Spiking neural P systems (SN P systems, for short) are a class of parallel and distributed computation models inspired from the way the neurons process and communicate information by means of spikes. In this paper, we consider a new variant of SN P systems, where each synapse instead of neuron has a set of spiking rules, and the neurons contain only spikes; when the number of spikes in a given neuron is “recognized” by a rule on a synapse leaving from it, the rule is enabled; at a computation step, at most one enabled spiking rule is applied on a synapse, and spikes are removed from a neuron if the maximum number of spikes that the applied spiking rules on the synapses starting from this neuron consume is . The computation power of this variant of SN P systems is investigated. Specifically, we prove that such SN P systems can generate or accept any set of Turing computable natural numbers. This result gives an answer to an open problem formulated in Theor. Comput. Sci., vol. 529, pp. 82–95, 2014.