The bifurcation study of 1:2 resonance in a delayed system of two coupled neurons ∗ Guihong Fan a , Sue Ann Campbell b , Gail Wolkowicz c and Huaiping Zhu d a School of Mathematical and Statistical Sciences Arizona State University, Tempe, AZ 85287, USA b Department of Applied Mathematics University of Waterloo, Waterloo, Ontario N2L 3G1 Canada c Department of Mathematics & Statistics McMaster University, Hamilton, Ontario L8S 4K1, Canada d Department of Mathematics and Statistics York University, Toronto,ON, M3J 1P3, Canada. Abstract In this paper, we consider a delayed system of differential equations modeling two neurons: one is excitatory, the other is inhibitory. We study the stability and bifurcations of the trivial equilibrium. Using center manifold theory for delay differential equations, we develop the uni- versal unfolding of the system when the trivial equilibrium point has a double zero eigenvalue. In particular, we show a universal unfolding may be obtained by perturbing any two of the parameters in the system. Our study shows that the dynamics on the center manifold are char- acterized by a planar system whose vector field has the property of 1:2 resonance, also frequently referred as the Bogdanov-Takens bifurcation with Z 2 symmetry. We show that the unfolding of the singularity exhibits Hopf bifurcation, pitchfork bifurcation, homoclinic bifurcation, and fold bifurcation of limit cycles. The symmetry gives rise to a “figure-eight” homoclinic orbit. Key words. Delay, 1:2 resonance, Bogdanov-Takens bifurcation, Center manifold, Normal form, Universal unfolding, Neurons, Hopf Bifurcation, Homoclinic orbit. 1 Introduction In 1984, Hopfield [17] introduced a simplified neural network model which is described by a system of first order differential equations, assuming instantaneous updating of each neuron * The research of Campbell and Wolkowicz is partially supported by NSERC, Zhu is supported by NSERC and an Early Researcher Award, Ministry of Research & Innovation of Ontario, Canada. 1