  We discuss synchronization in networks of neuronal oscillators which are
interconnected via diffusive coupling, i.e. linearly coupled via gap junctions.
In particular, we present sufficient conditions for synchronization in these
networks using the theory of semi-passive and passive systems. We show that the
conductance-based neuronal models of Hodgkin-Huxley, Morris-Lecar, and the
popular reduced models of FitzHugh-Nagumo and Hindmarsh-Rose all satisfy a
semi-passivity property, i.e. that is the state trajectories of such a model
remain oscillatory but bounded provided that the supplied (electrical) energy
is bounded. As a result, for a wide range of coupling configurations, networks
of these oscillators are guaranteed to possess ultimately bounded solutions.
Moreover, we demonstrate that when the coupling is strong enough the
oscillators become synchronized. Our theoretical conclusions are confirmed by
computer simulations with coupled \HR and \ML oscillators. Finally we discuss
possible "instabilities" in networks of oscillators induced by the diffusive
coupling.
