The integrate-and-fire and piecewise-linear (PL) (or FitzHugh-Nagumo) neuron models, which have been used in the study of synchronization phenomena, were often regarded to be quite different types. Periodically and spontaneously firings of such analytical tractable models are dynamically different from or similar to the physiological relevant neuron models. A qualitatively new relationship between the different types of the neuron models is proposed to introduce a coupling-dependence in the PL models according to whether the coupling term is added to the $x$- or $y$-dynamics. Next, with a return map analysis and a phase reduction method, synchronization behavior of two neurons with excitatory or inhibitory synaptic coupling, including some irregularly synchronous behavior that has never been reported thus far, can be systematically explored in terms of three parameters in the model (synaptic strength, decaying relaxation rate of the synaptic coupling and exponentially-decaying relaxation rate during the firing). We then show that the duration time of an impulse plays an important role in synchronization phenomena.

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