Stability analysis and Hopf bifurcation based on time delay neurons under electromagnetic fields

Abstract

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.

Bifurcation Analysis and Synchronous Patterns between Field Coupled Neurons with Time Delay

“…In particular, these improved neuronal circuits by embedding special functional devices such as piezoelectric ceramics, phototubes, and thermistors can sense special physical signals from the outside world, resulting in various types of dimensionless functional neuronal models such as thermosensitive neurons [14], photosensitive neurons [15], and auditory neurons [16]. Moreover, any neuron can be considered as a complex charged body from a physical point of view, whose static charged ions can induce a static electric field, while the intracellular transport and transmembrane flow of ions induce a magnetic field, so that any neuron is in the electromagnetic field induced by other neurons [17][18][19][20][21][22][23].…”