Precision of neural timing: The small ɛ limit

We explore the precision of neural timing in a model neural system with n identical input neurons whose firing time in response to stimulation is chosen from a density f. These input neurons stimulate a target cell which fires when it receives m hits within ɛ msec. We prove that the density of the firing time of the target cell converges as ɛ→0 to the input density f raised to the mth and normalized. We give conditions for convergence of the density in L1, pointwise, and uniformly as well as conditions for the convergence of the standard deviations.