  We derive exact analytical expressions for the cumulants of any orders of
neuronal membrane potentials driven by spike trains in a multivariate Hawkes
process model with excitation and inhibition. Such expressions can be used for
the prediction and sensitivity analysis of the statistical behavior of the
model over time, and to estimate the probability densities of neuronal membrane
potentials using Gram-Charlier expansions. Our results are shown to provide a
better alternative to Monte Carlo estimates via stochastic simulations, and
computer codes based on combinatorial recursions are included.
