Anomalous Random Neural Networks: a Special Renewal Process

In this paper we propose an open anomalous semi-Markovian random neural networks model with negative and positive signals with arbitrary random waiting times. We investigate the signal flow process in the anomalous random neural networks based on renewal process, and obtain the corresponding master equation for time evolution of the probability of the potential of the neurons. As examples, we discuss the special cases of exponential waiting times and power law ones, and find the fractional memory effect of the probability of the system state on its history evolution. Besides, the closed random neural networks model is introduced and the corresponding rate equation is given.