{{short description|Constant describing the distance a membrane potential passively travels along a neuron}}
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:
: <math> \lambda = \sqrt{\frac{r_m}{r_i + r_o}} </math>
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).<ref name=":0">{{cite journal |last1=Meffin |first1=Hamish |last2=Kameneva |first2=Tatiana |title=The electrotonic length constant: A theoretical estimate for neuroprosthetic electrical stimulation |journal=Biomedical Signal Processing and Control |date=April 2011 |volume=6 |issue=2 |pages=105–111 |doi=10.1016/j.bspc.2010.09.005 }}</ref> In calculation, the effects of ''r''<sub>''o''</sub> are negligible,<ref name=":0" /> so the equation is typically expressed as:
: <math> \lambda = \sqrt {\frac{r_m}{r_i}}</math>
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
: <math> V(x) = V_{\max} \left(1 - e^{-x /\lambda}\right)</math>
The fall of voltage can be expressed as:
: <math> V(x) = V_{\max} e^{-x /\lambda}</math>
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''<sub>max</sub> is defined as the maximum voltage attained in the action potential, where:
: <math>V_{\max} = r_m I</math>
where ''r''<sub>''m''</sub> 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''<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'' = ''λ'' 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.
== By resistivity == Expressed with resistivity rather than resistance, the constant ''λ'' is (with negligible ''r''<sub>''o''</sub>):<ref name=boron202>Page 202 in: {{cite book |author=Walter F., PhD. Boron |title=Medical Physiology: A Cellular And Molecular Approach |publisher=Elsevier/Saunders |year=2003 |pages=1300 |isbn=1-4160-2328-3 }}</ref>
:<math> \lambda = \sqrt{\frac {r \rho_m} {2 \rho_i}} </math>
Where <math> r </math> is the radius of the neuron.
The radius and number 2 come from these equations:
*<math> r_m = \frac{\rho_m}{2\pi r} </math> *<math> r_i = \frac{\rho_i}{\pi r^2} </math>
Expressed in this way, it can be seen that the length constant increases with increasing radius of the neuron.
== See also == * Isopotential muscle * Time constant
== References == {{reflist}}
Category:Electrophysiology