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<title>FitzHugh-Nagumo-Modell</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">FitzHugh-Nagumo-Modell</span></h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Das <b>FitzHugh-Nagumo-Modell</b> (nach Richard FitzHugh (1922–2007)<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> und J. Nagumo, die das Modell unabhängig voneinander entwickelten) beschreibt einen Prototyp eines anregbaren Systems, zum Beispiel eines <a href="Neuron" class="mw-redirect" title="Neuron">Neurons</a> oder <a href="Axon" title="Axon">Axons</a>.

Wenn die äußere Anregung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\rm {ext}}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle I_{\rm {ext}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1b8e0dcf6ad0780e535032a1ec267101e81075e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.493ex; height:2.509ex;" alt="{\displaystyle I_{\rm {ext}}}" loading="lazy"></span> einen Schwellenwert überschreitet, führt das System eine charakteristische Exkursion im <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (v,w)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle (v,w)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe1ba70e3b2d314bc63be62afe1e5c3a80b95834.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.635ex; height:2.843ex;" alt="{\displaystyle (v,w)}" loading="lazy"></span>-Phasenraum aus,

bevor die Variablen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

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<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w}">

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<annotation encoding="application/x-tex">{\displaystyle w}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/88b1e0c8e1be5ebe69d18a8010676fa42d7961e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}" loading="lazy"></span> zu ihren Ruhewerten <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (v_{0},w_{0})}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle (v_{0},w_{0})}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cc6d1fb66ef936de6dbf1a9f4af43e1ad3e30c32.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.743ex; height:2.843ex;" alt="{\displaystyle (v_{0},w_{0})}" loading="lazy"></span> zurückkehren.

Dieses Verhalten ist modellhaft für die Generation von Spikes (=kurzzeitige Erhöhung der Membranspannung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"></span>) in einem Neuron nach Stimulation durch einen externen Strom <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\rm {ext}}}">

<msub>

</mrow>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle I_{\rm {ext}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1b8e0dcf6ad0780e535032a1ec267101e81075e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.493ex; height:2.509ex;" alt="{\displaystyle I_{\rm {ext}}}" loading="lazy"></span>.

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<p>Notation:

</p>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"></span> bezeichnet das Membranpotential</li>

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<annotation encoding="application/x-tex">{\displaystyle w}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/88b1e0c8e1be5ebe69d18a8010676fa42d7961e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}" loading="lazy"></span> ist eine Erholungs-Variable, die einen zur Erholung nach der elektrischen Anregung notwendigen negativen Feedback mit linearer Dynamik beschreibt.</li>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle I_{\rm {ext}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1b8e0dcf6ad0780e535032a1ec267101e81075e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.493ex; height:2.509ex;" alt="{\displaystyle I_{\rm {ext}}}" loading="lazy"></span> ist der externen Strom.</li>

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<annotation encoding="application/x-tex">{\displaystyle a,b}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/181523deba732fda302fd176275a0739121d3bc8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.261ex; height:2.509ex;" alt="{\displaystyle a,b}" loading="lazy"></span> sind Konstanten.</li></ul>

<p>Die Gleichungen dieses dynamischen Systems lauten

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<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\dot {v}}=v-{\frac {1}{3}}v^{3}-w+I_{\rm {ext}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8e4e2ec86aea2c6a24983445b7f508cb2123d61a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.312ex; height:5.176ex;" alt="{\displaystyle {\dot {v}}=v-{\frac {1}{3}}v^{3}-w+I_{\rm {ext}}}" loading="lazy"></span></dd>

<mover>

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<annotation encoding="application/x-tex">{\displaystyle \tau {\dot {w}}=v-a-bw}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5f9d12706c57e9cf8d3b8ec5c6254af43dd4ab35.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.664ex; height:2.343ex;" alt="{\displaystyle \tau {\dot {w}}=v-a-bw}" loading="lazy"></span></dd></dl>

<p>Die Anregungs-Dynamik kann mithilfe der <a href="Phasenraum#Beispiel_einer_Phasenraumanalyse" title="Phasenraum">Nullklinen</a> anschaulich dargestellt werden. Der stationäre Punkt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (v_{0},w_{0})}">

<msub>

</mrow>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle (v_{0},w_{0})}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cc6d1fb66ef936de6dbf1a9f4af43e1ad3e30c32.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.743ex; height:2.843ex;" alt="{\displaystyle (v_{0},w_{0})}" loading="lazy"></span> (Ruhewerte) ist der Schnittpunkt der <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {v}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\dot {v}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bbeddb00c15fd98c71c617920828faaeafd5e9e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.227ex; height:2.176ex;" alt="{\displaystyle {\dot {v}}}" loading="lazy"></span>- und der <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {w}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\dot {w}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01571c4ce983c482f685ef753840a25c83607425.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:2.176ex;" alt="{\displaystyle {\dot {w}}}" loading="lazy"></span>-Nullklinen. Wird das System für kurze Zeit angeregt (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\rm {ext}}\neq 0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I_{\rm {ext}}\neq 0}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/418d42a0dba7019e76acd3d0a809e58dc540627c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.754ex; height:2.676ex;" alt="{\displaystyle I_{\rm {ext}}\neq 0}" loading="lazy"></span>), beschreibt es eine Exkursion im Phasenraum, die sich in vier Stadien einteilen lässt: zunächst beschreibt die Trajektorie eine fast horizontale Trajektorie, da wegen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau \gg 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau \gg 1}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/41083b19f81bb8e6da4a131bbdbf9a9ec3912aeb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.978ex; height:2.176ex;" alt="{\displaystyle \tau \gg 1}" loading="lazy"></span> gilt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {v}}\gg {\dot {w}}}">

