  We try to design a simple model exhibiting self-organized criticality, which
is amenable to a rigorous mathematical analysis. To this end, we modify the
generalized Ising Curie-Weiss model by implementing an automatic control of the
inverse temperature. With the help of exact computations, we show that, in the
case of a centered Gaussian measure with positive variance $\sigma^{2}$, the
sum $S_n$ of the random variables has fluctuations of order $n^{3/4}$ and that
$S_n/n^{3/4}$ converges to the distribution $C \exp(-x^{4}/(4\sigma^4))\,dx$
where $C$ is a suitable positive constant.
