Numerical Spiking Neural P Systems.

Spiking neural P (SN P) systems are a class of discrete neuron-inspired computation models, where information is encoded by the numbers of spikes in neurons and the timing of spikes. However, due to the discontinuous nature of the integrate-and-fire behavior of neurons and the symbolic representation of information, SN P systems are incompatible with the gradient descent-based training algorithms, such as the backpropagation algorithm, and lack the capability of processing the numerical representation of information. In this work, motivated by the numerical nature of numerical P (NP) systems in the area of membrane computing, a novel class of SN P systems is proposed, called numerical SN P (NSN P) systems. More precisely, information is encoded by the values of variables, and the integrate-and-fire way of neurons and the distribution of produced values are described by continuous production functions. The computation power of NSN P systems is investigated. We prove that NSN P is Turing universal as number generating devices, where the production functions in each neuron are linear functions, each involving at most one variable; as number accepting devices, NSN P systems are proved to be universal as well, even if each neuron contains only one production function. These results show that even if a single neuron is simple in the sense that it contains one or two production functions and the production functions in each neuron are linear functions with one variable, a network of simple neurons are still computationally powerful. With the powerful computation power and the characteristic of continuous production functions, developing learning algorithms for NSN P systems is potentially exploitable.