**Acknowledgements**

rest state. As *γ* increases, so does the number of cells that become noise-activated during roughly half a cycle of the external signal. For *γ* even higher, the cells' activity enhances its coherence with the external signal as a consequence of its coupling-mediated self-organization: as one neuron activates, it usually inhibits its nearest neighbors. The outcome of this phenomenon is the APS, which partially arises along the ring during the stage of activation by noise. In this scenario, noise (together with coupling and signal) plays a constructive role. Nonetheless for *γ* too

*Solid State Physics - Metastable, Spintronics Materials and Mechanics of Deformable…*

We exploit the knowledge of the NEP in Eq. (20) to attempt an analytical description of the problem in Section 4.3. The case of *perfect* spatiotemporal selforganization would be equivalent to a two-neuron system with variables *u*1, *u*2, *v*1, and *v*<sup>2</sup> and PBC. This simple model allows the formation of an *antiphase* state. Since a NEP cannot be easily found for this system—and with the only purpose of calculating barrier heights—we further reduce this description by *projecting the*

The projected two-variable system turns out to be gradient, a situation in which a NEP can always be found. As a consequence of the PBC, the NEP landscape along the slow manifolds is symmetric with respect to the *u*<sup>1</sup> ¼ *u*<sup>2</sup> line. For *E* ¼ 0*:*5 and maximum signal amplitude, the system has two *uniform* attractors (both cells inhibited, both cells activated), two APS (with one cell activated and one inhibited) with the *same* value of Φð Þ *u*1*; u*<sup>2</sup> , four saddles, and one maximum. For *S* ¼ 0 instead, the uniform attractor with both cells activated has collapsed with the maximum,

When the value of Φð Þ *u*1*; u*<sup>2</sup> at the uniform attractor, either APS and either corresponding saddle, is plotted as a function of *S*, one can see the following:

• Near maximum signal, the uniform attractor yields its stability to the APS. From this value of *S* on, the NEP barrier for the uniform attractor to decay into

One then understands the picture: as *S* increases, whatever of the APS is chosen. As *S* decreases past the collapse, only the uniform attractor survives. However, the neuron which was activated before has not recovered completely. Hence in the next

In Sections 2 and 3, we have discussed the influence of quantum correlations on

the formation of tightly bound solids. Section 2 is devoted to the effects of the overlaps and neglected multicenter integrals on tight-binding band spectra. An exact calculation in the framework of a simple atomic model has shown that they shift *unevenly* the top and bottom of the band spectrum (their effects are more pronounced at the top). Section 3 introduced a quantum Monte Carlo method specific for strongly correlated fermion systems. Section 4 addressed the stochastic

• Way before minimum signal, each APS collapses with its own saddle.

the APS (a noise-activated process) is small enough.

signal cycle, the other APS is more likely to appear.

**5. Conclusions**

**14**

*ϵβu*1*,* <sup>2</sup> � *v*1*,*<sup>2</sup> þ 2*E v*ð 2 þ *v*1 � *v*1*,* <sup>2</sup>Þ ¼ 0*:* (21)

**4.4 Spatiotemporal pattern formation in arrays of FHN neurons**

large, the sync becomes eventually degraded.

*dynamics along the corresponding slow manifolds*:

and, hence, two saddles have disappeared.

The author is deeply indebted with his coauthors G.G. Izús, A.D. Sánchez, and M.G. dell'Erba from IFIMAR-CONICET (Faculty of Exact and Natural Sciences) and D.A. Mirabella and C.M. Aldao from INTEMA-CONICET (Faculty of Engineering) of the National University of Mar del Plata (UNMdP), Argentina, with whom he undertook part of the work referred to here. Support by UNMdP, through Grant EXA826–15/E779, is acknowledged.
