In this work we give a small extended spiking neural P system that is weakly universal. This system is significantly smaller than the smallest strongly universal spiking neural P systems. Strong universality has strict conditions regarding the encoding of input and decoding of output. Weak universality has more relaxed conditions regarding the encoding of input and decoding of output. Păun and Păun [10] gave a strongly universal spiking neural P system with 84 neurons and another that has extended rules with 49 neurons. Subsequently, the number of neurons used for strong universality was reduced from 84 to 67 and from 49 to 41 by Zhang et al. [11]. Here we give a weakly universal spiking neural P system that uses extended rules and has only 12 neurons. 1.

Abstract. We solve the problem of finding the smallest possible universal spiking neural P system with extended rules. We give a universal spiking neural P system with extended rules and only 4 neurons. This is the smallest possible 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 NL, and so there exists no such universal system with 3 neurons (for any reasonable definition of universality). Finally, we show that if we generalise the output technique we can give a universal spiking neural P system with extended rules that has only 3 neurons. This is also the smallest possible universal system of its kind. 1

Abstract. In this work we offer a significant improvement on the previous smallest spiking neural P system. Păun and Păun [3] gave a universal spiking neural P system with 84 neurons. Subsequently, Zhang et al. [18] reduced the number of neurons used to give universality to 67. Here we give a small universal spiking neural P system that has only 11 neurons and uses rules without delay. 1