Dynamics of Disordered Network of Coupled Hindmarsh–Rose Neuronal Models

Abstract

We investigate the effects of disorder on the synchronized state of a network of Hindmarsh–Rose neuronal models. Disorder, introduced as a perturbation of the neuronal parameters, destroys the network activity by wrecking the synchronized state. The dynamics of the synchronized state is analyzed through the Kuramoto order parameter, adapted to the neuronal Hindmarsh–Rose model. We find that the coupling deeply alters the dynamics of the single units, thus demonstrating that coupling not only affects the relative motion of the units, but also the dynamical behavior of each neuron; Thus, synchronization results in a structural change of the dynamics. The Kuramoto order parameter allows to clarify the nature of the transition from perfect phase synchronization to the disordered states, supporting the notion of an abrupt, second order-like, dynamical phase transition. We find that the system is resilient up to a certain disorder threshold, after that the network abruptly collapses to a desynchronized state. The loss of perfect synchronization seems to occur even for vanishingly small values of the disorder, but the degree of synchronization (as measured by the Kuramoto order parameter) gently decreases, and the completely disordered state is never reached.