Spiking neural P systems with anti-spikes (ASN P systems, for short) are a variant of spiking neural P systems, which was inspired by inhibitory impulses/spikes or inhibitory synapses. In this work, we consider normal forms of ASN P systems. Specifically, we prove that ASN P systems with pure spiking rules of categories (a, a) and (a, ¯a) without forgetting rules are universal as number generating devices. In an ASN P system with spiking rules of categories (a, ¯a) and (¯a, a) without forgetting rules, the neurons change spikes to anti-spikes or change anti-spikes to spikes; such systems are proved to be universal. We also prove that ASN P systems with inhibitory synapses using pure spiking rules of category (a, a) and forgetting rules are universal. These results answer an open problem and improve a corresponding result from [IJCCC, IV(3), 2009, 273- 282].