Stability of synchronous oscillations in a system of Hodgkin-Huxley neurons with delayed diffusive and pulsed coupling

We study the synchronization dynamics for a system of two Hodgkin-Huxley (HH) neurons coupled diffusively or through pulselike interactions. By calculating the maximum transverse Lyapunov exponent, we found that, with diffusive coupling, there are three regions in the parameter space, corresponding to qualitatively distinct behaviors of the coupled dynamics. In particular, the two neurons can synchronize in two regions and desynchronize in the third. When excitatory and inhibitory pulse coupling is considered, we found that synchronized dynamics becomes more difficult to achieve in the sense that the parameter regions where the synchronous state is stable are smaller. Numerical simulations of the coupled system are presented to validate these results. The stability of a network of coupled HH neurons is then analyzed and the stability regions in the parameter space are exactly obtained.