Goldman-Hodgkin-Katz and Nernst formalisms in NeuroML2

======================================================

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.

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)

Running tests

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The following tests can be run to compare the usage of GHK/Nernst from *NEURON/LEMS*:

     git clone git@github.com:OpenSourceBrain/ghk-nernst.git

     cd ghk-nernst/NEURON/

     nrnivmodl

     ./ghk_na_k_ca.py

     ./nernst_na_k_ca.py

     cd ../NeuroML2

     jnml LEMS_ghk_na_k_ca.xml

     jnml LEMS_nernst_na_k_ca.xml

     cd ../

     python compare.py

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

of Ohm’s law for a particular mechanism, set an extra parameter GHK\_permeability (in m/s) using the **Edit Density Mechs** dialog.

![](nc_ghk.png)

Definitions

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From [Wikipedia](http://en.wikipedia.org/wiki/GHK_flux_equation)

![](http://mathurl.com/ngps7px.png)

From Steuber et al DCN model:

     A = exp(-23.20764929 * v / T) = exp((-z * F * (0.001) * v) / (R * T))                                        

     ical = perm * m*m * h * (4.47814e6 * v / T) * ((cali/1000) - (calo/1000) * A) / (1 - A)                      

     (z^2 * F^2 * (0.001) * v) / (R * T)

From [GENESIS](http://www.genesis-sim.org/GENESIS/gum-tutorials/beeman/Hyperdoc/Manual-26.html#ss26.30)

              valency * F                                                                                          

     K     =  - - - - - - -                                                                                        

              R * (T + 273)                                                                                        

                               Cin * exp(K * Vm) - Cout                                                           

     Ik    =  -p * F * K * Vm  - - - - - - - - - - - -                                                            

                                   exp(K * Vm) - 1                                                                

     (it is  missing one valency factor in comparison to the others, check genesis source)