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| The '''Morris–Lecar model''' is a [[biological neuron model]] developed by Catherine Morris and [[Harold Lecar]] to reproduce the variety of [[Neural oscillation|oscillatory]] behavior in relation to [[Calcium in biology|Ca<sup>++</sup>]] and [[Potassium in biology|K<sup>+</sup>]] conductance in the [[giant barnacle]] muscle fiber.<ref name = ml81>{{Citation| last = Morris| first = Catherine| last2 = Lecar| first2 = Harold| title = Voltage Oscillations in the barnacle giant muscle fiber| journal = Biophys J.| volume = 35| issue = 1| pages = 193–213|date=July 1981| url = http://jaguar.biologie.hu-berlin.de/downloads/Fachkurs/SS2010/Morris_Lecar_1981| doi = 10.1016/S0006-3495(81)84782-0| pmid=7260316| pmc=1327511}}</ref> Morris-Lecar neurons exhibit both class I and class II [[neuron]] excitability. | | The burgeoning world of mobile phones is an fascinating platform for technological development exactly where possibilities are limitless! A new Android mobile app has been developed which makes reading newspaper electronically quite effortless. Every single particular person these days owns a mobile phone. You can even go to a Television quiz show about music and use this game in your preparation course of action. That is why some men and women can't be in a position to catch up with the latest updates about the finest and newest goods of technology. Not only is it sleek and approachable, the LG InTouch Max GW620 also boasts a 5. megapixel digital camera comprehensive with LED flash. The best cheats for FarmVille on Facebook are the legal guides, ideas and secret tactics you can learn from other additional seasoned and probably much more devious players.<br><br> |
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| ==Experimental method==
| | The notion is special and attracts the focus of users immediately. They play mobile games day and night, willing to indulge themselves into the games. Lately, devices like the iPhone and Android Industry smartphones changed the mobile planet with their application platforms. In extreme cases, EVE On the net will close down the accounts of people who are repeat offenders at this. Your Best Bet To Monetize Mobile Website traffic Features of the Galaxy S: Mobile multiplayer on the web role playing games is a variation of this notion that has been creatively designed by a exceptional company for the Chinese audience.<br><br>A reporter at the scene of an advent can now write a story, upload images to it and send it off, all with out needing to go back to their workplace or residence to do it. Develop a Strong Plan Let's see what attributes in Android four.four KitKat are generating the entire planet crazy [http://cleversplitter.com/finding-painless-solutions-associated-farmheroes-hile/ farmheroes hile] about it. Let us have a brief look at the prevailing Mobile operating systems. This permits mobile companies to release handsets powered by android as opposed to the iOS that is exclusive for the iPhone. Their goods on gaming ought to be innovative enable to hold imagination. You can play Scrabble with all your pals on Facebook by means of the Facebook Connect function.<br><br>Individuals are discovering the joys of playing these games and are downloading the latest and the greatest in mobile phone games with a lot of enthusiasm. As a matter of fact, games being played on mobile phones are no longer confined to a single static screen. This LG mobile phone has an 8MP camera made with Schneider-Kreuznach optics and has all the capabilities of a typical camera. Class 10 GPRS and class ten EDGE retain the Acer Stream connected to cellular networks permitting telephone calls and messages to be carried out, while Web access relies primarily on 3G coverage. Samsung Epic 4G Price Many don't comprehend why this is so beneficial. Some of the ideal players, who know or have invented a fantastic quantity of cheats, post this information in their blogs and internet websites for free or in exchange for little favors such as back-up or support in the game. |
| The Morris-Lecar experiments relied on the [[current clamp]] method established by [[Richard Keynes|Keynes]] ''et al.'' (1973).<ref>{{Citation | last = Keynes | first = RD | author-link = [[Richard Keynes]] | last2 = Rojas | first2 = E | last3 = Taylor | first3 = RE | last4 = Vergara | first4 = J | title = Calcium and potassium systems of a giant barnacle muscle fibre under membrane potential control | journal = The Journal of Physiology (London) | volume = 229 | pages = 409–455 |date=March 1973 | url = http://www.pubmedcentral.gov/articlerender.fcgi?artid=1350315 | id = | pmid=4724831}}</ref> | |
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| {{Quotation|Large specimens of the [[barnacle]] ''Balanus nubilus'' (Pacific Bio-Marine Laboratories Inc., Venice, Calif.) were used. The barnacle was sawed into lateral halves, and the depressor scutorum rostralis muscles were carefully exposed. Individual fibers were dissected, the incision starting from the tendon. The other end of the muscle was cut close to its attachment on the shell and ligatured. Isolated fibers were either used immediately or kept for up to 30 min in standard artificial seawater (ASW; see below) before use. Experiments were carried out at room temperature of 22 [[Celsius|C]].<ref name = ml81 /> }}
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| ==The principal assumptions underlying the Morris-Lecar model include==
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| #Equations apply to a spatially iso-potential patch of membrane. There are two persistent (non-inactivating) voltage-gated currents with oppositively biased reversal potentials. The depolarizing current is carried by Na+ or Ca2+ ions (or both), depending on the system to be modeled, and the hyperpolarizing current is carried by K+.
