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 {\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.

^ ^a ^b 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.

[: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.

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