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

Abstract

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 η 3 + aη + b = 0 deduced from a chemical network model which is applied to Leuchtag's ferroelectric hypothesis for Na channels, where −η 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 1 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.