ENTRAINMENT IN NERVE BY A FERROELECTRIC MODEL .2. QUASI-PERIODIC OSCILLATION AND THE PHASE-LOCKING

A nonlinear state equation for membrane excitation can be simplified b y Leuchtag's ferroelectric model which is applied to a chemical networ k theory. A dissipative structure of such a membrane is described by a n equilibrium space, eta(3) + a eta + b = 0, giving a cusp catastrophe , and the membrane is self-organized in the resting state under the co ndition, a < 0 (T < T-c), where eta corresponds to the membrane potent ial, and a and b imply dipole-dipoIe and dipole-ion interactions of ch annel proteins embedded in the membrane, respectively. As well known, a specific characteristic of nonlinear electrical phenomena in the mem brane is a limit cycle arising through the entrainment by periodical s timuli or chaos. A phase transition between the equilibrium and the no n-equilibrium states (a dissipative structure without the resting stat e) is described by a parameter giving the difference from thermal equi librium. In this dynamic system, quasi-periodic oscillations which ari se in periodic external fields and the phase locking, that is, entrain ment, caused by changing l(0) at omega not equal omega(n) (omega(n) - the natural frequency of the membrane) are studied with parameters int roduced into Zeeman's formulas of (a) overdot and (b) overdot.