Entrainment in nerve by a ferroelectric model: Bifurcation and limit cycles

The excitation in nerve that is self-organized in a dissipative structure with the resting membrane potential (an equilibrium structure) occurs on an equilibrium space for the cusp catastrophe. The space is given by a nonlinear state equation eta(3) + a eta + b = 0 deduced from a chemical network model which is applied to Leuchtag's ferroelectric hypothesis for Na channels, where -eta corresponds to the membrane potential, a and b are control parameters related to the dipole-dipole and dipole-ion interactions, respectively. A phase transition of the membrane organized in a region, a < 0 (T < T-c), can be determined by a parameter which describes the difference from equilibrium. When the membrane in a self-oscillation is disturbed by a periodical Na current with the natural frequency of the membrane or near one, a stable limit cycle of the potential arises through an entrainment. With modified Zeeman's formulas for the movements of a and b in the equation, the transitions are calculated to arise at two points (lowest s(l) and highest s(h) limits) discontinuously, so s(h) which is the subcritical point differs from the result by the modified Hodgkin-Huxley theory. This seems to show a characteristic of the catastrophe.