Many networks used in machine learning and as models of biological neural networks make use of stochastic neurons or neuron-like units. We show that stochastic artificial neurons can be realized on silicon chips by exploiting the quasi-periodic behavior of mismatched analog oscillators to approximate the neuron's stochastic activation function. We represent neurons by finite state machines (FSMs) that communicate using digital events and whose transitions are event-triggered. The event generation times of each neuron are controlled by an analog oscillator internal to that neuron/FSM and the frequencies of the oscillators in different FSMs are incommensurable. We show that within this quasi-periodic system, the transition graph of a FSM can be interpreted as the transition graph of a Markov chain and we show that by using different FSMs, we can obtain approximations of different stochastic activation functions. We investigate the quality of the stochastic interpretation of such a deterministic system and we use the system to realize and sample from a restricted Boltzmann machine. We implemented the quasi-periodic event-based system on a custom silicon chip and we show that the chip behavior can be used to closely approximate a stochastic sampling task.