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| #Activation gates follow changes in membrane potential sufficiently rapidly that the activating conductance can instantaneously relax to its steady-state value at any voltage.
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| #The dynamics of the recovery variable can be approximated by a first-order linear differential equation for the probability of channel opening.
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| <ref name = scholar>This assumption is never exactly true, since channel proteins are composed of subunits, which must act in concert, to reach the open state. Despite missing delays in the onset of recovery, the model appears to be adequate for phase-plane considerations for many excitable systems.{{Citation| last = Lecar| first = Harold| title = Morris-Lecar model| journal = [[Scholarpedia]]| volume = 2| issue = 10| pages = 1333| year = 2007| doi = 10.4249/scholarpedia.1333}}</ref> | |
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| ==Physiological description==
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| The Morris–Lecar model is a two-dimensional system of [[Nonlinear system|nonlinear]] [[differential equations]]. It is considered a simplified model compared to the four-dimensional [[Hodgkin-Huxley model]].
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| Qualitatively, this system of equations describes the complex relationship between membrane potential and the activation of ion channels within the membrane: the potential depends on the activity of the ion channels, and the activity of the ion channels depends on the voltage. As bifurcation parameters are altered, different classes of neuron behavior are exhibited. {{math|''τ''<sub>''N''</sub>}} is associated with the relative time scales of the firing dynamics, which varies broadly from cell to cell and exhibits significant temperature dependency.<ref name = scholar />
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| Quantitatively:
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| :<math>
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| \begin{align}
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| C \frac{dV}{dt} & ~=~ I - g_\mathrm{L} (V-V_\mathrm{L}) - g_\mathrm{Ca} M_\mathrm{ss} (V-V_\mathrm{Ca}) - g_\mathrm{K} N (V-V_\mathrm{K}) \\
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| \frac{dN}{dt} & ~=~ \frac{N-N_\mathrm{ss}}{\tau_{N}}
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| \end{align}
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| </math>
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| where
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| <!-- :<math> M_{ss} = (1 + \tanh[(V-V_1)/V_2])/2 </math> :<math> N_{ss} = (1 + \tanh[(V-V_3)/V_4])/2 </math> :<math> \tau_N = 1/[\phi \cosh[(V-V_3)/(2V_4)]</math> :{{math|''M''<sub>ss</sub> {{=}} (1 + tanh[(''V'' - ''V''<sub>1</sub>) / ''V''<sub>2</sub>]) / 2}} :{{math|''N''<sub>ss</sub> {{=}} (1 + tanh[(''V'' - ''V''<sub>3</sub>) / ''V''<sub>4</sub>]) / 2}} :{{math|''τ''<sub>''N''</sub> {{=}} (''φ''·cosh[(''V'' - ''V''<sub>3</sub>) / 2''V''<sub>4</sub>])<sup>-1</sup>}} -->
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| :<math>
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| \begin{align}
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| M_\mathrm{ss} & ~=~ \tfrac{1}{2} \cdot (1 + \tanh [\tfrac{V-V_1}{V_2} ]) \\
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| N_\mathrm{ss} & ~=~ \tfrac{1}{2} \cdot (1 + \tanh [\tfrac{V-V_3}{V_4} ]) \\
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| \tau_N & ~=~ 1 / ( \phi \cosh [\tfrac{V-V_3}{2V_4} ] )
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| \end{align}
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| </math>
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| Note that the {{math|''M''<sub>ss</sub>}} and {{math|''N''<sub>ss</sub>}} equations may also be expressed as {{math|''M''<sub>ss</sub> {{=}} (1 + Exp[-2(''V'' - ''V''<sub>1</sub>) / ''V''<sub>2</sub>])<sup>-1</sup>}} and {{math|''N''<sub>ss</sub> {{=}} (1 + Exp[-2(''V'' - ''V''<sub>3</sub>) / ''V''<sub>4</sub>])<sup>-1</sup>}}, however most authors prefer the form using the hyperbolic functions.
