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:
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:
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
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 λ is the length constant (in millimeters).
V_{max} is defined as the maximum voltage attained in the action potential, where:
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.
Expressed with resistivity rather than resistance, the constant λ is (with negligible r_{o}):^{[1]}
Where $r$ 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.