The theory of Compressive Sensing (CS) exploits a well-known concept used in signal compression – sparsity – to design new, efficient techniques for signal acquisition. CS theory states that for a length-N signal x with sparsity level K, M = O(K log(N/K)) random linear projections of x are sufficient to robustly recover x in polynomial time. However, richer models are often applicable in real-world settings that impose additional structure on the sparse nonzero coefficients of x. Many such models can be succinctly described as a union of K-dimensional subspaces. In recent work, we have developed a general approach for the design and analysis of robust, efficient CS recovery algorithms that exploit such signal models with structured sparsity. We apply our framework to a new signal model which is motivated by neuronal spike trains. We model the firing process of a single Poisson neuron with absolute refractoriness using a union of subspaces. We then derive a bound on the number of random projections M needed for stable embedding of this signal model, and develop a algorithm that provably recovers any neuronal spike train from M measurements. Numerical experimental results demonstrate the benefits of our model-based approach compared to conventional CS recovery techniques.