4.2 A couple of neurons

We also confirm the availability of our method to continuous-time system. The following differential equation is a mathematical model of the neural oscillator:

$$\begin{array}{rcl} \dot{x}_1 &=& 10 f(x_1) - 10 f(x_2) - 2.5... ...- 9.0\\ f(x) &=& \displaystyle\frac{1}{1 + e^{-x}} \end{array}$$ (10)

The system has a limit cycle in the autonomous system; i.e., $B=0$. By changing $B$ or $\omega$, generation of quasi-periodic solution, chaotic motion, and frequency entrainment are observed. Then we construct the Poincaré mapping $T$ to the periodic solution, calculate isoclines and bifurcation curves in $\omega$-$B$ plane. Figure 4 shows a bifurcation diagram of this system. It is clear that same structures are found with Fig. 1.

図 4: Bifurcation diagram of Eq. (10).

図 5: Phase portrait: Poincaré mapping on the NS bifurcation with $\theta = 2\pi /7$. $\omega = 0.846039$, $B = 2.394399$, $\mbox{\boldmath $\ u $}_0 = (2.263091, -0.720162)$.