The role of the phase response curve (PRC) shape on the synchrony of synaptically coupled oscillating neurons is examined. If the PRC is independent of the phase, because of the synaptic form of the coupling, synchrony is found to be stable for both excitatory and inhibitory coupling at all rates, whereas the antisynchrony becomes stable at low rates. A faster synaptic rise helps extend the stability of antisynchrony to higher rates. If the PRC is not constant but has a profile like that of a leaky integrate-and-fire model, then, in contrast to the earlier reports that did not include the voltage effects, mutual excitation could lead to stable synchrony provided the synaptic reversal potential is below the voltage level the neuron would have reached in the absence of the interaction and threshold reset. This level is controlled by the applied current and the leakage parameters. Such synchrony is contingent on significant phase response (that would result, for example, by a sharp PRC jump) occurring during the synaptic rising phase. The rising phase, however, does not contribute significantly if it occurs before the voltage spike reaches its peak. Then a stable near-synchronous state can still exist between type 1 PRC neurons if the PRC shows a left skewness in its shape. These results are examined comprehensively using perfect integrate-and-fire, leaky integrate-and-fire, and skewed PRC shapes under the assumption of the weakly coupled oscillator theory applied to synaptically coupled neuron models.

We investigate why electrically coupled neuronal oscillators synchronize or fail to synchronize using the theory of weakly coupled oscillators. Stability of synchrony and antisynchrony is predicted analytically and verified using numerical bifurcation diagrams. The shape of the phase response curve (PRC), the shape of the voltage time course, and the frequency of spiking are freely varied to map out regions of parameter spaces that hold stable solutions. We find that type 1 and type 2 PRCs can hold both synchronous and antisynchronous solutions, but the shape of the PRC and the voltage determine the extent of their stability. This is achieved by introducing a five-piecewise linear model to the PRC and a three-piecewise linear model to the voltage time course, and then analyzing the resultant eigenvalue equations that determine the stability of the phase-locked solutions. A single time parameter defines the skewness of the PRC, and another single time parameter defines the spike width and frequency. Our approach gives a comprehensive picture of the relation of the PRC shape, voltage time course, and stability of the resultant synchronous and antisynchronous solutions.