Patterns of interaction of coupled reaction–diffusion systems of the FitzHugh–Nagumo type

Motivated by the dynamics of neuronal responses, we analyze the dynamics of the FitzHugh–Nagumo slow–fast system with diffusion and coupling. In the case of diffusion, the system provides a canonical example of Turing–Hopf bifurcation. By analyzing the linear stability of the local equilibrium, the occurrence of Turing–Hopf bifurcation, Turing–Turing bifurcation and coupled Turing–Hopf bifurcation are obtained. The normal form associated with the Turing–Hopf bifurcation is obtained by using the procedure of Song for calculating the normal form of PDEs. Further, in the case of two coupled FitzHugh–Nagumo reaction–diffusion, the Turing–Hopf–Turing bifurcation occurs, and we also find the case about the spatial resonance of Turing–Turing bifurcation arising, and two kinds spatially steady-state solutions are found which are synchronous or anti-phased. Finally, sample numerical results are reported.