The Matsuoka oscillator model is a neuronal network model which exhibits oscillatory activities due to adaptation property of each neuron and mutual inhibitions between neurons (Matsuoka, 1985; 1987). This is often used to model oscillatory neuronal circuits in the spinal cord called central pattern generators (CPGs) and to simulate biological locomotion such as human bipedal walking (Taga et al., 1991; Ogihara & Yamazaki, 2001). However, most previous studies have overlooked its convergent dynamics toward stationary states, corresponding to transient neuronal activities and non-oscillatory movements. In the present study, we conducted fixed point analysis on a two-neuron case of the Matsuoka oscillator model. Through this investigation, we (I) formulized the existence and stability of all possible fixed points, (II) depicted emergence of oscillatory solutions and bifurcation mechanisms between oscillatory and convergent dynamics, and (III) made predictions of a logarithmic scaling law of oscillation period and noise-induced oscillation. Our results possibly suggest that central nervous systems might take advantage of CPGs not only for rhythmic locomotion but also for non-oscillatory or discrete movements. The discussion of limitations revealed in this report will likely be followed by future extension of the Matsuoka oscillator model to understand an integrative mechanism for neural control of both rhythmic and discrete movements.Phase diagram of the model with respect to model parameters r and a. The model exhibits diverse oscillation patterns (a)--(d) when (r, a) is inside the white area; outside this area, the system state of the model converges to several stable fixed points.