We define and develop a novel process, related to stochastic resonance, in which a particle experiences three forces: a constant drift, a zero-mean white noise, and a time-periodic modulation. Upon reaching a threshold, the particle immediately returns to the starting point. The resulting process exhibits multiple maxima in the output power at the modulation frequency as a function of the white-noise variance, multimodal first-passage-time densities, and evidence of phase locking. Our model is an extension of the Gerstein-Mandelbrot model of neuron firing to the case of periodic stimuli, and therefore has applications in neural modeling.