Length constant

Length constant is a constant used in neurobiology signified by the Greek letter lambda (λ).

In an action potential (or in a passive spread of signal) in a neuron, the constant λ is

$\lambda \ =\ {\sqrt {\frac {r_{m}}{(r_{i}+r_{o})}}}$

where r_m is the resistance across the membrane, r_i is the resistance inside the membrane, and r_o is the resistance outside the membrane. In calculation, the effects of r_o are negligible, so the equation becomes

$\lambda \ =\ {\sqrt {\frac {r_{m}}{r_{i}}}}$

The resistance across the membrane is a function of the number of open ion channels and the resistance inside the membrane is generally a function of the diameter of the axon. A large diameter is related to a lower r_i.

The length constant is used to describe the rise of potential difference across the membrane

$V(x)\ =\ V_{max}(1-e^{-x/\lambda })$

The fall of voltage is described by

$V(x)\ =\ V_{max}(e^{-x/\lambda })$

Where voltage is typically in millivolts, x is distance in millimeters, and λ is in millimeters.

V_max is defined as the maximum voltage attained in the action potential, where

$V_{max}\ =\ r_{m}I$

where r_m is the resistance across the membrane and I is the current flow.

Setting for x= λ for the rise of voltage sets V(x) equal to .63 V_max. This means that the length constant is the distance at which 63% of V_max has been reached during the rise of voltage.

Setting for x= λ for the fall of voltage sets V(x) equal to .37 V_max, meaning that the length constant is the distance at which 37% of V_max has been reached during the fall of voltage.

The longer a length constant is, the bigger the effect of a potential (either an action potential or a current injected at the site) will have along the cell. A long length constant can result in spatial summation, or the algebraic summation of one potential with other potentials from other areas of the cell.

See also

• Time constant