  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.
