The stability and the Hopf bifurcation of a delayed neural network model with an inertial term are investigated in this paper. The characteristic equation of the linearized time delay equation is first considered about the trivial solution, and the condition for the existence of a Hopf bifurcation is obtained based on the study of the root location of the characteristic equation. Then, the newly developed method, the pseudo-oscillator analysis, is applied to study the local dynamics round the trivial solution and near the bifurcation point, including the bifurcation direction, the stability of the bifurcation-induced periodic solution, and the estimation of the amplitude of the periodic solution. Finally, two case studies are given to validate the theoretical prediction of the bifurcated periodic solution, which is checked numerically by the plots of the time history, the phase portraits and the bifurcation diagrams. It is shown that the Hopf bifurcation is supercritical, and the bifurcation-induced periodic solution is stable. In addition, the results show that the pseudo-oscillator analysis has several advantages over the center manifold reduction and the normal form theory used in literature: it involves easy calculation about integration of harmonic functions only, it results in simple computational results in terms of the system parameters, and it offers an estimation of the bifurcated periodic solution with a high computational accuracy.