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

The study of one-dimensional particle networks of Classical Mechanics, through Hamiltonian models, has taught us a lot about oscillations of particles coupled to each other by nearest neighbor (short range) interactions. Recently, however, a careful analysis of the role of long range interactions (LRI) has shown that several widely accepted notions concerning chaos and the approach to thermal equilibrium need to be modified, since LRI strongly affects the statistics of certain very interesting, long lasting metastable states. On the other hand, when LRI (in the form of non-local or all-to-all coupling) was introduced in systems of biological oscillators, Kuramoto's theory of synchronization 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 called chimera states were discovered where populations of synchronous and asynchronous oscillators are seen to coexist in the same system. Currently, their synchronization properties 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.