Length constant: Difference between revisions

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


In an [[action potential]] (or in a passive spread of signal) in a [[neuron]], the constant λ is
In an [[action potential]] (or in a passive spread of signal) in a [[neuron]], the constant λ is
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:<math> \lambda \ = \ \sqrt {\frac {r_{m}}{r_{i}}}</math>
:<math> \lambda \ = \ \sqrt {\frac {r_{m}}{r_{i}}}</math>


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<sub>i</sub>.
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<sub>i</sub>.


The length constant is used to describe the rise of [[potential difference]] across the membrane
The length constant is used to describe the rise of [[potential difference]] across the membrane
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==See also==
==See also==


*[[Time constant]]
*[[Time constant]][[Category:Suggestion Bot Tag]]

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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


where rm is the resistance across the membrane, ri is the resistance inside the membrane, and ro is the resistance outside the membrane. In calculation, the effects of ro are negligible, so the equation becomes

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 ri.

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

The fall of voltage is described by

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

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

where rm 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 Vmax. This means that the length constant is the distance at which 63% of Vmax has been reached during the rise of voltage.

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