$b=0.8$, and $\varepsilon=0.08$. The reversal potential is taken as $v_{r}=2$ for excitatory coupling. Similar model equations have been investigated in [@Perlikowski2010; @Yanchuk2011] for unidirectional rings.

We demonstrate that the destabilization of the homogeneous steady state, the set of plane waves as well as their stability properties possess the same qualitative features which we observed in the previous section on Stuart-Landau oscillators. However, the apparent difficulty for the analysis of nonlinear plane waves is that they are not known analytically. Therefore we use numerical bifurcation analysis with <span style="font-variant:small-caps;">DDE-Biftool</span> [@Engelborghs2002] and have to restrict ourselves to relatively small lattice size and delay values in Sec. \[sub:FHNH\].

Homogeneous steady state and its stability
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The system (\[eq:FHN\]) has a homogeneous steady state $\bar{\mathbf{u}}=\left(\bar{v},\bar{w},\bar{s}\right)$. The value for the membrane resting potential $\bar{v}$ can be obtained as a solution of the scalar equation $$0=\bar{v}-\frac{1}{3}\bar{v}^{3}-\frac{\bar{v}+a}{b}+I+C(v_{r}-\bar{v})\frac{\alpha(v)}{\alpha(v)+0.6}.\label{eq:FHNstst}$$ The steady-state values of the remaining variables follow as $$\bar{w}=\frac{\bar{v}+a}{b},\qquad\mathrm{and}\qquad\bar{s}=\frac{\alpha(v)}{\alpha(v)+0.6}.$$ In the case of weak coupling strength the homogeneous stationary state is unique, but for $C_{\mathrm{SN}}=1.46475$ a saddle-node bifurcation of the equilibrium takes place. For strong coupling $C>C_{\mathrm{SN}}$ there is a domain of the control parameter $I$ with three coexisting stationary states, see Fig. \[fig:FHMststhyb\]. In order to analyze the 