Stability of anti-phase and in-phase locking by electrical coupling but not fast inhibition alone

We consider a model network consisting of two identical neurons with inhibitory and electrical coupling and find conditions under which a particular type of coupling promotes stable in-phase locking, anti-phase behavior, or some other type of firing pattern. A traditional view is that fast inhibition leads to stable anti-phase behavior, while electrical coupling leads to stable in-phase locking. Here, we follow up with rigorous analysis our previous computational demonstration [T. Bem and J. Rinzel, J. Neurophysiol., 91 (2004), pp. 693-703] that this is not always the case. We give precise conditions, which depend on the intrinsic properties of the cells involved, for when this traditional view is not valid. In particular, if the cells have short duty cycles, then fast inhibitory coupling leads to an almost-in-phase solution in which short active phases of the two cells occur subsequently, one immediately after the other. Moreover, if the cells have short duty cycles, then a network with weak electrical coupling exhibits bistability of in-phase and anti-phase behavior. The cells are modeled as relaxation oscillators, and we analyze the network using geometric singular perturbation methods. This allows us to construct maps, fixed points of which correspond to a particular type of solution. Rigorous analysis of these maps leads to precise conditions for when a particular type of solution exists and when it is stable for either pure electrical or pure inhibitory coupling. We also demonstrate how adding the other type of coupling may change the network behavior.