  We analyze the dynamics of large networks of pulse-coupled quadratic
integrate-and-fire neurons driven by Cauchy noise and non-Cauchy heterogeneous
inputs. Two types of heterogeneities defined by families of $q$-Gaussian and
flat distributions are considered. Both families are parametrized by an integer
$n$, so that as $n$ increases, the first family tends to a normal distribution,
and the second tends to a uniform distribution. For both families, exact
systems of mean-field equations are derived and their bifurcation analysis is
carried out. We show that noise and heterogeneity can have qualitatively
different effects on the collective dynamics of neurons.
