Periodic breathing is often associated with heart disease or stroke, and commonly Cheyne-Stokes breathing has a period of about a minute. Periodic breathing also commonly occurs in healthy subjects at high altitude, and here the periods may be much shorter, of the order of 15-20 s. In this paper we study such periodic breathing using the classical model of Grodins et al. (1967, J. Appl. Physiol. 22, 260-276), together with a prescription for the dependence of ventilation on the blood CO2 concentration, modulated by the reduced oxygen pressure (the 'Oxford fan'). The model focusses on the fast dynamics of the arterial blood CO2, and differs in this respect from our previous work which emphasised the brain CO2 concentration; in this sense our model is in fact a generalization of the conceptually simpler Mackey-Glass model.