  Motivated by considerations from neuroscience (macroscopic behavior of large
ensembles of interacting neurons), we consider a population of mean field
interacting diffusions in $\mathbf {R}^m$ in the presence of a random
environment and with spatial extension: each diffusion is attached to one site
of the lattice $\mathbf {Z}^d$, and the interaction between two diffusions is
attenuated by a spatial weight that depends on their positions. For a general
class of singular weights (including the case already considered in the
physical literature when interactions obey to a power-law of parameter
$0<\alpha<d$), we address the convergence as $N\to\infty$ of the empirical
measure of the diffusions to the solution of a deterministic McKean-Vlasov
equation and prove well-posedness of this equation, even in the degenerate case
without noise. We provide also precise estimates of the speed of this
convergence, in terms of an appropriate weighted Wasserstein distance,
exhibiting in particular nontrivial fluctuations in the power-law case when
$\frac{d}{2}\leq\alpha<d$. Our framework covers the case of polynomially
bounded monotone dynamics that are especially encountered in the main models of
neural oscillators.
