The classical cable equation, in which membrane conductance is considered constant, is modified by including the linearized effect of membrane potential on sodium and potassium ionic currents, as formulated in the Hodgkin-Huxley equations for the squid giant axon. The resulting partial differential equation is solved by numerical inversion of the Laplace transform of the voltage response to current and voltage inputs. The voltage response is computed for voltage step, current step, and current pulse inputs, and the effect of temperature on the response to a current step input is also calculated.The validity of the linearized approximation is examined by comparing the linearized response to a current step input with the solution of the nonlinear partial differential cable equation for various subthreshold current step inputs.All the computed responses for the squid giant axon show oscillatory behavior and depart significantly from what is predicted on the basis of the classical cable equation. The linearization procedure, coupled with numerical inversion of the Laplace transform, proves to be a convenient approach which predicts at least qualitatively the subthreshold behavior of the nonlinear system.
We have investigated background and bleaching adaptation in vertebrate rods by intracellular recording in the retina of Bufo marinus. Backgrounds and bleaching produce adaptation in photoreceptors and lead to a shift and a compression of the response operating range. Threshold elevation due to backgrounds follows the Rose-DeVries rule at low intensities and the Weber-Fechner rule at high intensities. Threshold elevation due to bleaching is linear almost up to 17% bleached pigment and exponential thereafter. An equivalence can be established between bleaching and backgrounds with respect to threshold elevation, on the one hand, and with respect to response compression, on the other. These equivalences are the same within experimental error. The equivalence, moreover, appears to extend to the complete response curve. These results have implications for psychophysics as well as for photoreceptor transduction.
The effect of varying membrane capacitance, conductance, and rate constants on the properties of the nerve impulse is considered in terms of the degree of regeneration in the Hodgkin-Huxley model for the squid giant axon. It is shown through computer simulation that reducing regeneration generally increases the duration of the action potential and decreases its amplitude, rate of rise, and conduction velocity. The threshold becomes much less sharp and the amplitude of the response of a patch of membrane grades with stimulus strength. A second stimulus, applied shortly after a first stimulus, considerably perturbs the membrane potential from its original time-course. Under certain conditions, the nerve signal can propagate with a small decrement.
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