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 by Leuchtag's ferroelectric model which is applied to a chemical network theory. A dissipative structure of such a membrane is described by an equilibrium space, eta(3) + a eta + b = 0, giving a cusp catastrophe, and the membrane is self-organized in the resting state under the condition, a < 0 (T < T-c), where eta corresponds to the membrane potential, and a and b imply dipole-dipoIe and dipole-ion interactions of channel proteins embedded in the membrane, respectively. As well known, a specific characteristic of nonlinear electrical phenomena in the membrane is a limit cycle arising through the entrainment by periodical stimuli or chaos. A phase transition between the equilibrium and the non-equilibrium states (a dissipative structure without the resting state) is described by a parameter giving the difference from thermal equilibrium. In this dynamic system, quasi-periodic oscillations which arise in periodic external fields and the phase locking, that is, entrainment, caused by changing l(0) at omega not equal omega(n) (omega(n) - the natural frequency of the membrane) are studied with parameters introduced into Zeeman's formulas of (a) overdot and (b) overdot.