We investigated the quantitative relation between neuronal activity and blood flow by means of a general parametric mathematical model which described the neuro-vascular system as being dynamic, linear, time-invariant, and subjected to additive noise. The model was constructed from measurements by means of system identification methods and validated across experiments. We sought to cover the system response to multiple stimulation frequencies and durations by a single model. We used the model to investigate the transport delay, the linear order, the deviations from linearity, and conditions for linearizability. We exercised the model on data from rat cerebellar cortex. In anesthetized rats, stimulation of the inferior olive caused climbing fiber activity and blood flow changes. Field potential amplitudes were used as an indicator of neuronal activity and blood flow was measured by laser-Doppler flowmetry. In one set of experiments, stimulation frequencies were in the range 2-20 Hz and the stimulation durations were 60 s and 600 s. The transport delay was estimated to be nearly zero, the linear order to be two. The deviations from linearity were consistently characterized as frequency saturation and dips in blood flow responses to stimulation for 60 s, and overgrowth of blood flow responses to stimulation for 600 s. In another set of experiments, stimulation frequencies were in the range 0.5-10 Hz and the stimulation duration was 15 s. The neuro-vascular system could be approximated by the linear model when the stimulation frequencies were restricted to the range 0.5-7 Hz. In conclusion, our model could predict blood flow responses to stimuli of low frequency and short duration.