Editing Action potential (section)

===Cable theory===
{{Main|Cable theory}}
[[File:Cable theory Neuron RC circuit v3.svg|alt=A diagram showing the resistance and capacitance across the cell membrane of an axon. The cell membrane is divided into adjacent regions, each having its own resistance and capacitance between the cytosol and extracellular fluid across the membrane. Each of these regions is in turn connected by an intracellular circuit with a resistance.|thumb|300x300px|Cable theory's simplified view of a neuronal fiber. The connected [[RC circuit]]s correspond to adjacent segments of a passive [[neurite]]. The extracellular resistances ''r<sub>e</sub>'' (the counterparts of the intracellular resistances ''r<sub>i</sub>'') are not shown, since they are usually negligibly small; the extracellular medium may be assumed to have the same voltage everywhere.]]
The flow of currents within an axon can be described quantitatively by [[cable theory]]<ref name="rall_1989">[[Wilfrid Rall|Rall, W]] in {{harvnb|Koch|Segev|1989|loc=''Cable Theory for Dendritic Neurons'', pp. 9–62.}}</ref> and its elaborations, such as the compartmental model.<ref name="segev_1989">{{cite book | vauthors = Segev I, Fleshman JW, Burke RE | chapter = Compartmental Models of Complex Neurons | title = Methods in Neuronal Modeling: From Synapses to Networks. | veditors = Koch C, Segev I  | editor1-link = Christof Koch | date = 1989 | pages = 63–96 | publisher = The MIT Press | location = Cambridge, Massachusetts | isbn = 978-0-262-11133-1 | lccn = 88008279 | oclc = 18384545 }}</ref> Cable theory was developed in 1855 by [[William Thomson, 1st Baron Kelvin|Lord Kelvin]] to model the transatlantic telegraph cable<ref name="kelvin_1855" group=lower-alpha>{{cite journal | vauthors = Kelvin WT | year = 1855 | title = On the theory of the electric telegraph | journal = Proceedings of the Royal Society | volume = 7 | pages = 382–99 | doi = 10.1098/rspl.1854.0093| s2cid = 178547827 | author-link = William Thomson, 1st Baron Kelvin }}</ref> and was shown to be relevant to neurons by [[Alan Lloyd Hodgkin|Hodgkin]] and [[W. A. H. Rushton|Rushton]] in 1946.<ref name="hodgkin_1946" group=lower-alpha>{{cite journal | vauthors = Hodgkin AL, Rushton WA | title = The electrical constants of a crustacean nerve fibre | journal = Proceedings of the Royal Society of Medicine | volume = 134 | issue = 873 | pages = 444–79 | date = December 1946 | pmid = 20281590 | doi = 10.1098/rspb.1946.0024 | author-link1 = Alan Lloyd Hodgkin | bibcode = 1946RSPSB.133..444H | doi-access = free }}</ref> In simple cable theory, the neuron is treated as an electrically passive, perfectly cylindrical transmission cable, which can be described by a [[partial differential equation]]<ref name="rall_1989" />

:<math>
\tau \frac{\partial V}{\partial t} = \lambda^2 \frac{\partial^2 V}{\partial x^2} - V
</math>

where ''V''(''x'', ''t'') is the voltage across the membrane at a time ''t'' and a position ''x'' along the length of the neuron, and where λ and τ are the characteristic length and time scales on which those voltages decay in response to a stimulus. Referring to the circuit diagram on the right, these scales can be determined from the resistances and capacitances per unit length.{{sfn|Purves|Augustine|Fitzpatrick|Hall|2008|pp=52–53}}

:<math>
\tau =\ r_m c_m \,
</math>

:<math>
\lambda = \sqrt \frac{r_m}{r_\ell}
</math>

These time and length-scales can be used to understand the dependence of the conduction velocity on the diameter of the neuron in unmyelinated fibers. For example, the time-scale τ increases with both the membrane resistance ''r<sub>m</sub>'' and capacitance ''c<sub>m</sub>''. As the capacitance increases, more charge must be transferred to produce a given transmembrane voltage (by [[capacitance|the equation ''Q''&nbsp;=&nbsp;''CV'']]); as the resistance increases, less charge is transferred per unit time, making the equilibration slower. In a similar manner, if the internal resistance per unit length ''r<sub>i</sub>'' is lower in one axon than in another (e.g., because the radius of the former is larger), the spatial decay length λ becomes longer and the [[conduction velocity]] of an action potential should increase. If the transmembrane resistance ''r<sub>m</sub>'' is increased, that lowers the average "leakage" current across the membrane, likewise causing ''λ'' to become longer, increasing the conduction velocity.