Synchronized Hopf bifurcation analysis in a neural network model with delays

We consider the synchronized periodic oscillation in a ring neural network model with two different delays in self-connection and nearest neighbor coupling. Employing the center manifold theorem and normal form method introduced by Hassard et al., we give an algorithm for determining the Hopf bifurcation properties. Using the global Hopf bifurcation theorem for FDE due to Wu and Bendixson's criterion for high-dimensional ODE due to Li and Muldowney, we obtain several groups of conditions that guarantee the model have multiple synchronized periodic solutions when the transfer coefficient or time delay is sufficiently large.