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The multiple attractors induced by a non-resonant double Hopf bifurcation in a two-neuron network with self/neighbor-delayed-connections are investigated in this paper. Various dynamical behaviors are classified in the neighborhood of the bifurcating point and are expressed approximately in a closed form by the center manifold reduction. The solution induced by Hopf bifurcation is expressed approximately by the method of multiple scales. It follows that the delay induces the coexistences of silencing and periodic spiking, two periodic spiking, periodic and quasi-periodic spiking. The basins of attraction are classified numerically. These results have some potential applications in designing the neural network according to the memory pattern and storage.