  We consider a system of $N$ neurons, each spiking randomly with rate
depending on its membrane potential. When a neuron spikes, its potential is
reset to $0$ and all other neurons receive an additional amount $h/N$ of
potential, where $ h > 0$ is some fixed parameter. In between successive
spikes, each neuron's potential undergoes some leakage at constant rate $
\alpha. $ While the propagation of chaos of the system, as $N \to \infty$, to a
limit nonlinear jumping stochastic differential equation has already been
established in a series of papers, see De Masi et al. (2015) and Fournier and
L\"ocherbach (2016), the present paper is devoted to the associated central
limit theorem. More precisely we study the measure valued process of
fluctuations at scale $ N^{-1/2}$ of the empirical measures of the membrane
potentials, centered around the associated limit. We show that this fluctuation
process, interpreted as c\`adl\`ag process taking values in a suitable weighted
Sobolev space, converges in law to a limit process characterized by a system of
stochastic differential equations driven by Gaussian white noise. We complete
this picture by studying the fluctuations, at scale $ N^{-1/2}, $ of a fixed
number of membrane potential processes around their associated limit
quantities, giving rise to a mesoscopic approximation of the membrane
potentials that take into account the correlations within the finite system.
