Application of bifurcation theory to nerve excitations

An expression for the amplitude of stable steady self‐oscillatory solutions for the inverted bifurcation is obtained by applying bifurcation theory to the BVP equation, which is a model describing nerve excitations. Up to the fifth‐power term with respect to the amplitude is included in the calculation, assuming that the amplitude of the steady self‐oscillatory solution is of small order. The obtained expression for the inverted‐bifurcation can continuously be transformed, leading to an expression for the normal‐bifurcation by merely varying one parameter. Further, utilizing that expression and employing the reductive perturbation method in which the amplitude is regarded as an infinitesimal parameter, an expression describing the relaxation phenomenon in the neighborhood of the stable steady solution is derived for the inverted‐bifurcation as well as for the normal‐bifurcation, the nature of which is close to that of the inverted‐bi‐furcation. Since this expression has the form of the usual time‐dependent Ginzburg‐Landau(TDGL) equation and an additional term, the behavior of those bifurcations which considerably differ from the usual normal‐bifurcation can well be described by the derived expression.