Cable Theory - The Length Constant

The Length Constant

The length constant, (lambda), is a parameter that indicates how far a current will travel along the inside of an axon, and thereby influence the voltage along that distance. The larger the value of, the farther the current will flow. The length constant can be expressed as:

(13)

This formula makes sense intuitively, because the larger the membrane resistance, r_m, the greater the value of, and the more current will remain inside the axoplasm to travel longitudinally through the axon. The higher the axoplasmic resistance, the smaller the value of, the harder it will be for current to travel through the axoplasm, and the shorter the current will be able to travel. It is possible to solve equation (12) and arrive at the following equation (which is valid in steady-state conditions, i.e. -- when length approaches infinity):

(14)

Where is the depolarization at (point of current injection), e is the exponential constant (approximate value 2.71828) and is the voltage at a given distance x from x=0. When then

(15)

and

(16)

which means that when we measure at distance from we get

(17)

Thus is always 36.8 percent of .