Almost-synchronous solutions for mutually coupled excitatory neurons

We consider synchronization in a pair of neurons described by voltage-gated conductance equations and coupled by mutual excitation. Our model neurons have three time scales: the very fast transition between active and inactive states; an intermediate scale during the active portion of a cell’s trajectory; and the slowest during the inter-burst interval. We show that the interplay of time scales can lead to stable “almost-synchronous” solutions in which the jumps between active and inactive states of the two cells happen with a time difference that is a small fraction of the total period of the coupled system. Furthermore, modulation of parameters not affecting time scales can change the stable solution from almost-synchronous to synchronous. We use a geometric analysis that enables us to identify the parts of the trajectories over which the interactions move the coupled trajectory away from synchrony, the parameters responsible for this phenomenon and how the distance from synchrony depends on the time scales and can be modulated.