Analysis of a Canard Mechanism by Which Excitatory Synaptic Coupling Can Synchronize Neurons at Low Firing Frequencies

A population of oscillatory Hodgkin--Huxley (HH) model neurons is shown numerically to exhibit a behavior in which the introduction of excitatory synaptic coupling synchronizes and dramatically slows firing. This effect contrasts with the standard theory that recurrent synaptic excitation promotes states of rapid, sustained activity, independent of intrinsic neuronal dynamics. The observed behavior is not due to simple depolarization block nor to standard elliptic bursting, although it is related to these phenomena. We analyze this effect using a reduced model for a single, self-coupled HH oscillator. The mechanism explained here involves an extreme form of delayed bifurcation in which the development of a vortex structure through interaction of fast and slow subsystems pins trajectories near a surface that consists of unstable equilibria of a certain reduced system, in a canard-like manner. Using this vortex structure, a new passage time calculation is used to approximate the interspike time interval. We also consider how changes in the synaptic opening rate can modulate oscillation frequency and can lead to a related scenario through which bursting may occur for the HH equations as the synaptic opening rate is reduced.