  Shot noise processes have been extensively studied due to their mathematical
properties and their relevance in several applications. Here, we consider
nonnegative shot noise processes and prove their weak convergence to
L\'evy-driven Ornstein-Uhlenbeck (OU), whose features depend on the underlying
jump distributions. Among others, we obtain the OU-Gamma and OU-Inverse
Gaussian processes, having gamma and inverse gaussian processes as background
L\'evy processes, respectively. Then, we derive the necessary conditions
guaranteeing the diffusion limit to a Gaussian OU process, show that they are
not met unless allowing for negative jumps happening with probability going to
zero, and quantify the error occurred when replacing the shot noise with the OU
process and the non-Gaussian OU processes. The results offer a new class of
models to be used instead of the commonly applied Gaussian OU processes to
approximate synaptic input currents, membrane voltages or conductances modelled
by shot noise in single neuron modelling.
