Dynamical behavior of the Hodgkin-Huxley equations for muscles

In order to reveal the mechanism behind the ion channel diseases of muscles (icdm), bifurcation phenomena in the space-clamped Hodgkin-Huxley equations for muscles (HHM) is studied. Parameters relate to icdm, the leakage conductance g_l and a parameter of sodium permeability (V¯_m¯ or V¯_h¯) are chosen as bifurcation parameters. As results, three types of behavior: a normal action potential; repetitive firing; and a depolarized resting potential, are observed in HHM as a function of the parameters. The parameter region of the three types of behaviors meets at the codimension-two bifurcation point called an inclination-flip bifurcation point (if), around which complicated dynamics including a period doubling bifurcation root to chaos occurs. The three types of behaviors in HHM correspond qualitatively well to the electrical activities of real muscles of (i) normal, (ii) myotonia (muscle stiffness) and (iii) paralysis, respectively. The results imply that a relatively small change of the parameters might cause drastic change of electrical activity of muscles. Such dynamics explains the physiological and medical observations of icdm qualitatively well