Motivated by neuroscience discoveries during the last few years, many studies consider pulse-coupled neural networks with spike-timing as an essential component in information processing by the brain. There also exists some technical challenges while simulating the networks of artificial spiking neurons. The existing studies use a Hodgkin–Huxley (H–H) model to describe spiking dynamics and neuro-computational properties of each neuron. But they fail to address the effect of specific non-Gaussian noise on an artificial H–H neuron system. This paper aims to analyze how an artificial H–H neuron responds to add different types of noise using an electrical current and subunit noise model. The spiking and bursting behavior of this neuron is also investigated through numerical simulations. In addition, through statistic analysis, the intensity of different kinds of noise distributions is discussed to obtain their relationship with the mean firing rate, interspike intervals, and stochastic resonance.