Semi-passivity and synchronization of neuronal oscillators*

We discuss synchronization in networks of neuronal oscillators which are linearly coupled via gap junctions. We show that the neuronal models of Hodgkin-Huxley, Morris-Lecar, 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 will posses ultimately bounded solutions. Moreover, when the coupling is strong enough the oscillators become synchronized. We demonstrate the synchronization of Hindmarsh-Rose oscillators by means of a computer simulation.