the fixed point. In this case, $a$ in Eq.(\[eq:delta\]) has to be substituted by a larger value $\tilde{a}$ with $a<\tilde{a}<1.7$ in order to get a better estimate of $\delta$. Note that without the phase-shift $\delta$ the coupling term $C[x_2(t-\tau)-x_1(t)]$ would always vanish in the $2\tau$-periodic state.

Next, we shall investigate conditions upon the coupling parameters $C$ and $\tau$ allowing for limit cycle oscillations. On one hand, if $\tau$ becomes smaller than some $\tau_{min}$, the impulse from the excitatory neuron arrives too early to trigger a spike, since the system is still early in its refractory phase. On the other hand, if $C$ becomes too small, the coupling force of the excitatory neuron is too weak to excite the system above its threshold and pull it far enough towards $B$.

![\[fig:phaseportraits\] Phase portraits of delay-coupled excitable system $(x_1,y_1)$ for different delay times $\tau$ (trajectories: solid blue, nullclines: dashed black). (a) $\tau \!=\! 3$ ($\delta =0.009$), (b) $\tau\!=\!0.8$ ($\delta =0.015$). Other parameters: $a\!=\!1.05$, $\epsilon\!=\!0.01$, $C\!=\!0.5$.](fig09.eps){width="50.00000%"}

![\[fig:hauplot\] Regime of oscillations in the $(\tau, C)$ parameter plane for initial conditions corresponding to single-pulse-excitation in one system. The oscillation period $T$ is color coded. The transition between black and color marks the bifurcation line. Inset 