Calculations are reported of the time-dependent Nernst-Planck equations for a thin permeable membrane between electrolytic solutions. Charge neutrality is assumed for the time-dependent case. The response of such a membrane system to step current input is measured in terms of the time and space changes in concentration, electrical potential, and effective conductance. The report also includes discussion of boundary effects that occur when charge neutrality does not hold in the steady-state case.

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This report will present a discussion of computational results from two mathematical models which have been applied to various problems on the excitable nerve membrane. To both of these models, the electro-diffusion theory and the empirical Hodgkin-Huxley model, K. S . Cole has made important theoretical, as well as experimental contributions. We shall discuss numerical calculations from each of these models. We would like to be able to say that we have solved the problems that Cole has posed ('65) regarding the relationship between the two mathematical approaches; in fact, we have only supplied a few more quantitative details in each case.In the first section of the report the calculation of the transient response of a Nernst-Planck liquid junction to step changes in current will be discussed. What seems to be its deficiencies as a biological membrane model will be pointed out. In the second section some extensions of the numerical characterization of HodgkinHuxley theory will be given. These include the strength-duration curve as a function of temperature, a calculation suggested by Cole, some results on repetitive firing for propagated impulses, and some quantitative material on the stability of the propagated impulse.1. In the Cole paper referred to earlier ('65), a discussion of the transient response of the time-dependent NernstPlanck equations is presented by means of a series of appropriate linearizations. Of importance is the delineation of two time scales. In the nondimensional system (1)-( 8 ) , we have set out the equations and referred to similar time scales, T, and Td which correspond to electric charging time and diffusion time:The quantities and their dimensional normalizations are explained in the appendix. As can be seen, there are only two kinds of parameters: the ratio of time scales, K, and the mobilities. For typical choices of the quantities,where 1 is the membrane width. Thus, with 1 = cm, K = 1.65 X lo-', and with 1 = lo-' cm, K = 1.65 X lo-''. AS one can see from equations ( 4 ) and (5), for small K, the assumptions of electroneutrality and absence of displacement current are valid when 6E/6x and 6E/6t are not large. The assumption K = 0 says that the electric charging time is negligible. Note that K is independent of the mobilities. In the Cole paper referred to, a dimensional number K~ is defined; it is the in- 99

The linear stability of a Stefan-like problem for moving steps is analyzed within the context of Burton, Cabrera, and Frank’s theory of crystal growth [Philos. Trans. R. Soc. London Ser. A 243, 299 (1951)]. Asymmetry and departures from equilibrium at steps are included. The equations for regular perturbations around the steady state are solved analytically. The stability criterion depends on supersaturation and average step spacing, both experimentally accessible, and on dimensionless combinations of surface diffusivity, surface diffusion length, and adatom capture probabilities at steps, which can be estimated from bond models. This stability criterion is analyzed and presented graphically in terms of these physical parameters.

Solutions for the growth rate of perturbations in the locations of moving steps on a growing or evaporating crystal are presented. They are obtained by solving an equation derived by R. Ghez, H. G. Cohen, and J. B. Keller [J. Appl. Phys. 73, 3685 (1993)] based upon the Burton–Cabrera–Frank theory of crystal growth. They agree with the results derived via the adiabatic approximation when the dimensionless growth rate is small, which shows that those results are correct. However, when the growth rate is large the present exact results differ from those of the adiabatic approximation, as might be expected.