  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 heavy tailed random
variable belonging to the domain of attraction of an $\alpha-$ stable law and
scaled by $N^{-1/\alpha},$ where $0 < \alpha <2 .$ We call these jumps
collateral jumps. Moreover, in case $ 0 < \alpha < 1, $ the jumping particle
itself undergoes a macroscopic, main jump. Such systems appear in the modeling
of large neural networks, such as the human brain.
  The particular scaling of the collateral jumps implies that the limit of the
empirical measures of the system is random and equals the conditional
distribution of one typical particle in the limit system, given the source of
common noise. Thus the system exhibits the conditional propagation of chaos
property. The limit system turns out to be solution of a non-linear SDE, driven
by an $ \alpha-$stable process. We prove strong unique existence of the limit
system and introduce a suitable coupling to obtain the strong convergence of
the finite to the limit system, together with precise error bounds for finite
time marginals.
