The objective of system identification methods is to construct a mathematical model of a dynamical system in order to describe adequately the input-output relationship observed in that system. Over the past several decades, mathematical models have been employed frequently in the oculomotor field, and their use has contributed greatly to our understanding of how information flows through the implicated brain regions. However, the existing analyses of oculomotor neural discharges have not taken advantage of the power of optimization algorithms that have been developed for system identification purposes. In this article, we employ these techniques to specifically investigate the "burst generator" in the brainstem that drives saccadic eye movements. The discharge characteristics of a specific class of neurons, inhibitory burst neurons (IBNs) that project monosynaptically to ocular motoneurons, are examined. The discharges of IBNs are analyzed using different linear and nonlinear equations that express a neuron's firing frequency and history (i.e., the derivative of frequency), in terms of quantities that describe a saccade trajectory, such as eye position, velocity, and acceleration. The variance accounted for by each equation can be compared to choose the optimal model. The methods we present allow optimization across multiple saccade trajectories simultaneously. We are able to investigate objectively how well a specific equation predicts a neuron's discharge pattern as well as whether increasing the complexity of a model is justifiable. In addition, we demonstrate that these techniques can be used both to provide an objective estimate of a neuron's dynamic latency and to test whether a neuron's initial firing rate (expressed as an initial condition) is a function of a quantity describing a saccade trajectory (such as initial eye position).