We study the stochastic kinetics of a signaling module consisting of a two-state stochastic point process with negative feedback. In the active state, a product is synthesized which increases the active-to-inactive transition rate of the process. We analyze this simple autoregulatory module using a path-integral technique based on the temporal statistics of state flips of the process. We develop a systematic framework to calculate averages, autocorrelations, and response functions by treating the feedback as a weak perturbation. Explicit analytical results are obtained to first order in the feedback strength. Monte Carlo simulations are performed to test the analytical results in the weak feedback limit and to investigate the strong feedback regime. We conclude by relating some of our results to experimental observations in the olfactory and visual sensory systems.
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