The approximate lag and anticipating synchronization between two unidirectionally coupled Hindmarsh-Rose neurons with uncertain parameters

This research presents an adaptive synchronization approach crafted to facilitate exact lag synchronization between a pair of unidirectionally linked Hindmarsh-Rose (HR) neurons, taking into account both explicit propagation delays and the existence of uncertain parameters. The precise condition for lag synchronization is deduced analytically, utilizing the Laplace transform and convolution theorem, alongside the iterative approach within the framework of Volterra integral equations theory. The established criterion guarantees robust stability irrespective of the propagation delay's magnitude, facilitating the realization of approximate lag and anticipating synchronization in a pair of HR neurons. The approximate synchronizations are realized in the absence of direct time-delay coupling, with the Taylor series expansion serving as an alternative to the precise time-delay component. Numerical simulations are executed to validate the effectiveness of the suggested approximate synchronization approach. The research demonstrates that employing the current state of an HR neuron, despite having uncertain parameters, enables the accurate prediction of future states and the reconstruction of past states. This study provides a novel perspective for comprehending neural processes and the advantageous attributes inherent in nonlinear and chaotic systems.