The self-pulsing regime of a microhollow cathode discharge in argon is reported. The plasma is generated inside the hole drilled in an anode-dielectric-cathode device. The hole dimension ranges from 200 to 400 mu m and the gas pressure ranges from 40 to 200 Torr. It is shown by optical spectroscopy and fast CCD imaging that the current pulse is related to a fast expansion of the plasma outside the microhole on the cathode backside. The pulse current duration ranges from 0.4 to 2 mu s depending on the gas pressure. The self- pulsing regime occurs at medium current range (0.1-1mA). At lower current the discharge is steady and the plasma is confined inside the hole (abnormal regime); at higher current, the plasma is steady and the plasma expands outside the hole on the cathode backside. The self-pulsing frequency is a linear function of the averaged discharge current and decreases with the device capacitance. The dependence of the self-pulsing characteristics (frequency, light emission, power deposition, etc) on the gas pressure follows a Paschen-like law; this is interpreted in considering that the fast expansion of the plasma outside the hole is similar to a gas breakdown. A simple electrical model, using a bistable voltage-controlled variable resistor to simulate the evolution of the plasma impedance, provides qualitative results in good agreement with the experiments.