Existence and bifurcation of canards in R-3 in the case of a folded node

We give a geometric analysis of canards of folded node type in singularly perturbed systems with two-dimensional (2D) folded critical manifold using the blow-up technique. The existence of two primary canards is known provided a nonresonance condition mu is not an element of N is satisfied, where mu = lambda(1)/lambda(2) denotes the ratio of the eigenvalues of the associated folded singularity of the reduced flow. We show that, due to resonances, bifurcation of secondary canards occurs. We give a detailed geometric explanation of this phenomenon using an extension of Melnikov theory to prove a transcritical bifurcation of canards for odd mu. Furthermore, we show numerically the existence of a pitchfork bifurcation for even mu and a novel turning point bifurcation close to mu is an element of N. We conclude the existence of [(mu - 1)/2] secondary canards away from the resonances. Finally, we apply our results to a network of Hodgkin Huxley neurons with excitatory synaptic coupling and explain the observed slowing of the. ring rate of the synchronized network due to the existence of canards of folded node type.