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

-------------

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

    </pre>

    <pre>

     cd ../NeuroML2

     jnml LEMS_ghk_na_k_ca.xml

     jnml LEMS_nernst_na_k_ca.xml

    </pre>

    <pre>

     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

-----------

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

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

-   From Steuber et al DCN model:

     {{latex(A = exp(~~23.20764929 \* v / T) = exp \* v) / ))}}

     ical = perm \* m\*m \* h \* \*~~ (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)

    \<pre\>

     valency \* F

     K = - - - - - - -

     R \* (T + 273)

Cin \* exp(K \* Vm) - Cout

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

 exp(K \* Vm) - 1

</pre>

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