the real parts of the eigenvalues: (a) stable stationary state, (b) destabilization via Hopf-bifurcation, (c) saddle-node bifurcation (SN), where an eigenvalue with $\Omega=0$ becomes unstable, (d) weakly unstable stationary state, (e) cusp bifurcation (C) and appearance of a strongly unstable spectrum, (f) saddle-node bifurcation and lower boundary of the Hopf-domain $H_{2}.$ \[fig:FHMststhyb\] ](FHN_lattice_stst_paper){width="1\linewidth"}

### Large delay approximation {#sub:FHNlong}

### Strongly unstable spectrum {#strongly-unstable-spectrum .unnumbered}

The strongly unstable spectrum results from considering only the instantaneous part of Eq. (\[eq:FHNcheq\]) $$0=\det{\left|A-\lambda\operatorname{Id}\right|}=(a_{33}-\lambda)\left[(a_{11}-\lambda)(a_{22}-\lambda)-a_{21}a_{12}\right].$$ There is always one real solution $\lambda_{0}=a_{33}\leq-0.6$ of this 3rd-order polynomial, which is strictly negative. The remaining eigenvalues are $$\lambda_{\pm}=\frac{1}{2}\left(a_{11}-b\varepsilon\pm\sqrt{(a_{11}+b\varepsilon)^{2}-4\varepsilon}\right).$$ Note that all relevant parameters are contained in $a_{11}=1-\bar{v}^{2}-C\bar{s}$, involving the current $I$ and coupling strength $C$ directly or via the corresponding homogeneous steady state, respectively.

The strongly unstable spectrum exists when the real part of the largest eigenvalue $\lambda_{+}$ is positive. This is the case when $a_{11}>b\varepsilon$. The appearance of the strongly unstable spectrum occurs at the cusp-bifurcation of the asymptotic continuous spectrum and labled as “C” in Fig. \[fig:FHMststhyb\]. Moreover, there exists a pair of complex conjugate eigenvalues for $a_{11}<2\sqrt{\varepsilon}-b\varepsilon$, otherwise the eigenvalues are real. The corresponding boundary is labeled with “S” in Fig. \[fig:FHMststhyb\] and mediates the transition 