of $M\times N$ delay-coupled systems (delay differential equations) of the form $$\begin{aligned}
\frac{\mathrm{d}}{\mathrm{d}t} \mathbf{u}_{m,n}(t) &= \mathbf{F}\left(\mathbf{u}_{m,n}(t),\mathbf{u}_{m-1,n}(t-\tau_{m,n}^{\downarrow})+\mathbf{u}_{m,n-1}(t-\tau_{m,n}^{\rightarrow})\right),\label{eq: general system}\\
 &\qquad m=1,\ldots ,M; \quad n=1,\dots, N, \nonumber \end{aligned}$$ where $\mathbf{F}:\mathbb{R}^{d}\times\mathbb{R}^{d}\to\mathbb{R}^{d}$ is a nonlinear function determining the dynamics of $\mathbf{u}_{m,n}\in\mathbb{R}^{d}$ in the lattice. The indices $m$ and $n$ determine the position of the node, see Fig. \[pic: 2-dim lattice coupling structure\]. We assume periodic boundary conditions $\mathbf{u}_{M+1,n}\equiv\mathbf{u}_{1,n}$ and $\mathbf{u}_{m,N+1}\equiv\mathbf{u}_{m,1}$ such that the system has translation invariance. Time delays $\tau_{m,n}^{\downarrow}$ and $\tau_{m,n}^{\rightarrow}$ denote the connection delays between the corresponding nodes. Since each node has two incoming connections, the arrows $\downarrow$ and $\rightarrow$ correspond to the coupling from the node located above, respectively left, see Fig. \[pic: 2-dim lattice coupling structure\]. Here we restrict the analysis to two systems: Stuart-Landau (SL) oscillators as a simplest dynamical system exhibiting limit cycle behavior and FitzHugh-Nagumo (FHN) systems as a representative of conductance based, excitable neuronal models [@FitzHugh1961; @Nagumo1962]. While the first model allows for a deeper analytical insight, the second one can be studied mainly numerically and shows similar qualitative results.

![Coupling scheme. The dynamics of each node $\mathbf{u}_{m,n}(t)$ is described by system (\[eq: general system\]). Each coupling connection possesses a delay $\tau_{m,n}^{\downarrow}$ or $\tau_{m,n}^{\rightarrow}$. All edges 