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Spiking neural P systems with anti-spikes (ASN P systems, for short) are a variant of spiking neural P systems, which were 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, a̅) without forgetting rules are universal as number generating devices. In an ASN P system with spiking rules of categories (a, a̅) and (a̅, 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].