  We consider the criticality for firing structures of a simplified
integrate-and-fire neural model on the regular network, small-world network,
and random networks. We simplify an integrate-and-fire model suggested by
Levina, Herrmann and Geisel (LHG). In our model we set up the synaptic strength
as a constant value. We observed the power law behaviors of the probability
distribution of the avalanche size and the life time of the avalanche. The
critical exponents in the small-world network and the random network were the
same as those in the fully connected network. However, in the regular
one-dimensional ring, the model does not show the critical behaviors. In the
simplified LHG model, the short-cuts are crucial role in the self-organized
criticality. The simplified LHG model in three types of networks such as the
fully connected network, the small-world network, and random network belong to
the same universality class.
