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

有肌强直和周期性瘫痪的疾病。其中一些在寒冷中会变得更糟，一些不会受到温度的影响。这种现象是冷性肌强直悖论，其发生的原因尚不清楚。我们的研究目的是利用肌肉纤维的Hodgkin-Huxley方程(下称HHM)对其发生机制进行两方面的研究。在我们的调查中，它不适合人类疾病的模型。因此，我们利用文献中的哺乳动物肌肉数据构建了哺乳动物的HHM。利用非线性动力系统理论和数值计算对哺乳动物HHM的分叉现象进行了研究。分叉参数是氯离子电导和钠离子通道的一些已知参数，在氯离子电导和钠离子参数的变化中，HHM没有观察到温度依赖性。在HHM两参数平面上倾斜-翻转分岔点附近的分岔图上，参数平面被分为三个区域：正常激发、重复激发(肌强直)和静息电位去极化(周期性瘫痪)。正常区域和其他两个区域用同宿分叉分开，肌强直区和瘫痪区域用双循环分叉分开，一些钠通道缺陷的疾病在低温下会恶化，但我们在HHM中的结果并非如此。这可能是因为我们在HHM中发现的钠通道缺陷是错误的，或者是因为我们在HHM中发现的温度依赖性与真正的肌肉不匹配。

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.