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

Semi-passivity and synchronization of diffusively coupled neuronal oscillators