From mechanical to biological oscillator networks: The role of long range interactions

In studying one-dimensional particle networks of Classical Mechanics, through Hamiltonian models, we have learned a lot about oscillations under nearest (short range) interactions. Recently, however, through the introduction of long range interactions (LRI), several widely accepted notions concerning chaos and the approach to thermal equilibrium have been challenged based on the discovery of very interesting long lasting metastable states. On the other hand, when LRI (in the form of non-local or all-to-all coupling) were introduced in systems of biological oscillators, Kuramoto’s theory of synchronization in phase oscillators was developed and soon thereafter researchers studied amplitude and phase oscillations in networks of FitzHugh Nagumo and Hindmarsh Rose (HR) neuron models. In these models certain fascinating phenomena of coexistence of synchronous and asynchronous oscillations were discovered, called chimera states. Currently, the synchronization and metastability properties of these states are being widely investigated in HR mathematical models as well as realistic neural networks, similar to what one finds in simple living organisms like the C.elegans worm.