String diagram

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). In calculation, the effects of r_o are negligible, 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 diameter of the axon, the lower the r_i.

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

$V(x)\ =\ V_{max}(1-e^{-x/\lambda })$

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):

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

Where $r$ is the radius of the neuron.

The radius and number 2 come from that:

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

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

References

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See also

• Isopotential muscle
• Time constant