we have shown that the location of the reso- na nce peaks can be predicted by a spectral analysis of th e Laplacian matrix. In heuristic terms [29], the posi- tive eigenvalues of the Laplacian can be seen to express the contribution of the coupling term to the vulnerabil- ity of the system to perturbations. We conclude that the location of the amplification region, for a given system size and coupling constant, is reasonably independent of the particular dynamical system. In broad terms, it cor- responds to the point where the positive eigenvalues of the Laplacian matrix become localized, signaling a tran- sition to a region where perturbations can accumulate in an incremental manner. The more precise location would depend on the particular dynamical system by means of a condition on open modes. Competitive interactions are widespread in nature, no- tably in biological systems. In those systems and oth- ers, there has been some studies highlighting their role in achieving a coherent behavior in the absence of forc- ing: increasing synchronization [37] or enabling a collec- tive firing [36]. In the present study, we saw they can also help to enhance perception, something that can be potentially relevant in sensory systems. Acknowledgments: We acknowledge financial sup- port by the MEC (Spain) and FEDER (EU) through project FIS2007-60327. TVM is supported by FCT (Portugal) through Grant No.SFRH/BD/23709/2005. VNL acknowledges financial support of NERC (project NE/F005474/1) and postdoctoral fellowship of the AXA Research Fund. We have chosen a very generic double-well model, and have shown that the optimal disorder is the one that destroys the ordered system bistability. The resulting multistable effective potential allows for the amplifica- tion of very weak or fast signals. There is not a need to match specific levels of disorder with specific frequencies. Our main result is that the system collective response is enhanced by the presence of an intermediate fraction of repulsive links. Hence, competitive interactions are taken as a source of disorder, as an alternative to pre- vious studies where disorder was induced by noise [4] or diversity [21], and a similar amplification was verified. In this work, we have analyzed the response to a weak period signal, of a model composed by bistable units cou- pled through both attractive and repulsive links. fraction of repulsive links, increasing the number of open modes, can overcome this situation to a certain degree, allowing for resonances at a smaller coupling strength. V. CONCLUSIONS FIG. 11: Measure of localization M: close to the optimal region there is an increase in the fraction of localized open eigenstates. The small dependance of M with K and C is as expected. Parameters: N = 200. Inset: A = 0 . 2, T = 300. 0.3 0.4 0.5 0.6 p 0 0.01 0.02 0.03 0.04 M K=0 K=0.2, C=15 K=0.2, C=8 0.3 0.4 0.5 0.6 p 0 5 10 15 20 25 30 R C=15 C=8 [8] A. Pikovsky, A. Zaikin, M. A. de la Casa, Phys. Rev. Lett. 88 , 050601 (2002). [9] B. von Haeften, G. Iz´ us, H. S. Wio, Phys. Rev. E, 72 , 021101 (2005). [10] C. J. Tessone, R. Toral, Physica A 351 , 106 (2005). [11] G. Schmidt, I. Goychuk, P. H¨ anggi, Europhys. Lett. 56 , 22 (2001); Phys. Biol. 1 , 61 (2004). [12] P. Jung, U. Behn, E. Pantazelou, F. Moss, Phys. Rev. A 46 , R1709 (1992). [13] M. Morillo, J. G´ omez-Ord´ o˜ nez, J. M. Casado, Phys. Rev. E 52 , 316 (1995). [1] R. Benzi, A. Sutera, A. Vulpiani, J. Phys. A 14 , 453 (1981). [2] C. Nicolis and G. Nicolis, Tellus 33 , 225 (1981). [3] H. A. Kramers, Physica 7 , 284 (1940). [4] L. Gammaitoni, P. H¨ anggi,. P.Jung,. F.Marchesoni,, Rev. Mod. Phys. 70 , 223 (1998). [5] Topical Issue on Stochastic Resonance, Eur. Phys. J. B, 69 , No. 1, (2009), edited by P. H¨ anggi and F. Marchesoni. [6] H. S. Wio, J. A. Revelli, M. A. Rodriguez, R. R. Deza, G. G. Iz´ us, Eur. Phys. J. B, 69 , 71, (2009). [7] R. Benzi, J. Stat. Mech., P01052 (2009).