Periodic solutions in an array of coupled FitzHugh-Nagumo cells

We analyse the dynamics of an array of $N^2$ identical cells coupled in the shape of a torus. Each cell is a 2-dimensional ordinary differential equation of FitzHugh-Nagumo type and the total system is $\mathbb{Z}_N\times\mathbb{Z}_N-$symmetric. The possible patterns of oscillation, compatible with the symmetry, are described. The types of patterns that effectively arise through Hopf bifurcation are shown to depend on the signs of the coupling constants, under conditions ensuring that the equations have only one equilibrium state.