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Spiking neural P systems (SN P systems, for short) are a class of distributed parallel computing devices inspired from the way neurons communicate by means of spikes, where each neuron can have several spiking rules and forgetting rules and neurons work in parallel in the sense that each neuron that can fire should fire at each computation step. In this work, we consider SN P systems with the restrictions: 1) systems are simple (resp. almost simple) in the sense that each neuron has only one rule (resp. except for one neuron); 2) at each step the neuron(s) with the maximum number of spikes among the neurons that can spike will fire. These restrictions correspond to that the systems are simple or almost simple and a global view of the whole network makes the systems sequential. The computation power of simple SN P systems and almost simple SN P systems working in the sequential mode induced by maximum spike number is investigated. Specifically, we prove that such systems are Turing universal as both number generating and accepting devices. The results improve the corresponding ones in Theor. Comput. Sci., 410 (2009), 2982-2991.