M. Krupa (USA) and C.M.A. Pinto (Portugal)
Canards, relaxation oscillations, mixed-mode oscillations, localization, symmetry.
We show numerical and analytical evidence of curious phenomena that can occur when two neurons are coupled. We model the internal dynamics of each neuron by the space clamped Hodgkin-Huxley equations. We consider linear diffusive symmetric and asymmetric coupling. The symmetric coupled system is studied by Labouriau and Pinto using bifurcation theory. They find that the two neurons synchronize perfectly (show the same behaviour at all times) for positive values of the coupling constant. As the coupling is decreased towards negative values the two neurons synchronize in a somewhat subtle way, like the movements of the legs in a human walk. For the asymmetric system a variety of curious phenomena arise. The two neurons show relaxation oscillations, localized solutions and mixed mode oscillations. These patterns seem to depend on the values of the coupling. When the ratio between the coupling strengths of the two neurons is far from 1 localization arises. If this ratio is almost 1 then the two neurons show relaxation oscillations. Mixed mode oscillations are the intermediate state.
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