Analysis Of Hopf Bifurcation In The Hodgkin-Huxley Equations For Muscles

It is reported that many diseases are related with the aberrance of celltrans-membrane ion channels. For instances, the chloride conductance decreases inmyotonia congenital, while the sodium ions permeability is aberrated in certainparalysis. This paper employed the physiological data obtained in rats musclesvoltage-clamped experiments carried by Duval,Adrian and Pappone, and took theHodgkin-Huxley model for muscles(HHM model) as study object.Firstly, with the aid of algebra criterion in high dimension equations, the dynamicHopf-bifurcation analysis about single parameter of leakage conductance as well assodium ions anti-electromotive force was performed on the model.Secondly, in this paper, the Hodgkin-Huxley model is studied with the leakageconductance and the sodium reversal potential selected as parameters for a two-parameterHopf-bifurcation analysis.Thirdly, this paper also discussed the multi-parameter Hopf-bifurcation ofmodified HHM model exposed to external electrical fields. Based on results ofspace-time characteristics of trans-membrane voltage, the variation of celltrans-membrane voltage exposed to extremely low frequency electrical field wasanalyzed. The traditional HHM model was modified by importing new parametersdenoting external electrical fields. According to analysis, when the parameter of leakage conductance and sodiumions anti-electromotive force varies, the dynamic Hopf bifurcation would emerge.Simulations showed that cell membrane would produce continuous waveforms ofaction potential under bifurcation condition, which is coincident with the aberrancephenomenon of action potential observed in medical experiments. The analysis ofmodified HHM model exposed to external electrical fields also suggested bifurcationwould appear with the variation of parameters about external electrical fields, whichmay be accounted for diseases caused by electro-magnetic exposure.