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Adrian Quintana, 27 Apr 2015 17:02

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Goldman-Hodgkin-Katz and Nernst formalisms in NeuroML2
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Sample [LEMS & NeuroML 2](https://github.com/OpenSourceBrain/ghk-nernst/tree/master/NeuroML2) files implementing a single compartment model, containing *HH* type Na<sup>+</sup>/K<sup>+</sup> channels, a Ca<sup>++</sup> channel and pool. The Calcium current through the channel can be calculated either through the [GHK flux equation](http://en.wikipedia.org/wiki/GHK_flux_equation), or using the [Nernst Equation](http://en.wikipedia.org/wiki/Nernst_equation) to approximate its reversal potential.
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The same mechanisms are implemented in *NEURON* as a comparison. Notice that *NEURON* has built in support for calculating reversal potentials via the Nernst equation, which can be compared to our implementation written from scratch by running [Native NEURON](https://github.com/OpenSourceBrain/ghk-nernst/blob/master/NEURON/nernstnat_na_k_ca.py) , [from scratch](https://github.com/OpenSourceBrain/ghk-nernst/blob/master/NEURON/nernst_na_k_ca.py) ; see also [native Nernst modfile](https://github.com/OpenSourceBrain/ghk-nernst/blob/master/NEURON/cachan_nernst_native.mod) and [from scratch modfile](https://github.com/OpenSourceBrain/ghk-nernst/blob/master/NEURON/cachan_nernst.mod)
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Running tests
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The following tests can be run to compare the usage of GHK/Nernst from *NEURON\_/\_LEMS*:
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     git clone git@github.com:OpenSourceBrain/ghk-nernst.git
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     cd ghk-nernst/NEURON/
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     nrnivmodl
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     ./ghk_na_k_ca.py
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     ./nernst_na_k_ca.py
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    </pre>
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    <pre>
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     cd ../NeuroML2
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     jnml LEMS_ghk_na_k_ca.xml
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     jnml LEMS_nernst_na_k_ca.xml
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    </pre>
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    <pre>
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     cd ../
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     python compare.py
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neuroConstruct
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Recent versions of neuroConstruct support GHK, generating NeuroML2 and NEURON code accordingly. In order to use the GHK flux equation instead
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of Ohm’s law for a particular mechanism, set an extra parameter GHK\_permeability (in m/s) using the **Edit Density Mechs** dialog.
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![](nc_ghk.png)
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Definitions
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From [Wikipedia](http://en.wikipedia.org/wiki/GHK_flux_equation)
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![](http://mathurl.com/ngps7px.png)
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From Steuber et al DCN model:
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     A = exp(-23.20764929 * v / T) = exp((-z * F * (0.001) * v) / (R * T))                                        
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     ical = perm * m*m * h * (4.47814e6 * v / T) * ((cali/1000) - (calo/1000) * A) / (1 - A)                      
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     (z^2 * F^2 * (0.001) * v) / (R * T)
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From [GENESIS](http://www.genesis-sim.org/GENESIS/gum-tutorials/beeman/Hyperdoc/Manual-26.html#ss26.30)
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              valency * F                                                                                          
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     K     =  - - - - - - -                                                                                        
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              R * (T + 273)                                                                                        
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                               Cin * exp(K * Vm) - Cout                                                           
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     Ik    =  -p * F * K * Vm  - - - - - - - - - - - -                                                            
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                                   exp(K * Vm) - 1                                                                
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     (it is  missing one valency factor in comparison to the others, check genesis source)