  With increasing requirements for greater precision, it becomes essential to
describe the effects of isospin breaking induced by both quark masses and
electro-magnetic effects. In this work we perform a lattice analysis of the
weak decay constants of the neutral pseudoscalar mesons including such isospin
breaking effects, with particular consideration being given to the state mixing
of the $\pi^0$, $\eta$ and $\eta^\prime$. We also detail extensions to the
non-perturbative RI$^\prime$-MOM renormalization scheme for application to
non-degenerate flavour-neutral operators which are permitted to mix, and
present initial results. Using flavour-breaking expansions in terms of quark
masses and charges we determine the leptonic decay constants for the $\pi^0$
and $\eta$ mesons, demonstrating in principle how precision determinations of
all neutral pseudoscalar decay constants could be reached in lattice QCD with
QED and strong isospin-breaking accounted for.
