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| In [[neurobiology]], the '''length constant''' ('''λ''') is a mathematical constant used to quantify the distance that a [[membrane potential#Graded potentials|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.
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| The length constant can be defined as:
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| :<math> \lambda \ = \ \sqrt{\frac {r_{m}} {(r_{i}+ r_{o})}} </math>
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| where r<sub>m</sub> is the membrane [[electrical resistance and conductance|resistance]] (the force that impedes the flow of [[electric current]] from the outside of the membrane to the inside, and vice versa), r<sub>i</sub> is the axial resistance (the force that impedes current flow through the [[axoplasm]], parallel to the membrane), and r<sub>o</sub> is the extracellular resistance (the force that impedes current flow through the extracellular fluid, parallel to the membrane). In calculation, the effects of r<sub>o</sub> are negligible, so the equation is typically expressed as:
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| :<math> \lambda \ = \ \sqrt {\frac {r_{m}}{r_{i}}}</math>
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| 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 diameter of the axon, the lower the r<sub>i</sub>.
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| The length constant is used to describe the rise of [[potential difference]] across the membrane
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| :<math> V(x) \ = \ V_{max} (1 - e^{-x /\lambda})</math>
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| The fall of voltage can be expressed as:
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| :<math> V(x) \ = \ V_{max} (e^{-x /\lambda})</math> | |
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| 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).
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| V<sub>max</sub> is defined as the maximum voltage attained in the action potential, where:
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| :<math>V_{max} \ = \ r_{m}I</math> | |
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| where r<sub>m</sub> is the resistance across the membrane and I is the current flow.
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| Setting for x= λ for the rise of voltage sets V(x) equal to .63 V<sub>max</sub>. This means that the length constant is the distance at which 63% of V<sub>max</sub> has been reached during the rise of voltage.
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| Setting for x= λ for the fall of voltage sets V(x) equal to .37 V<sub>max</sub>, meaning that the length constant is the distance at which 37% of V<sub>max</sub> has been reached during the fall of voltage.
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| ==By resistivity==
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| Expressed with [[resistivity]] rather than resistance, the constant λ is (with negligible r<sub>o</sub>):<ref name=boron202>Page 202 in: {{cite book |author=Walter F., PhD. Boron |title=Medical Physiology: A Cellular And Molecular Approach |publisher=Elsevier/Saunders |location= |year=2003 |pages=1300 |isbn=1-4160-2328-3 |oclc= |doi=}}</ref>
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| :<math> \lambda = \sqrt{\frac {r \times \rho_{m}} {2 \times \rho_{i}}} </math>
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| Where <math> r </math> is the radius of the neuron.
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| The radius and number 2 come from that:
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| *<math> \rho_{m} = r_{m} \times 2\pi r </math>
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| *<math> \rho_{i} = r_{i} \times \pi r^2 </math>
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| Expressed in this way, it can be seen that the length constant increases with increasing radius of the neuron.
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| ==References==
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| {{reflist}}
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| ==See also==
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| *[[Isopotential muscle]]
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| *[[Time constant]]
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| [[Category:Electrophysiology]]
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Jayson Berryhill is how I'm known as and my spouse doesn't like it at all. For a while I've been in Alaska but I will have to move in a year or two. Credit authorising is how she makes a living. Doing ballet is something she would by no means give up.
My homepage: real psychic