Hiroyuki Kushibe

Self-Organized chemical model and approaches to membrane excitation

By applying our self-organized theory to the ferroelectric hypothesis for the channel gating mechanism by Leuchtag, we obtained a simple cubic state equation, z3 + Az + B = θ, for the behavior or instability of a dissipative structure like an excitable membrane, where — z corresponds to membrane potential and control variables A and B are assigned to dipole-dipole interaction between channel molecules and to dipole-ion interaction, respectively. This represents that, under the condition of A < 0 (T < βTc), membranes are excited with a cusp catastrophe on the bifurcation sets, 4A3 + 27B2 = 0, by the movement of A and B and that it returns to the initial metastable state, that is, the resting state. In the resting state Na+ channels are closed with the ferroelectricity and in the excited state they open with the paraelectricity. By using the functions for A˙ and B˙ which are modified from Zeemans formulas, we have calculated and analyzed some of membrane excitation; action potentials generally observed in ...

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 Leuchtags 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 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 Zeemans 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.

Entrainment in nerve by a ferroelectric model (II): quasi-periodic oscillation and the phase locking

Abstract A nonlinear state equation for membrane excitation can be simplified by Leuchtags ferroelectric model which is applied to a chemical network theory. A dissipative structure of such a membrane is described by an equilibrium space, η3 + aη + b = 0, giving a cusp catastrophe, and the membrane is self-organized in the resting state under the condition, a

Beyond-All-Orders Asymptotics and Heteroclinic Structures

We analyze the chaotic dynamics, by developing the asymptotic expansion beyond all orders exploited for the analysis of separatrices-crossing angle. Choosing the standard map for illustration, we determine the Stokes multiplier (SM) by sharpening the idea on an analytic continuation in crossing the Stokes line. σ-dependent higher-order corrections to the primary (n = 1) SM and the nature of SM for higher-order singularities are revealed. The asymptotic analytical result for the unstable manifold impinging upon a hyperbolic fixed point is shown to reproduce Birkhoff and Smale’s horse-shoe.

2704 Nonlinear responses of nerve membranes by ferroelectric model

HIROKI NAKAE Oscillation of neural networks may play a role in various neural functions because it has been observed during development, perception, and memory. The network oscillation has been pointed out to depend on the oscillation of intracellular conditions. In order to explain macroscopic oscillation, however, some mechanisms other than the oscillation of individual neurons must be assumed. In this study, a neural network consisting of excitatory and inhibitory neurons was modeled by connecting neighboring cells. Each cell was designed to have properties of temporal summation and spontaneous firing. As a result of simulation under asynchronous conditions, the network showed an oscillatory behavior, although the frequency was not strictly constant. It was not observed with neurons without the property of temporal summation or under synchronous conditions. These results suggest that collaboration of excitatory and inhibitory neurons could be a cause of neural network oscillation.

Efficiencies of diffraction gratings by an electromagnetic theory

Abstract Diffraction efficiencies of several different types of gratings are estimated by using an electromagnetic theory. The result shows that they are in good agreement with observed ones.