We use the formalism of Quantitative Hydrodynamics to improve the quantitative hydrodynamic limit obtained in Chariker, De Masi, Lebowitz and Presutti (2023) for an interacting particle system inspired by integrate-and-fire neuron models. Our estimates are good enough to show that the typical fluctuations around the aforementioned hydrodynamic limit are Gaussian, and governed by an inhomogeneous stochastic linear equation. Joint work with Julian Amorim (IMPA) and Yangrui Xiang (Louisiana State University)