We propose a stochastic model for the firing activity of a neuronal unit. It includes the decay effect of the membrane potential in absence of stimuli, and the occurrence of time-varying excitatory inputs governed by a Poisson process. The sample-paths of the membrane potential are piecewise exponentially decaying curves with jumps of random amplitudes occurring at the input times. An analysis of the probability distributions of the membrane potential and of the firing time is performed. In the special case of time-homogeneous stimuli the firing density is obtained in closed form, together with its mean and variance.