Analysis of Myotonia Paradox on the Hodgkin-Huxley Equations for Muscle Fibers

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There are diseases named myotonia and periodic paralysis. Some of those get worse in coldness and some are not affected by temperature. This phenomena is cold myotonia paradox and why it occurs is not known. The purpose of our study is two research on the mechanism it occurs by using the Hodgkin-Huxley Equations for muscle fibers (denoted HHM below).Experimentary derived HHM exist only for frogs. In our investigation, it is not adapt for a model of human diseases. So we construct HHM for mammalian from data of mammalian muscle on literature.Bifurcation phenomena of HHM for mammalian are investigated by the theory of nonlinear dynamical systems and numerical calculations. Bifurcation parameters are the chloride conductance and some parameters of sodium channel which are known to change in the diseases.As results, both in changes in chloride conductance and the sodium parameters, no temperature dependency are not observed in HHM.On the bifurcation diagram on two-parameter plane of HHM around an inclination-flip bifurcation point, the parameter plane is separated into three regions : normal firing, repetitive firing (myotonia) and depolarization of the resting potential (periodic paralysis). The normal region and the two other regions are separated by homoclinic bifurcations, and the myotonia region and the paralysis region are separated by double cycle bifurcations.Some diseases of sodium channel defect are known to get worse in low temperature, but our results in HHM are not. It might be because the sodium channel defects we took in HHM are wrong or because the temperature dependence we took in HHM are not fit with real muscle.