Two-parameter bifurcations in the Hodgkin-Huxley equations for muscle fibers

ddIn order to reveal the mechanism behind the ion channel diseases of muscl es, we numerically analyzed bifurcations in the space-clamped Hodgkin-Huxle y equations for muscles of frogs (HHM). The leakage conductance g(l) and th e sodium channel effective bias voltage <(V-m)over bar> were selected as bi furcation parameters. We continued codimension-one bifurcation branches of saddle node, homoclinic, Hopf, double cycle on the <(V-m)over bar> x g(l) p lane. We showed the existence of codimension-two bifurcation points: a twis ted resonant, a Bogdanov-Takens, an inclination-flip (if), a degenerate Hop f, and a cusp. The first three were found for HHM with parameters within a realistic range. The if point is found to be a key point in the behavior of HHM, around which the parameter plane is separated into three regions: nor mal action potential, repetitive firing (myotonia), and depolarized resting potential (periodic paralysis). The characteristics of if are not as yet c ompletely revealed theoretically, so our results are also of mathematical i nterest. (C) 2000 Scripta Technica.