  We consider a threshold-crossing spiking process as a simple model for the
activity within a population of neurons. Assuming that these neurons are driven
by a common fluctuating input with Gaussian statistics, we evaluate the
cross-correlation of spike trains in pairs of model neurons with different
thresholds. This correlation function tends to be asymmetric in time,
indicating a preference for the neuron with the lower threshold to fire before
the one with the higher threshold, even if their inputs are identical. The
relationship between these results and spike statistics in other models of
neural activity are explored. In particular, we compare our model with an
integrate-and-fire model in which the membrane voltage resets following each
spike. The qualitative properties of spike cross correlations, emerging from
the threshold-crossing model, are similar to those of bursting events in the
integrate-and-fire model. This is particularly true for generalized
integrate-and-fire models in which spikes tend to occur in bursts as observed,
for example, in retinal ganglion cells driven by a rapidly fluctuating visual
stimulus. The threshold crossing model thus provides a simple, analytically
tractable description of event onsets in these neurons.
