Editing Action potential (section)

==Quantitative models==
{{Main|Quantitative models of the action potential}}

[[Image:MembraneCircuit.svg|thumb|336px|right|Equivalent electrical circuit for the Hodgkin–Huxley model of the action potential. ''I<sub>m</sub>'' and ''V<sub>m</sub>'' represent the current through, and the voltage across, a small patch of membrane, respectively. The ''C<sub>m</sub>'' represents the capacitance of the membrane patch, whereas the four ''g''{{'s}} represent the [[electrical conductance|conductances]] of four types of ions. The two conductances on the left, for potassium (K) and sodium (Na), are shown with arrows to indicate that they can vary with the applied voltage, corresponding to the [[voltage-gated ion channel|voltage-sensitive ion channels]]. The two conductances on the right help determine the [[resting membrane potential]].|alt=Circuit diagram depicting five parallel circuits that are interconnected at the top to the extracellular solution and at the bottom to the intracellular solution.]]

Mathematical and computational models are essential for understanding the action potential, and offer predictions that may be tested against experimental data, providing a stringent test of a theory. The most important and accurate of the early neural models is the [[Hodgkin–Huxley model]], which describes the action potential by a coupled set of four [[ordinary differential equation]]s (ODEs).<ref name="hodgkin_1952" group=lower-alpha /> Although the Hodgkin–Huxley model may be a simplification with few limitations<ref>{{cite journal | vauthors = Baranauskas G, Martina M | title = Sodium currents activate without a Hodgkin-and-Huxley-type delay in central mammalian neurons | journal = The Journal of Neuroscience | volume = 26 | issue = 2 | pages = 671–84 | date = January 2006 | pmid = 16407565 | pmc = 6674426 | doi = 10.1523/jneurosci.2283-05.2006 }}</ref> compared to the realistic nervous membrane as it exists in nature, its complexity has inspired several even-more-simplified models,{{sfn|Hoppensteadt|1986}}<ref group=lower-alpha>* {{cite journal | vauthors = Fitzhugh R | title = Thresholds and plateaus in the Hodgkin-Huxley nerve equations | journal = The Journal of General Physiology | volume = 43 | issue = 5 | pages = 867–96 | date = May 1960 | pmid = 13823315 | pmc = 2195039 | doi = 10.1085/jgp.43.5.867 }}<br />* {{cite journal | vauthors = Kepler TB, Abbott LF, Marder E | title = Reduction of conductance-based neuron models | journal = Biological Cybernetics | volume = 66 | issue = 5 | pages = 381–7 | year = 1992 | pmid = 1562643 | doi = 10.1007/BF00197717 | s2cid = 6789007 }}</ref> such as the [[Morris–Lecar model]]<ref name="morris_1981" group=lower-alpha>{{cite journal | vauthors = Morris C, Lecar H | title = Voltage oscillations in the barnacle giant muscle fiber | journal = Biophysical Journal | volume = 35 | issue = 1 | pages = 193–213 | date = July 1981 | pmid = 7260316 | pmc = 1327511 | doi = 10.1016/S0006-3495(81)84782-0 | bibcode = 1981BpJ....35..193M }}</ref> and the [[FitzHugh–Nagumo model]],<ref name="fitzhugh" group=lower-alpha>{{cite journal | vauthors = Fitzhugh R | title = Impulses and Physiological States in Theoretical Models of Nerve Membrane | journal = Biophysical Journal | volume = 1 | issue = 6 | pages = 445–66 | date = July 1961 | pmid = 19431309 | pmc = 1366333 | doi = 10.1016/S0006-3495(61)86902-6 | bibcode = 1961BpJ.....1..445F }}<br />* {{cite journal | vauthors = Nagumo J, Arimoto S, Yoshizawa S | year = 1962 | title = An active pulse transmission line simulating nerve axon | journal = Proceedings of the IRE | volume = 50 | pages = 2061–2070 | doi = 10.1109/JRPROC.1962.288235 | issue = 10 | s2cid = 51648050 }}</ref> both of which have only two coupled ODEs. The properties of the Hodgkin–Huxley and FitzHugh–Nagumo models and their relatives, such as the Bonhoeffer–Van der Pol model,<ref name="bonhoeffer_vanderPol" group=lower-alpha>{{cite journal | vauthors = Bonhoeffer KF | title = Activation of passive iron as a model for the excitation of nerve | journal = The Journal of General Physiology | volume = 32 | issue = 1 | pages = 69–91 | date = September 1948 | pmid = 18885679 | pmc = 2213747 | doi = 