Three small universal spiking neural P systems

In this work we give three small spiking neural P systems. We begin by constructing a universal spiking neural P system with extended rules and only 4 neurons. This is the smallest possible number of neurons for a universal system of its kind. We prove this by showing that the set of problems solved by spiking neural P systems with 3 neurons is bounded above by NLNL, and so there exists no such universal system with 3 neurons. If we generalise the output technique we immediately find a universal spiking neural P system with extended rules that has only 3 neurons. This is also the smallest possible number of neurons for a universal system of its kind. Finally, we give a universal spiking neural P system with standard rules and only 7 neurons. In addition to giving a significant improvement in terms of reducing the number of neurons, our systems also offer an exponential improvement on the time and space overheads of the small universal spiking neural P systems of other authors.