This thesis evaluates a method of computing highly accurate solutions for network simulations of integrate-and-fire (IAF) neurons. Simulations are typically evolved using time-stepping, but since the IAF model is composed of linear first-order ODEs with hard thresholds, explicit solutions in terms of integrals of exponentials exist and can be approximated using quadrature. The technique presented here utilizes Clenshaw-Curtis quadrature to approximate these integrals to high accuracy. It uses the secant method to more precisely identify spike times, thus yielding more accurate solutions than do time-stepping methods. Additionally, modeling synaptic input with delta functions permits the quadrature method to be practical for simulating largescale networks. I determine general conditions under which the quadrature method is faster and more accurate than time-stepping methods. In order to make these methods accessible to other researchers, I introduce and develop software designed for simulating networks of IAF hippocampal cells.