The transition towards the brain state induced by psychedelic drugs is frequently neglected in favor of a static description of their acute effects. We use a time-dependent whole-brain model to reproduce large-scale brain dynamics measured with fMRI from 15 volunteers under 20 mg intravenous N,N-Dimethyltryptamine (DMT), a short-acting psychedelic. To capture its transient effects, we parametrize the proximity to a global bifurcation using a pharmacokinetic equation. Simulated perturbations reveal a transient of heightened reactivity concentrated in fronto-parietal regions and visual cortices, correlated with serotonin 5HT2a receptor density, the primary target of psychedelics. These advances suggest a mechanism to explain key features of the psychedelic state and also predicts that the temporal evolution of these features aligns with pharmacokinetics. Our results contribute to understanding how psychedelics introduce a transient where minimal perturbations can achieve a maximal effect, shedding light on how short psychedelic episodes may extend an overarching influence over time.