Loss of synchronization in partially coupled Hodgkin-Huxley equations

We study the loss of synchronization of two partially coupled space-clamped Hodgkin-Huxley equations, with symmetric coupling. This models the coupling of two cells through an electrical synapse. For strong enough coupling it is known that all solutions of the equations approach a state where the two cells are perfectly synchronized, having the same behaviour at each moment. We describe the local bifurcations that arise when the coupling strength is reduced, using a mixture of analytical and numerical methods. We find that perfect synchrony is retained for very small positive values of the coupling strength, for almost all initial conditions. Although perfect synchrony is lost for negative values of the coupling constant, the system always retains some degree of synchronization until it becomes totally unstable. This happens in two ways: in many cases for almost all initial conditions the solutions still approach a perfectly synchronized state. Even when this is not true, the attracting solutions are still synchronized, with a half-period phase shift.