Digital computer solutions for excitable membrane models

Abstract

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

Ion Channel Stochasticity May Be Critical in Determining the Reliability and Precision of Spike Timing

“…We conclude that the nonfiring stable state in the deterministic HH model becomes a key player in the stochastic HH model. Experimentally, the coexistence of the two stable solutions in the the squid giant axon, as well as in the corresponding HH model, was demonstrated by Guttman et al (1980) (see also Cooley et al, 1965).…”

“…Channel stochasticity has such a dramatic effect on the voltage dynamics because it exploits a peculiar, and largely neglected, aspect of the deterministic HH equations: its two stable states for suprathreshold current input (see the discussion of the bistability in the HH equations in Cooley, Dodge, andCohen, 1965, andGuttman, Lewis, andRinzel, 1980). For a DC input, one state is the well-known repetitive firing behavior (the light trace in Figure 6A) whereas the other state is a nonfiring behavior of early damped voltage oscillations that converges to a steady voltage (see Figure 6A, dark trace).…”

Computer simulation of arrhythmias in a network of coupled excitable elements.

“…This does not mean that the mechanism of triggered activity may not reside in altered cellular electrophysiological properties, irrespective of coupling. It has been shown theoretically (Cooley et al, 1965) that "hard" oscillations (that is, oscillatory activity in a system that also has a stable, nonoscillatory stationary state) may be found in Hodgkin-Huxley elements. Our results do show, however, that coupling of elements that lack this property themselves may easily result in this type of behavior, suggesting that the conditions for triggerable focal activity may be less restricted than commonly assumed.…”

The bifurcation of periodic solutions in the Hodgkin-Huxley equations

“…The results of Cole, Antosiewicz and Rabinowitz [1] and Cooley, Dodge and Cohen [2] indicate that for each I £ (6.8, 9.8) there is a stable large-amplitude periodic orbit, and an unstable small-amplitude periodic solution which disappears at / = 9.8. Thus, in accordance with our theorem, it appears that there is a bifurcation of small-amplitude unstable periodic solutions on the interval (6.8, 9.8) with I = 9.8 being the bifurcation point.…”

“…Cooley, Dodge and Cohen [2] investigated a suggestion of FitzHugh that repetitive firing in the space clamped axon may be associated with instability of the steady state solution. For each / > 0 their numerical calculations indicate that (1.6)-(1.9) has exactly one steady-state solution, ir, = (vi, n"(v,), h"(v,)).…”