are unidirectional.\[pic: 2-dim lattice coupling structure\]](2dtorus_w_delays)

An example of a stable spatio-temporal pattern in a lattice of $100\times150$ FHN neurons with non-homogeneous delays, the “Mona Lisa”-pattern is shown in Figure \[fig:ML\]. Each frame corresponds to a snapshot at a fixed time $t$ and the different level of gray at a point $(m,n)$ corresponds to the value of the voltage component of $\mathbf{u}_{m,n}(t)$ at this time $t$. More details on how such patterns can be created are given in the following sections, especially in Sec. \[sec:non\].

![\[fig:ML\]Snapshots of the spatio-temporal behavior in a system of $100\times150$ identical FHN neurons Eq. (\[eq:FHN\]) with appropriately adjusted time delays $\tau_{m,n}^{\downarrow}$ and $\tau_{m,n}^{\rightarrow}$. At each grid point with the coordinate $(m,n)$, the level of gray (see colorbar) corresponds to the membrane voltage $v_{m,n}(t)$ at this time moment. The pattern reappears periodically with a time period $T=21.95$. More details are given in Sec. \[sec:non\]. ](FHN_lattice_MonaLisa){width="100.00000%"}

The structure of the remaining part of the paper is as follows: In Sec. \[sec:SLhom\] we consider SL systems with homogeneous time delays $\tau_{m,n}^{\rightarrow}=\tau_{m,n}^{\downarrow}=\tau$. We investigate the stability of the homogeneous steady state as well as various plane wave solutions in the system. The number of stable plane wave solutions is 