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 farther 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 {\frac {r_{m}}{r_{i}+r_{o}}}}

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).[1] In calculation, the effects of r_o are negligible,[1] so the equation is typically expressed as:

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

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_{\max }\left(1-e^{-x/\lambda }\right)

The fall of voltage can be expressed as:

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

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_{\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.

By resistivity

Expressed with resistivity rather than resistance, the constant λ is (with negligible r_o):[2]

\lambda ={\sqrt {\frac {r\rho _{m}}{2\rho _{i}}}}

Where $r$ is the radius of the neuron.

The radius and number 2 come from these equations:

$r_{m}={\frac {\rho _{m}}{2\pi r}}$
$r_{i}={\frac {\rho _{i}}{\pi r^{2}}}$

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

References

Meffin, Hamish; Kameneva, Tatiana (2011-04-01). "The electrotonic length constant: A theoretical estimate for neuroprosthetic electrical stimulation". Biomedical Signal Processing and Control. 6 (2): 105–111. doi:10.1016/j.bspc.2010.09.005. ISSN 1746-8094 – via ScienceDirect.
Page 202 in: Walter F., PhD. Boron (2003). Medical Physiology: A Cellular And Molecular Approach. Elsevier/Saunders. p. 1300. ISBN 1-4160-2328-3.

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[:0-1] Meffin, Hamish; Kameneva, Tatiana (2011-04-01). "The electrotonic length constant: A theoretical estimate for neuroprosthetic electrical stimulation". Biomedical Signal Processing and Control. 6 (2): 105–111. doi:10.1016/j.bspc.2010.09.005. ISSN 1746-8094 – via ScienceDirect.

[boron202-2] Page 202 in: Walter F., PhD. Boron (2003). Medical Physiology: A Cellular And Molecular Approach. Elsevier/Saunders. p. 1300. ISBN 1-4160-2328-3.

Length constant

By resistivity

See also

References