Spiking neural P systems with rules on synapses (RSSN P systems, for short) are a class of distributed and parallel computation models inspired by the way in which neurons process and communicate information with each other by means of spikes, where neurons only contain spikes and the evolution rules are on synapses. RSSN P systems have been proved to be Turing universal, using the strategy that restricts all the applied rules to consume the same number of spikes from the given neuron, termed as equal spikes consumption strategy. In this work, in order to avoid imposing the equal spikes consumption restriction on the application of rules, a new strategy for rule application, termed as sum spikes consumption strategy, is considered in RSSN P systems, where a maximal set of enabled rules from synapses starting from the same neuron is nondeterministically chosen to be applied, in the sense that no further synapse can use any of its rules, and the sum of these numbers of spikes that all the applied rules consume is removed from the neuron. In this way, the proposed strategy avoids checking whether all the applied rules consume the same number of spikes from the given neuron. The computation power of RSSN P systems working in the proposed strategy is investigated, and it is proved that such systems characterize the semilinear sets of natural numbers, i.e., such systems are not universal. Furthermore, RSSN P systems with weighted synapses working in the proposed strategy are proved to be Turing universal. These results show that the weight on synapses is a powerful ingredient of RSSN P systems in terms of the computation power, which makes RSSN P systems working in sum spikes consumption strategy become universal from non-universality.