10.1085/jgp.32.1.69 }}<br />* {{cite journal | vauthors = Bonhoeffer KF | year = 1953 | title = Modelle der Nervenerregung | journal = Naturwissenschaften | volume = 40 | pages = 301–311 | doi = 10.1007/BF00632438|bibcode = 1953NW.....40..301B | issue = 11 | s2cid = 19149460 }}<br />* {{cite journal | vauthors = Van der Pol B | year = 1926 | title = On relaxation-oscillations | journal = Philosophical Magazine | volume = 2 | pages = 977–992| author-link = Balthasar van der Pol }}<br />* {{cite journal | year = 1928 | title = The heartbeat considered as a relaxation oscillation, and an electrical model of the heart | journal = Philosophical Magazine | volume = 6 | pages = 763–775| vauthors = Van der Pol B, Van der Mark J| author-link1 = Balthasar van der Pol | doi=10.1080/14786441108564652}}<br />* {{cite journal | year = 1929 | title = The heartbeat considered as a relaxation oscillation, and an electrical model of the heart | journal = Arch. Neerl. Physiol. | volume = 14 | pages = 418–443| vauthors = Van der Pol B, van der Mark J| author-link1 = Balthasar van der Pol }}</ref> have been well-studied within mathematics,<ref name="math_studies">Sato, S; Fukai, H; Nomura, T; Doi, S in {{harvnb|Reeke|Poznanski|Sporns|Rosenberg|2005|loc=''Bifurcation Analysis of the Hodgkin-Huxley Equations'', pp. 459–478.}}<br />* FitzHugh, R in {{harvnb|Schwann|1969|loc=''Mathematical models of axcitation and propagation in nerve'', pp. 12–16.}}<br />* {{harvnb|Guckenheimer|Holmes|1986|pp=12–16}}</ref><ref group=lower-alpha>{{cite journal | vauthors = Evans JW | year = 1972 | title = Nerve axon equations. I. Linear approximations | journal = Indiana Univ. Math. J. | volume = 21 | pages = 877–885 | doi = 10.1512/iumj.1972.21.21071 | issue = 9| doi-access = free }}<br />* {{cite journal | vauthors = Evans JW, Feroe J | year = 1977 | title = Local stability theory of the nerve impulse | journal = Math. Biosci. | volume = 37 | pages = 23–50 | doi = 10.1016/0025-5564(77)90076-1 }}</ref> computation<ref name="computational_studies">Nelson, ME; Rinzel, J in {{harvnb|Bower|Beeman|1995|loc=''The Hodgkin-Huxley Model'', pp. 29–49.}}<br />* Rinzel, J & Ermentrout, GB; in {{harvnb|Koch|Segev|1989|loc=''Analysis of Neural Excitability and Oscillations'', pp. 135–169.}}</ref> and electronics.<ref name="keener_1983" group=lower-alpha>{{cite journal | vauthors = Keener JP | year = 1983 | title = Analogue circuitry for the Van der Pol and FitzHugh-Nagumo equations | journal = IEEE Transactions on Systems, Man, and Cybernetics | volume = 13 | issue = 5 | pages = 1010–1014 | doi = 10.1109/TSMC.1983.6313098 | s2cid = 20077648 }}</ref> However the simple models of generator potential and action potential fail to accurately reproduce the near threshold neural spike rate and spike shape, specifically for the [[mechanoreceptors]] like the [[Pacinian corpuscle]].<ref>{{cite journal | vauthors = Biswas A, Manivannan M, Srinivasan MA | title = Vibrotactile sensitivity threshold: nonlinear stochastic mechanotransduction model of the Pacinian Corpuscle | journal = IEEE Transactions on Haptics | volume = 8 | issue = 1 | pages = 102–13 | year = 2015 | pmid = 25398183 | doi = 10.1109/TOH.2014.2369422 | s2cid = 15326972 | url = https://zenodo.org/record/894772 }}</ref> More modern research has focused on larger and more integrated systems; by joining action-potential models with models of other parts of the nervous system (such as dendrites and synapses), researchers can study [[neural computation]]{{sfnm|1a1=McCulloch|1y=1988|1pp=19–39, 46–66, 72–141|2a1=Anderson|2a2=Rosenfeld|2y=1988|2pp=15–41}} and simple [[reflex]]es, such as [[escape reflex]]es and others controlled by [[central pattern generator]]s.<ref name="cpg">Getting, PA in {{harvnb|Koch|Segev|1989|loc=''Reconstruction of Small Neural Networks'', pp. 171–194.}}</ref><ref name="pmid10713861" group=lower-alpha>{{cite journal | vauthors = Hooper SL | title = Central pattern generators | journal = Current Biology | volume = 10 | issue = 5 | pages = R176–R179 | date = March 2000 | pmid = 10713861 | doi = 10.1016/S0960-9822(00)00367-5 | bibcode = 2000CBio...10.R176H | citeseerx = 10.1.1.133.3378 | s2cid = 11388348 }}</ref>
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