Mathematical Perspective of Hodgkin-Huxley Model and Bifurcation Analysis

Abstract

Hodgkin-Huxley model (HH model) qualitatively describes the generation of the action potential of squid giant axons. The resting state or oscillatory phase state of the membrane potential (voltage) in the HH model depends mainly on applied stimulus (external currents) to neurons. The firing of the action potential in neurons depends upon the depolarization or repolarization of ions. The probability of channel gates (to be open or close) determines the movement of ions across the cell membrane. The term of K+ ionic currents (related to several activation gates) in external current contains exponential power 4 in HH model in which we propose a modified HH model by considering the higher power (5 and 6) of K activation in potassium ionic currents and studied the behavior of all three models comparatively. The modified HH model with a higher power of potassium activation reached resting-state sooner and gains stability (after oscillatory) at a high external current. The qualitative behavior of the modified model (with the higher exponential power) is different as there is a shifting of Hopf bifurcation points in comparison with the original HH model. Moreover, a larger periodic region was observed in most of the parameter phase spaces (external current I versus parameters) except against the Na conductance and Na potential. The modified HH model which determines that higher power of K activation is more significant for action potential in neurons.