  We derive a quantitative version of the hydrodynamic limit for an interacting
particle system inspired by integrate-and-fire neuron models. More precisely,
we show that the $L^2$-speed of convergence of the empirical density of states
in a generalized contact process defined over a $d$-dimensional torus of size
$n$ is of the optimal order $\mathcal O(n^{d/2})$. In addition, we show that
the typical fluctuations around the aforementioned hydrodynamic limit are
Gaussian, and governed by a inhomogeneous stochastic linear equation.