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| ===Variables===
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| * {{math|''V''}} : membrane potential
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| * {{math|''N''}} : recovery variable: the probability that the K+ channel is conducting
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| ===Parameters and constants===
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| * {{math|''I''}} : applied current
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| * {{math|''C''}} : membrane capacitance
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| * {{math|''g''<sub>L</sub>}}, {{math|''g''<sub>Ca</sub>}}, {{math|''g''<sub>K</sub>}} : leak, Ca<sup>++</sup>, and K<sup>+</sup> conductances through membranes channel
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| * {{math|''V''<sub>L</sub>}}, {{math|''V''<sub>Ca</sub>}}, {{math|''V''<sub>K</sub>}} : equilibrium potential of relevant ion channels
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| * {{math|''V''<sub>1</sub>}}, {{math|''V''<sub>2</sub>}}, {{math|''V''<sub>3</sub>}}, {{math|''V''<sub>4</sub>}}: tuning parameters for steady state and time constant
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| ==Bifurcations==
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| {{main|Bifurcation theory}}
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| Bifurcation in the Morris–Lecar model have been analyzed with the applied current {{math|''I''}}, as the main bifurcation parameter and {{math|''φ''}}, {{math|''g''<sub>Ca</sub>}}, {{math|''V''<sub>3</sub>}}, {{math|''V''<sub>4</sub>}} as secondary parameters for [[phase plane]] analysis.<ref>{{Citation | last = Tsumoto | first = Kunichika | last2 = Kitajimab | first2 = Hiroyuki | last3 = Yoshinagac | first3 = Tetsuya | last4 = Aiharad | first4 = Kazuyuki | last5 = Kawakamif | first5 = Hiroshi | title = Bifurcations in Morris–Lecar neuron model | journal = Neurocomputing | volume = 69 | issue = 4–6 | pages = 293–316 |date=January 2006 | language = English & Japanese | url = http://pegasus.medsci.tokushima-u.ac.jp/~tsumoto/work/nolta2002_4181.pdf | doi = 10.1016/j.neucom.2005.03.006 }}</ref>
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| {{multiple image
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| <!-- Essential parameters -->
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| | align = center
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| | direction = horizontal
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| | width = 360
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| <!-- Extra parameters -->
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| | header = Possible Bifurcations
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| | header_align = center
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| | header_background =
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| | footer = [[Electrophysiology#Current_clamp|Current clamp]] simulations of the Morris-Lecar model. The injected current for the SNIC bifurcation and the homoclinic bifurcation is varied between 30 nA and 50 nA, while the current for the Hopf bifurcation is varied between 80nA and 100nA
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| | footer_align =
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| | footer_background =
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| | background color =
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| |image1=MorrisLecarHopfTimeDomain.gif
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| |width1=360
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| |caption1=<div class="center" style="width:auto; margin-left:auto; margin-right:auto;"> [[Hopf Bifurcation]] </div>
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| |alt1=Upper Darboux sum example
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| |image2=MorrisLecarSNICTimeDomain.gif
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| |width2=360
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| |caption2=<div class="center" style="width:auto; margin-left:auto; margin-right:auto;"> [[saddle-node bifurcation|SNIC bifurcation]] </div>
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| |alt2=Lower Darboux sum example
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| |image3=MorrisLecarHomoclincTimeDomain.gif
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| |width3=360
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| |caption3=<div class="center" style="width:auto; margin-left:auto; margin-right:auto;">[[Homoclinic bifurcation]] </div>
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| |alt3=Lower Darboux sum example
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| }}
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| ==See also==
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| * [[Computational neuroscience]]
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| * [[Biological neuron model]]
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| * [[Hodgkin-Huxley model]]
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| * [[FitzHugh-Nagumo model]]
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| * [[Neural oscillations]]
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| * [[Nonlinear dynamics]]
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| * [[Quantitative models of the action potential]]
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| ==References==
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| {{Reflist}}
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| ==External links==
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| *[http://math.jccc.net:8180/webMathematica/MSP/mmartin/morlecar.msp A Morris-Lecar simulator]
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| *[http://www.scholarpedia.org/article/Morris-Lecar_model Scholarpedia: Morris-Lecar Model]
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| *[http://www.ohri.ca/profiles/morris_summ.asp: Catherine Morris - Research Profile]
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| {{Neuroscience}}
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| {{DEFAULTSORT:Morris-Lecar model}}
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| [[Category:Neuroscience]]
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A reporter at the scene of an advent can now write a story, upload images to it and send it off, all with out needing to go back to their workplace or residence to do it. Develop a Strong Plan Let's see what attributes in Android four.four KitKat are generating the entire planet crazy farmheroes hile about it. Let us have a brief look at the prevailing Mobile operating systems. This permits mobile companies to release handsets powered by android as opposed to the iOS that is exclusive for the iPhone. Their goods on gaming ought to be innovative enable to hold imagination. You can play Scrabble with all your pals on Facebook by means of the Facebook Connect function.
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