leads to desynchronization of the coupled system (dark regions in Fig. \[fig:isi\_control\]).

Delay-coupled neurons
=====================

In this section we study the influence of a delay in the coupling of two neurons, rather than a delayed self-feedback. We set the noise terms in Eq. (\[eq:system\]) equal to zero, $D_1=D_2=0$, but consider a time-delay $\tau$ in the coupling. In the deterministic system the delayed coupling plays the role of a stimulus which can induce self-sustained oscillations in the coupled system even if the fixed point is stable. In this sense the delayed coupling has a similar effect as the noise term in the previous sections. Here the bifurcation parameters for delay-induced bifurcations are the coupling parameters $C$ and $\tau$.

Linear stability of fixed point
-------------------------------

In the following we shall choose symmetric timescales $\epsilon_1=\epsilon_2=\epsilon=0.01$ and fix $a=1.05$, where each of the two subsystems has a stable fixed point and exhibits excitability.

The unique fixed point of the system is symmetric and is given by ${\bf x}^* \equiv (x_1^*, y_1^*, x_2^*, y_2^*)$, where $x_i^* = -a$, $y_i^*= a^3/3 - a$. Linearizing Eq. (\[eq:system\]) around the fixed point $\bf{x}^*$ by setting ${\bf x}(t)={\bf x}^*+\delta{\bf x}(t)$, one obtains: $$\begin{aligned}
\delta\dot{\bf{x}}
= \frac{1}{\epsilon} \left( \begin{array}{cccc}
\xi 