by demonstrating that no delay-induced Hopf bifurcation can occur. Substituting the ansatz $\lambda=i \omega$ into Eq. (\[char\_eq\]) and separating into real and imaginary parts yields for the imaginary part $$\xi =\pm C \cos(\omega \tau)$$ This equation has no solution for $a>1$ since $|\xi|=a^2-1+C>C$, which proves that a Hopf bifurcation cannot occur.

Delay-induced antiphase oscillations
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Delay-induced oscillations in excitable systems are inherently different from noise-induced oscillations. The noise term continuously kicks the subsystems out of their respective rest states, and thus induces sustained oscillations. Instantaneous coupling without delay then produces synchronization effects between the individual oscillators [@HAU06; @HOE07; @HOE08]. For [*delayed*]{} coupling the case is entirely different. Here the impulse of one neuron triggers the other neuron to emit a spike, which in turn, after some delay, triggers the first neuron to emit a spike. Hence self-sustained periodic oscillations can be induced without the presence of noise (Fig. \[fig:ts\_comparison\]). It is evident that the oscillations of the two neurons have a phase lag of $\pi$. The period of the oscillations is given by $T=2(\tau + \delta)$ with a small quantity $\delta>0$.

![\[fig:ts\_comparison\] Delay-induced oscillations. (a), (b): Time series of both subsystems (red solid lines: activator $x_i$, green solid lines: inhibitor 