  Lorentzian distributions have been largely employed in statistical mechanics
to obtain exact results for heterogeneous systems. Analytic continuation of
these results is impossible even for slightly deformed Lorentzian
distributions, due to the divergence of all the moments (cumulants). We have
solved this problem by introducing a `pseudo-cumulants' expansion. This allows
us to develop a reduction methodology for heterogeneous spiking neural networks
subject to extrinsinc and endogenous noise sources, thus generalizing the
mean-field formulation introduced in [E. Montbri\'o et al., Phys. Rev. X 5,
021028 (2015)].
