Neurons contain a large number of ions inside and outside the cell, and the transmembrane currents formed by the movement of these ions cause membrane potential fluctuations and induce electromagnetism inside and outside the cell. In addition, any change in external electromagnetic fields can cause changes in the membrane potential of the neurons. Therefore, based on the three-dimensional Hindmarsh — Rose (HR) neuron model, a five-dimensional neuron model with time delay is developed in this paper by introducing flux and electric field variables and considering the resulting time delay. First, the Hopf bifurcation theory is used to demonstrate the local stability of the system at the equilibrium point at different time delays. Then, the stability of the Hopf bifurcation and its direction are proved by using the central flow shape theorem. Finally, the existence of the Hopf bifurcation is proved using the phase diagram and the bifurcation diagram, and the effects of several important parameters on the model are investigated by numerical simulations using time series plots, ISI bifurcation plots and two-parameter bifurcation plots. The model is found to be accompanied by chaotic and chaos-free plus-periodic bifurcation structures, mixed-mode discharges and other phenomena. Also, its discharge pattern can be controlled after adding time delay. The results of this paper provide help to the pathogenic mechanism and control of neurological diseases.
Hopf bifurcation and chaos in a single inertial neuron model with time delay
