Synchronization of two Hindmarsh–Rose neurons with unidirectional coupling

The Hindmarsh–Rose (HR) dynamical system is a well-known model of neuronal activity. It is known that two HR neurons can synchronize when coupled in the action potential variable. Here, I report on exponentially fast synchronization of two HR neurons with novel unidirectional coupling. Explicit proof of global stability is given where the Lyapunov function is found with single parameter bounds as sufficient criteria. Numerical explorations verify such synchronization yet reveal additional single-link unidirectional couplings enabling full or subsystem synchronization where the parameters of two HR neurons might differ. Despite the mathematical prediction, from neuroscience and molecular biology points of view such connectivity between neurons has no simple explanation. The result opens up a fundamental question on the valid interpretation of unidirectional links, their potential use, and if such synchronization is an active principle of biological neurons.