Adopting small-world neural networks of the Hodgkin-Huxley (HH) model, the stimulation parameters in desynchronisation and its possible implications for vagus nerve stimulation (VNS) are numerically investigated. With the synchronisation status of networks to represent epilepsy, then, adding pulse to stimulations to 10% of neurons to simulate the VNS, we obtain the desynchronisation status of networks (representing antiepileptic effects). The simulations show that synchronisation evolves into desynchronisation in the HH neural networks when a part (10%) of neurons are stimulated with a pulse current signal. The network desynchronisation appears to be sensitive to the stimulation parameters. For the case of the same stimulation intensity, weakly coupled networks reach desynchronisation more easily than strongly coupled networks. The network desynchronisation reduced by short-stimulation interval is more distinct than that of induced by long stimulation interval. We find that there exist the optimal stimulation interval and optimal stimulation intensity when the other stimulation parameters remain certain.

In this paper we consider the generalized Hodgkin-Huxley model introduced in Austin (2008). This model describes the propagation of an action potential along the axon of a neuron at the scale of ion channels. Mathematically, this model is a fully coupled piecewise-deterministic Markov process (PDMP) in infinite dimensions. We introduce two time scales in this model in considering that some ion channels open and close at faster jump rates than others. We perform a slow-fast analysis of this model and prove that, asymptotically, this ‘two-time-scale’ model reduces to the so-called averaged model, which is still a PDMP in infinite dimensions, for which we provide effective evolution equations and jump rates.