linearization breaks down, and (ii) the pulse duration $T_f$ is exceeded. For small $t$ Eq.(\[eq:fast1\]) can be expanded as $$\begin{aligned}
\label{eq:fast2}
   x_1(t) &=&  -a + \frac{2C}{\epsilon} t\end{aligned}$$ which is equivalent to neglecting the upstream flow field $-|\xi| \delta x_1$ in Eq.(\[eq:fast\]) near the stable fixed point $A$ compared to the pulling force $2C$ of the remote spike which tries to excite the system towards $B$. Once the system has crossed the middle branch of the $x_1$ nullisocline at $A'$, the intrinsic flow field accelerates the trajectory fast towards $B$, initiating the firing state. Therefore there is a turn-on delay $\delta$, given by the time the trajectory takes from $A$ to $A'$, i.e. $x_1(\delta)\approx a-2$, according to Eq.(\[eq:fast2\]): $$\begin{aligned}
\label{eq:delta}
   \delta &=&  (a-1)\frac{\epsilon}{C}\end{aligned}$$ Since the finite rise time of the impulse has been neglected in our estimate, the exact solution $\delta$ is slightly larger and does not vanish at $a=1$.

With increasing $a$ the distance $A-A'$ increases, and so does $\delta$. The small additional phase shift $\delta$ between the spike $x_1(t)$ and the delayed pulse $x_2(t-\tau)$ results in a non-vanishing coupling term at the beginning and at the end of the spike $x_1(t)$. It is the reason (i) that the spike 