stability of the stationary state, we derive the linearized evolution equation for small perturbations of the equilibrium and subsequently diagonalize it in Fourier-space, just as in the previous section. One obtains the system $$\frac{\mathrm{d}}{\mathrm{d}t}\mathbf{\delta\tilde{\mathbf{u}}}(t)=A\mathbf{\delta\tilde{u}}(t)+2B\cos{\left(k_{-}\right)}e^{ik_{+}}\mathbf{\delta\tilde{u}}(t-\tau)$$ with the real valued matrices $$A=\left(\begin{array}{ccc}
1-\bar{v}^{2}-C\bar{s} & -1 & 0\\
\varepsilon & -b\varepsilon & 0\\
5\alpha(\bar{v})(1-2\alpha(\bar{v}))(1-\bar{s}) & 0 & -\alpha(\bar{v})-0.6
\end{array}\right)$$ and $$B=\left(\begin{array}{ccc}
0 & 0 & \frac{C}{2}(v_{r}-\bar{v})\\
0 & 0 & 0\\
0 & 0 & 0
\end{array}\right).$$ The corresponding characteristic equation reads $$0=\prod_{\{(k_{+},k_{-})\}}\det{\left|-\lambda\operatorname{Id}+A+2B\cos{\left(k_{-}\right)}e^{ik_{+}}e^{-\lambda\tau}\right|},\label{eq:FHNcheq}$$ where $\operatorname{Id}$ represents the $3\times3$ identity matrix. Similarly to the previous analysis, the stability of the homogeneous steady state is completely determined by the Eq. (\[eq:FHNcheq\]), which can be studied numerically using e.g. Newton-Raphson iteration. An additional insight in the properties of the spectrum can be given using the large delay approximation, which is done in Sec \[sub:FHNlong\]. Further, in Sec. \[sub:FHNH\], we study the appearance of periodic solutions in Hopf bifurcations.

![Homogeneous stationary state and its bifurcations for $C=3$ for the FitzHugh-Nagumo model Eq. (\[eq:FHN\]). In the case of an infinitely large lattice, the spectrum is described by the hybrid dispersion relation (\[eq: FHNststhybrid\]), parametrized by two continuous parameters $\Omega$ and $k_{-}$, shown on the right. The color denotes 