We analyze a pair of excitable FitzHugh-Nagumo elements, each of which is coupled repulsively. While the rest state for each element is globally stable for a phase-attractive coupling, various firing patterns, including cyclic and chaotic firing patterns, exist in an phase-repulsive coupling region. Although the rest state becomes linearly unstable via a Hopf bifurcation, periodic solutions associated to the firing patterns is not connected to the Hopf bifurcation. This means that the solution

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... ranch emanating from the Hopf bifurcation is subcritical and unstable for any coupling strength. Various types of cyclic firing patterns emerge suddenly through saddle-node bifurcations. The parameter region in which different periodic solutions coexist is also found.