In this paper, we consider spiking neural P systems with antispikes. Because of the use of two types of objects, the system can encode the binary digits in a natural way and hence represent the formal models more efficiently and naturally than the standard SN P systems. This work deals with the computing power of spiking neural P system with anti-spikes. It is demonstrated that, as transducers, spiking neural P systems with anti-spikes can simulate any Boolean circuit and also computing devices such as finite automata and finite transducers. We also investigate how the use of anti-spikes in spiking neural P systems affect the capability to solve the satisfiability problem.