The paper presents a new artificial neural network (ANN) obtained by embedding a memristor into the self-connection synapse of one neuron from the Sprott ANN. The mathematical model of the new network is described by a five-dimensional (5D) dynamic system. A comprehensive analysis of its dynamic properties is carried out, including bifurcation diagrams, Lyapunov exponents, Kaplan–Yorke dimension, timing diagrams and phase portraits, multistability, and offset boosting control. The theoretical model is further verified by electronic simulation of a chaotic system using Multisim software. A synchronization model of two coupled memristive subneural Sprott networks is proposed to simulate synchronization between regions of biological neural systems. Linearization methods and Lyapunov stability theory are employed to prove synchronization. These results provide useful insights into the nonlinear dynamic characteristics of the new Sprott ANN.