  This paper presents an investigation of the dynamics of two coupled
non-identical FitzHugh-Nagumo neurons with quadratic term and delayed synaptic
connection. We consider coupling strength and time delay as bifurcation
parameters, and try to classify all possible dynamics. Bifurcation diagrams are
obtained numerically or analytically from the mathematical model, and the
parameter regions of different behaviors are clarified. The neural system
exhibits a unique rest point or three ones by employing the saddle-node
bifurcation, when strong coupling is applied in the system. Also the trivial
rest point shows transcritical bifurcation with one of the new rest points. The
asymptotic stability and possible Hopf and Bautin bifurcations of the trivial
rest point are studied by analyzing the corresponding characteristic equation.
Fold cycle, torus, fold-torus, and big homoclinic bifurcations of limit cycles,
together with $1:1$ and $1:4$ resonances are found. The delay-dependent
stability regions are illustrated in the parameter plane, through which the
double-Hopf, Hopf-transcritical, and double-zero bifurcation points can be
obtained from the intersection of different bifurcation branches. Various
patterns of multistability have been observed, both for small and large values
of delay. The system may exhibit synchronous and anti-phase periodic solutions,
which occur due to Hopf, fold cycle and torus bifurcations. By increasing the
range of parameter $\tau$ other branches of fold and torus bifurcations appear,
which lead to creation of more stable periodic solutions.
