$y_i$; black dashed lines: fixed point values of activator and inhibitor). (c), (d): Phase portraits of activators (c) and inhibitors (d). Parameters: $a\!=\!1.05$, $\epsilon=0.01$, $C=0.5$, $\tau=3$. ](fig08.eps){width="70.00000%"}

In order to understand this additional phase-shift $\delta$, we shall now consider in detail the different stages of the oscillation as marked in Fig. \[fig:single\_FHN\]. Due to the small value of $\epsilon \ll 1$ there is a distinct timescale separation between the fast activators and the slow inhibitors, and a single FitzHugh-Nagumo system performs a fast horizontal transition $A \to B$, then travels slowly approximately along the right stable branch of the $x_1$ nullcline $B \to C$ ([*firing*]{}), then jumps back fast to $D$, and returns slowly to the [*rest state $A$*]{} approximately along the left stable branch of the $x_1$ nullcline ([*refractory phase*]{}). If $a$ is close to unity, these four points are approximately given by $A=(-a, -a+\frac{a_1^3}{3})$, $B=(2,-\frac{2}{3})$, $C=(1,\frac{2}{3})$, $D=(-2,\frac{2}{3})$. A rough estimate for $A'$ is $(a-2,-a+\frac{a_1^3}{3})$. The two slow phases $B \to C$ and $D \to A$ can be approximated by $y_1 \approx x_1 - \frac{x_1^3}{3}$ and hence $\dot{y}_1 \approx \dot{x}_1(1-x_1^2)=x_1 + a$ which gives $$\begin{aligned}
\label{eq:slow}
   \dot{x}_1 &=&  \frac{x_1+a}{1-x_1^2}\end{aligned}$$ which can be solved analytically, describing the firing phase (+) 