<mover>

</mover>

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<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\dot {v}}\gg {\dot {w}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4a022df59240edfabed8139ae74c4db1c8164c14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.505ex; height:2.176ex;" alt="{\displaystyle {\dot {v}}\gg {\dot {w}}}" loading="lazy"></span>. Sobald die Trajektorie die kubische <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {v}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\dot {v}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bbeddb00c15fd98c71c617920828faaeafd5e9e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.227ex; height:2.176ex;" alt="{\displaystyle {\dot {v}}}" loading="lazy"></span>-Nullkline erreicht, sinkt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {v}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\dot {v}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bbeddb00c15fd98c71c617920828faaeafd5e9e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.227ex; height:2.176ex;" alt="{\displaystyle {\dot {v}}}" loading="lazy"></span> rapide und die Trajektorie folgt der <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {v}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\dot {v}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bbeddb00c15fd98c71c617920828faaeafd5e9e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.227ex; height:2.176ex;" alt="{\displaystyle {\dot {v}}}" loading="lazy"></span>-Nullklinen. Am oberen Scheitelpunkt der <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {v}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\dot {v}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bbeddb00c15fd98c71c617920828faaeafd5e9e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.227ex; height:2.176ex;" alt="{\displaystyle {\dot {v}}}" loading="lazy"></span>-Nullklinen, erfolgt eine weitere horizontale Passage zum linken Ast der <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {v}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\dot {v}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bbeddb00c15fd98c71c617920828faaeafd5e9e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.227ex; height:2.176ex;" alt="{\displaystyle {\dot {v}}}" loading="lazy"></span>-Nullkline, und anschließend eine erneute Phase, in der die Trajektorie dieser Nullklinen folgt.

</p><p>Das FitzHugh-Nagumo-Modell, welches detailliert die Aktivierungs- und Deaktivierungsdynamik in einem spikenden Neuron abbildet, ist eine vereinfachte Version des <a href="Hodgkin-Huxley-Modell" title="Hodgkin-Huxley-Modell">Hodgkin-Huxley-Modell</a>. In den Original-Artikeln von FitzHugh wird dies Modell auch als Bonhoeffer-van-der-Pol-Oszillator bezeichnet, da es den <a href="Van-der-Pol-Oszillator" class="mw-redirect" title="Van-der-Pol-Oszillator">Van-der-Pol-Oszillator</a> als Spezialfall für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=b=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a=b=0}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a9c78ffbb4fc115e8e2a630de5f90fa625effc69.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.587ex; height:2.176ex;" alt="{\displaystyle a=b=0}" loading="lazy"></span> enthält.

</p>

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li>R. FitzHugh: <i>Mathematical models of threshold phenomena in the nerve membrane.</i> Bull. Math. Biophysics, Band 17, 1955, S. 257–278</li>

<li>R. FitzHugh: <i>Impulses and physiological states in theoretical models of nerve membrane</i>. Biophysical J., Band 1, 1961, S. 445–466</li>

<li>R. FitzHugh: <i>Mathematical models of excitation and propagation in nerve,</i> in: H.P. Schwan, Hrsg., Biological Engineering, McGraw-Hill Book Co., N.Y. 1969, S. 1–85 (Kapitel 1)</li>

<li>J. Nagumo, S. Arimoto, S. Yoshizawa: <i>An active pulse transmission line simulating nerve axon</i>. Proc IRE., Band 50, 1962, S. 2061–2070.</li>

<li>Eugene Izhikevich, Richard Fitzhugh: FitzHugh-Nagumo model, <a rel="nofollow" class="external text" href="http://www.scholarpedia.org/article/FitzHugh-Nagumo_Model">Artikel</a>. In: <i><a href="Scholarpedia" title="Scholarpedia">Scholarpedia</a>.</i> (englisch, inkl. Literaturangaben), 2006</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20230105005558/http://www.scholarpedia.org/article/FitzHugh-Nagumo_Model">Artikel</a>. In: <i><a href="Scholarpedia" title="Scholarpedia">Scholarpedia</a>.</i> (englisch, inkl. Literaturangaben) von Eugene Izhikevich, Richard Fitzhugh 2006, Archivlink abgerufen am 11. Oktober 2025.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><span class="cite"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20220403142507/http://www.scholarpedia.org/article/User:Richard_FitzHugh"><i>User:Richard FitzHugh - Scholarpedia.</i></a> Archiviert vom <style data-mw-deduplicate="TemplateStyles:r250917974">

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</style><span class="dewiki-iconexternal"><a class="external text" href="https://redirecter.toolforge.org/?url=http%3A%2F%2Fwww.scholarpedia.org%2Farticle%2FUser%3ARichard_FitzHugh">Original</a></span> am <span style="white-space:nowrap;">3. April 2022</span><span>;</span><span class="Abrufdatum"> abgerufen am 10. Oktober 2025</span> (englisch).</span><span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AFitzHugh-Nagumo-Modell&rft.title=User%3ARichard+FitzHugh+-+Scholarpedia&rft.description=User%3ARichard+FitzHugh+-+Scholarpedia&rft.identifier=https%3A%2F%2Fweb.archive.org%2Fweb%2F20220403142507%2Fhttp%3A%2F%2Fwww.scholarpedia.org%2Farticle%2FUser%3ARichard_FitzHugh&rft.source=http://www.scholarpedia.org/article/User:Richard_FitzHugh&rft.language=en"> </span></span>

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