title: Length constant
text: # 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). 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 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 $r_o):$
  
      $\lambda ={\sqrt {\frac {r\rho _{m}}{2\rho _{i}}}}$
  
  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.
  
  