a formulation of the model problem that will be analyzed in Sec. III and numerically simulated in Sec. IV, while Sec. V will provide a discussion of the implications of the results obtained.

II. Formulation
===============

We consider the basic Kuramoto model supplemented by the addition of a $\delta $-correlated noise term $n(t)$ and two impulsive stimuli, one at time $t=0$, and the other at time $t=\tau $, $$d\theta _i/dt=\omega _i+K/N\sum ^N_{j=1}\sin (\theta _j-\theta
_i)-h(\theta _i)\Delta (t)+n(t) \ ,$$ $$\Delta (t)=\hat d_0\delta (t)+\hat d_1\delta (t-\tau ) \ ,$$ $$\langle n(t)n(t')\rangle =2\xi \delta (t-t') \ ,$$ $$h(\theta )=\sum _nh_ne^{in\theta } \ , \ \ h_n=h^*_{-n} \ , \ \
h_0=0 \ ,$$ where $h^*_{-n}$ denotes the complex conjugate of $h_{-n}$. In the above $\theta _i(t)$ represents the angular phase of oscillator $i$, where $i=1,2,\ldots ,N\gg 1$; and $\omega _i$ is the natural frequency of oscillator $i$ where we take $\omega _i$ for different oscillators (i.e., different $i$) to be distributed according to some given, time-independent distribution function $g(\omega )$, where $g(\omega )$ has an average frequency $\bar
\omega =\int \omega g(\omega )d\omega $, is symmetric about $\omega =\bar \omega $, and monotonically decreases as $|\omega
-\bar \omega |$ increases.

To motivate 