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.

ee also

*Time constant