A variety of nonlinear circuits can be tamed to reproduce the main dynamical properties in neural activities for biological neurons while designing reliable artificial synapses for connecting these neural circuits becomes a challenge. In this paper, a Josephson junction is used to build coupling channel for connecting two FitzHugh-Nagumo neural circuits driven by periodical voltage source. The hybrid synapse is designed by using a linear resistor paralleled with a Josephson junction, which can estimate the effect of external magnetic field by generating additive phase error between the junction. Indeed, it activates nonlinear coupling due to phase diversity in the Josephson junction. The coupled circuits are estimated in dimensionless dynamical systems by applying scale transformation on the variables and parameters in neural circuits. In fact, the intrinsic parameters of coupling channel are adjusted to detect the occurrence of synchronization. Bifurcation analysis is calculated to predict and confirm the occurrence of synchronization between two neural circuits. It is found that synchronization can be stabilized between two FHN neural circuits by selecting appropriate values for parameters in the Josephson junction involved in the coupling channel. It gives useful guidance for implementing artificial synapse for signal processing in neural circuits. © 2020, Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature.