does not vanish but still remains small, compared to other common control techniques using external periodic signals, for instance, in deep-brain stimulation to suppress neural synchrony in Parkinson’s disease [@TAS02].

The phase portrait and the null-isoclines of a single FitzHugh-Nagumo system without noise and feedback are shown in Fig. \[fig:single\_FHN\](a). The fixed point A is a stable focus or node for $a>1$ ([*excitable regime*]{}). If the system is perturbed well beyond point A’ (see inset), it performs a large excursion $A \to B \to C \to D \to A$ in phase space corresponding to the emission of a spike (Fig. \[fig:single\_FHN\](b)). At $a=1$ the system exhibits a Hopf bifurcation of a limit cycle, and the fixed point A becomes an unstable focus for $a<1$ ([*oscillatory regime*]{}).

![\[fig:single\_FHN\] Excitable dynamics of a single FitzHugh-Nagumo system: (a) Phase portrait $(x_1,y_1)$ (trajectory: solid blue, nullclines: dashed black), (b) Time series of activator $x_1(t)$ (red) and inhibitor $y_1(t)$ (green). The colored dots A, B, C, D mark corresponding points on panels (a) and (b). The inset in (b) shows a blow-up of the phase portrait near A. Parameters: $\epsilon_1=0.01$, $a=1.05$, $D_1=0$.](fig02.eps){width="\textwidth"}

In the following we choose the excitability parameter $a=1.05$ in the excitable regime 