Global Bifurcation Structure of Periodic Solutions in Hodgkin-Huxley Equations

The Hodgkin-Huxley equations (Hodgkin & Huxley 1952) are a neuron model describing electrical excitation of the squid giant axon membrane. By examining the global bifurcation structure of these equations, we found a degenerate Hopf bifurcation point. Several stable periodic orbits coexist in the neighborhood of this point. We determined parameter ranges where such multistability occurs and delimited regions where either two stable periodic solutions, or two stable periodic solutions and a stable equilibrium point coexist. We argue that comparison between the global bifurcation structure experimental data provides insight into the domain of validity of the Hodgkin-Huxley equations.