ENTRAINMENT IN NERVE BY A FERROELECTRIC MODEL - BIFURCATION AND LIMIT-CYCLES

The excitation in nerve that is self-organized in a dissipative struct ure with the resting membrane potential (an equilibrium structure) occ urs on an equilibrium space for the cusp catastrophe. The space is giv en by a nonlinear state equation eta(3) + a eta + b = 0 deduced from a chemical network model which is applied to Leuchtag's ferroelectric h ypothesis for Na channels, where -eta corresponds to the membrane pote ntial, a and b are control parameters related to the dipole-dipole and dipole-ion interactions, respectively. A phase transition of the memb rane organized in a region, a < 0 (T < T-c), can be determined by a pa rameter which describes the difference from equilibrium. When the memb rane in a self-oscillation is disturbed by a periodical Na current wit h the natural frequency of the membrane or near one, a stable limit cy cle of the potential arises through an entrainment. With modified Zeem an's formulas for the movements of a and b in the equation, the transi tions are calculated to arise at two points (lowest s(l) and highest s (h) limits) discontinuously, so s(h) which is the subcritical point di ffers from the result by the modified Hodgkin-Huxley theory. This seem s to show a characteristic of the catastrophe.