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1. The relation between the strength and duration of a just threshold stimulus (strength—duration curve) is analysed for an excitable membrane polarized uniformly and for an excitable cable polarized at one point.

2. The effect on the strength—duration curve of non-linearity in the membrane current—voltage curve has been analysed. The strength—duration curve can be derived if the membrane current—voltage relation is independent of time. The effects of changes in the current—voltage curve with time due to the existence of a finite membrane activation time and membrane accommodation are analysed.

3. The strength—duration curve for the Hodgkin—Huxley membrane equations (Hodgkin & Huxley, 1952) is compared to that of Hill's (1936) two-time constant model.

4. The relation between membrane current—voltage curves and those for a point-polarized cable are derived for the steady-state condition. The cable properties tend to linearize the current—voltage curve and to sharpen the voltage threshold.

5. The strength—duration curve for a point-polarized cable whose membrane obeys the Hodgkin—Huxley equations is computed numerically. There is an additional large effect on the cable strength—duration curve arising from the redistribution of charge during passage of a constant current; and the resulting strength—duration curve lies within the range of curves predicted by Hill's model.

6. The conditions required for a constant charge threshold (Hodgkin & Rushton, 1946) are shown to be satisfied for short, intense stimuli applied to a cable at one point.

7. The results are discussed with reference to the experimental studies available.

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