Spiking neural P systems are a class of distributed parallel computing devices inspired from the way neurons communicate by means of spikes. In this paper, the problem of looking for small universal computing devices is investigated in the framework of spiking neural P systems. A new approach is introduced to simulate register machines by spiking neural P systems, where only one neuron is used for all instructions of the simulated register machine; in this way, less neurons are used to construct universal spiking neural P systems working in exhaustive mode. Specifically, a universal spiking neural P system with 36 neurons is constructed, which works in exhaustive mode. This significantly improves the already known result, where 125 neurons are used.