Joint work with Dasha Loukianova and Elisa Marini We consider a system of N interacting particles, described by SDEs driven by Poisson random measures, where the coefficients depend on the empirical measure of the system. Every particle jumps with a jump rate depending on its position. When this happens, all the other particles of the system receive a small random kick which is distributed according to a random variable belonging to the domain of attraction of an alpha-stable law, suitably scaled, for some alpha in between 0 and 2. We call these jumps ''collateral jumps''. Moreover, the jumping particle itself undergoes a macroscopic, ''main'' jump. Such systems appear in the modeling of large neural networks, such as the human brain. I will discuss how the system behaves in the large population limit, and how this limiting behavior depends on the value of alpha. In particular, using a representation of the collateral jump sum as a time-changed random walk, I will explain how we obtain convergence in law, in Skorokhod space, of the system to a limit infinite-exchangeable system of SDEs driven by a common stable process (or a common Brownian motion) which arises due to the central limit theorem and constitutes a common noise term in the dynamics of all the limit particles. Finally, I will discuss how this is related to the property of ''conditional propagation of chaos''.