Spiking neural P systems with weights are a new class of distributed and parallel computing models inspired by spiking neurons. In such models, a neuron fires when its potential equals a given value (called a threshold). In this work, spiking neural P systems with thresholds (SNPT systems) are introduced, where a neuron fires not only when its potential equals the threshold but also when its potential is higher than the threshold. Two types of SNPT systems are investigated. In the first one, we consider that the firing of a neuron consumes part of the potential (the amount of potential consumed depends on the rule to be applied). In the second one, once a neuron fires, its potential vanishes (i.e., it is reset to zero). The computation power of the two types of SNPT systems is investigated. We prove that the systems of the former type can compute all Turing computable sets of numbers and the systems of the latter type characterize the family of semilinear sets of numbers. The results show that the firing mechanism of neurons has a crucial influence on the computation power of the SNPT systems, which also answers an open problem formulated in Wang, Hoogeboom, Pan, Păun, and Pérez-Jiménez ( 2010 ).

Spiking neural P systems (SN P systems) are a class of distributed parallel computing devices inspired by the way neurons communicate by means of spikes; neurons work in parallel in the sense that each neuron that can fire should fire, but the work in each neuron is sequential in the sense that at most one rule can be applied at each computation step. In this work, with biological inspiration, we consider SN P systems with the restriction that at each step, one of the neurons (i.e., sequential mode) or all neurons (i.e., pseudo-sequential mode) with the maximum (or minimum) number of spikes among the neurons that are active (can spike) will fire. If an active neuron has more than one enabled rule, it nondeterministically chooses one of the enabled rules to be applied, and the chosen rule is applied in an exhaustive manner (a kind of local parallelism): the rule is used as many times as possible. This strategy makes the system sequential or pseudo-sequential from the global view of the whole network and locally parallel at the level of neurons. We obtain four types of SN P systems: maximum/minimum spike number induced sequential/pseudo-sequential SN P systems with exhaustive use of rules. We prove that SN P systems of these four types are all Turing universal as number-generating computation devices. These results illustrate that the restriction of sequentiality may have little effect on the computation power of SN P systems.

Different biological processes take different times to be completed, which can also be influenced by many environmental factors. In this work, a realistic definition of nonsynchronized spiking neural P systems (SN P systems, for short) is considered: during the work of an SN P system, the execution times of spiking rules cannot be known exactly (i.e., they are arbitrary). In order to establish robust systems against the environmental factors, a special class of SN P systems, called time-free SN P systems, is introduced, which always produce the same computation result independent of the execution times of the rules. The universality of time-free SN P systems is investigated. It is proved that these P systems with extended rules (several spikes can be produced by a rule) are equivalent to register machines. However, if the number of spikes present in the system is bounded, then the power of time-free SN P systems falls, and in this case, a characterization of semilinear sets of natural numbers is obtained.