The network oscillations are ubiquitous across many brain regions. In the basal ganglia, oscillations are also present at many levels and a wide range of characteristic frequencies have been reported to occur during both health and disease.The striatum is the input nucleus of the basal ganglia that receives massive glutamatergic inputs from the cortex and is highly susceptible to cortical oscillations. However, there is limited knowledge about the exact nature of this routing process and therefore, it is of key importance to understand how time-dependent, periodic external stimuli propagate through the striatal circuitry. Using a large-scale network model of the striatum and corticostriatal projections, here we try to elucidate the importance of specific GABAergic neurons and their interactions in shaping striatal oscillatory activity. Our results show that fast-spiking interneurons, despite their uncorrelated firing, might have a crucial role in the emergence of high-frequency oscillations in the medium spiny neuron population, even if their activity is kept low. Rather, what matters is the firing time relative to just a few other neurons within an oscillation cycle. Finally, we show how the state of ongoing activity, the strengths of different types of inhibitions, the density of outgoing projections, and the overall activity of striatal cells influence network activity. These results suggest that the propagation of oscillatory inputs into the medium spiny neuron population is efficient, if indirect, through fast-spiking interneurons. Therefore, pharmaceuticals that target fast-spiking interneurons may provide a novel treatment for regaining the spectral characteristics of striatal activity that correspond to the healthy state.