Rate of excitation propagation in a reduced Hodgkins-Huxley model. II. Slow relaxation of the sodium current]

The influence of sodium current activation on the value of nerve excitation conduction velocity is investigated on the basis of Hodgkin-Huxley model. The potassium activation and sodium inactivation are considered as slow processes which do not develop to an appreciable extent in the region of conduction velocity formation. The system of equations was derived and solved analytically after neglecting the dependency of sodium relaxation time on potential; the approximation of steady-state sodium activation was also used with the help of Hevyside function. The algebraic equation for conduction velocity was obtained; its solution has a simple analytical form in two limits of rapid and slow sodium current relaxation. The comparison with the experimental data has shown that at not very high temperatures the slow (compared to the potential dynamics) sodium current relaxation approximation is more appropriate. The dependency of impulse velocity on capacitance and conductance of the fiber was analyzed.     

Rate of excitation propagation in a reduced Hodgkins-Huxley model. I. Rapid relaxation of the sodium current]

The question of calculating excitation propagation velocity is analyzed on the basis of the Hodgkin-Huxley model. The activation of the sodium current is assumed to be rapid as compared to the rate of potential variation. Because of slow variation of potassium activation and sodium inactivation the dynamics of these processes is assumed to be of negligible effect in the region of impulse velocity formation. By means of pieace-wise linear approximation of thus obtained voltage-current characteristics the characteristics the analytical solution of the problem was found. In two limiting cases this solution coincides with the solutions of Kolmogorov and Scott. The dependence of impulse velocity on parameters is analyzed and illustrated graphically.     

Electric breakdown of lipid vesicles

Theoretical expression for free energy F of spherical lipid vesicle containing through pore in the presence of diffusional potential difference is derived. It is assumed that the pore radius is small in comparison with vesicle size. According to estimation the variation of elastic energy of vesicle membrane with pore radius is small. Therefore electrical breakdown becomes reversible for reasonable region of r values. Conditions of equilibrium and dynamic modes of breakdown are analyzed. Random oscillation mode of intravesicular label discharge is shown for some region of vesicle parameters.     

Predictive optimality criterion for idealization of ion channel data and exact Akaike's criterion

Different statistical or low-pass filters may be used for the idealization of ion channel data. We address the problem of predicting optimal filter parameters, represented by a threshold test-value for statistical filters and by a cut-off frequency for low-pass filters. Optimal idealization is understood in the sense of maximal similarity between recovered and real signals. Special procedures are suggested to quantitatively characterize the difference between the recovered and the real signals, the latter being known for simulated data. These procedures, called objective criteria, play the role of referees in estimating the performance of different predictive optimality criteria. We have tested standard Akaike's AIC and its modification by Rissanen, MDL. Both gave unsatisfactory results. We have shown analytically, that the Akaike-type criterion, based on the use of a certain penalty for the log likelihood function per transition, indicates the correct optimum point only if the penalty is set equal to half the optimal threshold. As the latter varies significantly for different data sets, this criterion is not particularly helpful. A new universal predictive optimality criterion, valid for real data and any idealization method, is suggested. It is formally similar to AIC, but instead of log likelihood it uses the doubled number of false transitions. The predictive power of the new criterion is demonstrated with different types of data for Hinkley and 50% amplitude methods. Received: 23 July 1996 / Accepted: 9 May 1997     

Rate of excitation propagation in a reduced Hodgkins-Huxley model. III. Integrodifferential equations]

The Hodgkin - Huxley system of equations is reduced to single integral-differential equation in neglection of slow variables dynamics. Two limiting cases of fast and slow sodium activation processes are considered. The first case leads to a nonlinear differential equation for the potential, the second one - to an ordinary differential equation with a known source as a function of coordinate. Such a simplification is due to approximation of steady-state sodium activation variable with the help of Heviside function. The validity of this approximation is discussed; the corresponding error is estimated by calculation of the second approximation for the source function.