Length constant

In neurobiology, the length constant (''λ'') is a mathematical constant used to quantify the distance that a graded electric potential will travel along a neurite via passive electrical conduction. The greater the value of the length constant, the further the potential will travel. A large length constant can contribute to spatial summation—the electrical addition of one potential with potentials from adjacent areas of the cell.

The length constant can be defined as:

: $\lambda = \sqrt$

where ''r''_''m'' is the membrane resistance (the force that impedes the flow of electric current from the outside of the membrane to the inside, and vice versa), ''r''_''i'' is the axial resistance (the force that impedes current flow through the axoplasm, parallel to the membrane), and ''r''_''o'' is the extracellular resistance (the force that impedes current flow through the extracellular fluid, parallel to the membrane). In calculation, the effects of ''r''_''o'' are negligible, so the equation is typically expressed as:

: $\lambda = \sqrt$

The membrane resistance is a function of the number of open ion channels, and the axial resistance is generally a function of the diameter of the axon. The greater the number of open channels, the lower the ''r''_''m''. The greater the diameter of the axon, the lower the ''r''_''i''.

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

: $V(x) = V_ \left(1 - e^\right)$

The fall of voltage can be expressed as:

: $V(x) = V_ e^$

Where voltage, ''V'', is measured in millivolts, ''x'' is distance from the start of the potential (in millimeters), and ''λ'' is the length constant (in millimeters).

''V''_max is defined as the maximum voltage attained in the action potential, where:

: $V_ = r_m I$

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.

By resistivity

Expressed with resistivity rather than resistance, the constant ''λ'' is (with negligible ''r''_''o''):Page 202 in:

:$\lambda = \sqrt$

Where $r$ is the radius of the neuron.

The radius and number 2 come from these equations:

*$r_m = \frac$
*$r_i = \frac$

Expressed in this way, it can be seen that the length constant increases with increasing radius of the neuron.

See also

* Isopotential muscle
* Time constant

References

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Electrophysiology