The Role of Complex Stimulation in Dynamics of a Model of Spiking Neurons

Numerous models has been proposed to provide mathematical representation for neurons following Hodgkin-Huxley research. Among these models, the reduced version of Hodgkin-Huxley model proposed by Wilson could successfully replicate the dynamics of both fast sodium voltage gated channels and a slow gating variables. However, it is worth mentioning that although this model has been proposed in real domain, it has fixed points or other dynamical structures in complex domain. Moreover, having a certain amplitude and phase, biological variables such as membrane voltage could be considered as vector quantities in complex plane. We, therefore, chose to study the simplified Wilson model of spiking neurons in complex plane as an example to examine the role of the complex local equilibrium points in general dynamics of the system. The framework proposes that the highly complex and multidimensional states of bursting and complex spiking regimes observed in excitable cells can be appeared in complex plane of spiking 2-dimensional neuronal models throughout a vector quantity of stimulation.