A neuron that is stimulated repeatedly by the same time-dependent stimulus exhibits slightly different spike timing at each trial. We compared the exact solution of the time-dependent firing rate for a stochastically spiking neuron model with refractoriness (spike response model) with that of an inhomogeneous Poisson process subject to the same stimulus. To arrive at a mapping between the two models we used alternatively (i) a systematic parameter-free Volterra expansion of the exact solution or (ii) a linear filter combined with nonlinear Poisson rate model (linear-nonlinear Poisson cascade model) with a single free parameter. Both the cascade model and the second-order Volterra model showed excellent agreement with the exact rate dynamics of the spiking neuron model with refractoriness even for strong and rapidly changing input. Cascade rate models are widely used in systems neuroscience. Our method could help to connect experimental rate measurements to the theory of spiking neurons.