In neurobiology, the length constant (û) 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 summationâÂÂthe electrical addition of one potential with potentials from adjacent areas of the cell.
The length constant can be defined as:
where r<sub>m</sub> 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<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:
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<sub>m</sub>. The greater the diameter of the axon, the lower the r<sub>i</sub>.
The length constant is used to describe the rise of potential difference across the membrane
The fall of voltage can be expressed as:
Where voltage, V, is measured in millivolts, x is distance from the start of the potential (in millimeters), and û is the length constant (in millimeters).
V<sub>max</sub> is defined as the maximum voltage attained in the action potential, where:
where r<sub>m</sub> is the resistance across the membrane and I is the current flow.
Setting for x = û 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.
Setting for x = û 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.
Expressed with resistivity rather than resistance, the constant û is (with negligible r<sub>o</sub>):
Where is the radius of the neuron.
The radius and number 2 come from these equations:
Expressed in this way, it can be seen that the length constant increases with increasing radius of the neuron.