The collective dynamics of neural populations are often characterized in
terms of correlations in the spike activity of different neurons. Open
questions surround the basic nature of these correlations. In particular, what
leads to higher-order correlations -- correlations in the population activity
that extend beyond those expected from cell pairs? Here, we examine this
question for a simple, but ubiquitous, circuit feature: common fluctuating
input arriving to spiking neurons of integrate-and-fire type. We show that
leads to strong higher-order correlations, as for earlier work with discrete
threshold crossing models. Moreover, we find that the same is true for another
widely used, doubly-stochastic model of neural spiking, the linear-nonlinear
cascade. We explain the surprisingly strong connection between the collective
dynamics produced by these models, and conclude that higher-order correlations
are both broadly expected and possible to capture with surprising accuracy by
simplified (and tractable) descriptions of neural spiking