  The neural oscillator model proposed by Matsuoka is a piecewise affine
system, which exhibits distinctive periodic solutions. Although such typical
oscillation patterns have been widely studied, little is understood about the
dynamics of convergence to certain fixed points and bifurcations between the
periodic orbits and fixed points in this model. We performed fixed point
analysis on a two-neuron version of the Matsuoka oscillator model, the result
of which explains the mechanism of oscillation and the discontinuity-induced
bifurcations such as subcritical/supercritical Hopf-like, homoclinic-like, and
grazing bifurcations. Furthermore, it provided theoretical predictions
concerning a logarithmic oscillation-period scaling law and noise-induced
oscillations, which are both observed around those bifurcations. These results
are expected to underpin further investigations into both oscillatory and
transient neuronal activities with respect to central pattern generators.
