Semi-passivity and synchronization of diffusively coupled neuronal oscillators

TL;DR

This paper establishes conditions for synchronization in networks of neuronal oscillators interconnected via diffusive coupling, demonstrating that semi-passivity properties ensure boundedness and synchronization, supported by theoretical analysis and simulations.

Contribution

It introduces semi-passivity as a key property for ensuring boundedness and synchronization in neuronal networks with diffusive coupling, covering various neuron models.

Findings

Neuronal models like Hodgkin-Huxley, Morris-Lecar, FitzHugh-Nagumo, and Hindmarsh-Rose satisfy semi-passivity.

Strong enough coupling leads to oscillator synchronization.

Networks exhibit ultimately bounded solutions under certain conditions.

Abstract

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.