The relation between spontaneous and stimulated global brain activity is a fundamental problem in the understanding of brain functions. This question is investigated both theoretically and experimentally within the context of nonequilibrium fluctuation-dissipation relations. We consider the stochastic coarse-grained Wilson-Cowan model in the linear noise approximation and compare analytical results to experimental data from magnetoencephalography of the human brain. The short-time behavior of the autocorrelation function for spontaneous activity is characterized by a double-exponential decay, with two characteristic times, differing by two orders of magnitude. Conversely, the response function exhibits a single-exponential decay in agreement with experimental data for evoked activity under visual stimulation. Results suggest that the brain response to weak external stimuli can be predicted from the observation of spontaneous activity and pave the way to controlled experiments on the brain response under different external perturbations.