  Describing the collective dynamics of large neural populations using
low-dimensional models for averaged variables has long been an attractive task
in theoretical neuroscience. Recently developed reduction methods make it
possible to derive such models directly from the microscopic dynamics of
individual neurons. To simplify the reduction, the Cauchy distribution is
usually assumed for heterogeneous network parameters. Here we extend the
reduction method for a wider class of heterogeneities defined by the
$q$-Gaussian distribution. The shape of this distribution depends on the
Tsallis index $q$ and gradually changes from the Cauchy distribution to the
normal Gaussian distribution as this index changes. We derive the mean-field
equations for an inhibitory network of quadratic integrate-and-fire neurons
with a $q$-Gaussian distributed excitability parameter. It is shown that the
dynamic modes of the network significantly depend on the form of the
distribution determined by the Tsallis index. The results obtained from the
mean-field equations are confirmed by numerical simulation of the microscopic
model.
