  In [CG16], Cerf and Gorny constructed a model of self-organized criticality,
by introducing an automatic control of the temperature parameter in the
generalized Ising Curie-Weiss model. In this article, we build upon this model
by replacing the mean-field interaction of [CG16] with a one-dimensional
interaction with a certain range d(n) which varies as a function of the number
n of particles. In the Gaussian case, we show that, for a very long range of
interaction (d(n) of order n), the model exhibits the same behaviour as in the
mean-field case, whereas in the case of a nearest neighbour interaction (d(n) =
1), the behaviour highlighted by Cerf and Gorny breaks out.
