Exercise 1: Membrane potential and stochastic, voltage dependent ion channels

In this exercise you will investigate how the membrane potential of a cell is determined and how voltage dependent ion channels operate. The goal of the lab is to get you acquainted with

·    The laws of ion movement along concentration and potential gradients and across cell membranes, through leakage and voltage dependent ion channels.

·    The Goldmann-Hodgkin-Katz equation, which is used to calculate the resting potential of a cell, for single and multi-ion, conditions, as well as the currents through the cell membrane.

·    The current-voltage differential equation of a single membrane compartment (“cell”). This is the basic component of multi-compartment cell models.

·    How stochastic ion channels operate and how their voltage dependence may be modeled by the Hodgkin-Huxley equations.

·    How a population of such channels can produce a sodium (Na) spike.

1.      Tasks

You should implement most parts of this lab on your own. This includes the voltage and current versions of the Goldman- Hodgkin-Katz equation, a membrane compartment, and a stochastic Na channel. For the last part, which involves putting varying numbers of such channels on a tiny membrane compartment, however, you run a ready-made Java program.

At the examination of the lab, plots and questions marked with a “·” should be presented and answered respectively.

2.      Lab environment

The tools you use in this lab are a matter of your own preference. You can use Matlab or the programming language of your choice for implementation and graphic output. Gnuplot is an easy to use Unix plot program for those who don't use Matlab.

The last part will involve running the program Membcomp which can be found in the course catalog /info/biomod01/, together with a short description of the program. Directions for how to set up your environment to be able to run this program is given in section 6 below.

Already from the start you should create a separate directory in which you run the lab.

3. The Goldmann-Hodgkin-Katz equation

Start by implementing the GHK voltage equation  (JW 2.7.21) with three different ions, potassium (K), sodium (Na), and chloride (Cl). We assume a temperature of 293 oK. Use SI units in computations as much as possible! Values for the gas constant R and Faradays constant F are:

                    F = 96480 Coulomb/mol

                    R = 8.314 Joule/Kelvin/mol

Hint: Use arrays for the ion related variables, it saves writing code later.



P (m/s)

Cin (mM)

Cout (mM)


2.00 10-8




0.06 10-8




0.20 10-8




·    Calculate the membrane resting potential for the values in the table above of ion specific permeabilities (P) and intracellular (Cin) and extracellular (Cout) ion concentrations (in mol/l, “molar” designated by M).

·    What is the resting potential if only the K permeability is non-zero?

·    Interchange the Cin with Cout and calculate resting potential again. Do this for PNa = 0.03 10-5 and for PNa = 4 10-5, too.

Next, implement the GHK current equation (JW 2.7.17) for the same conditions as in the table above. Note that the currents carried by the individual ions should be calculated separately and then summed to get the total current.

·    What is Im at a Vm of –70mV and 0 mV?

·    What is the dimension of Im?

·    Calculate the I-V relationship of the membrane for the same conditions as above (use GHK-I and step V from –80 to 80 mV in steps of 5 mV). Draw the I(V) curve.

·    What is the slope conductance at –50 mV?

·    Interchange the Cin with Cout for all ions and draw the curve for this situation.

4.  A spherical membrane compartment

Now, a spherical membrane compartment will represent a cell or a small part of it. A typical diameter of a neuron is 100mm whereas a spine head might be only 1mm. We assume a membrane thickness of 100 Angstrom, and a specific capacitance (cm) of 1 mF/cm2 (0.01 F/m2). Implement the current-voltage differential equation for this compartment (JW eqn. 3.1.1, with Im derived from GHK-I). Note that the current should be positive inwards, which is opposite to how GHK-I was first derived.

Integrate this equation using the Euler forward method: Vm(t+1) = Vm(t) + Im/Cm Dt

Hint: Remember that the current now relates to a much smaller membrane area, and that the capacitance, too, is dependent on membrane area.

·    What is INa at Vm  = –50 mV?

A timestep of 0.1 ms will be fine. Set up to run a simulation of 0.05 s, i.e. 500 timesteps, starting at Vm (0) = -50 mV.

·    Plot the time course of Vm!

·    Estimate by inspection an approximate value of tm during these conditions.

Hint: V(t) has the form of e-t/t.

·    What is the total membrane conductance of the cell at –50 mV? What is its input resistance?

Now, during the simulation, change permeabilities P according to:





0.000 <= t < 0.015

0.20 10-8

0.06 10-8

0.015 <= t < 0.020

0.20 10-8

P1* = 0.40 10-8

0.020 <= t < 0.030

0.20 10-8

0.06 10-8

0.030 <= t < 0.040

2.00 10-8

0.06 10-8

0.040 <= t

0.20 10-8

0.06 10-8

* P1 relates to section 6 below.

·    Plot the time course of Vm under these conditions. Verify that your dynamic values approaches those calculated from GHK_V!

·    What, if any, difference do you see between the whole cell and the spine head with regard to e.g. INa and Vm (t)?

5.  Stochastic, voltage dependent ion channels

You should now implement a voltage dependent stochastic sodium channel with inactivation. We start with a single gating particle with two states, open and closed. Such a particle switches between open and closed as:




where a and b (dimension s-1) are voltage dependent rates and P(O®C) = bDt and P(C®O) = aDt (given small enough Dt).

Implement a function update() such that when called it updates the particle’s state according to the above. Use a timestep of 0.1 ms as above.

Now, assume a and b depend on transmembrane voltage Vm:

a(V) = -105(V+0.035)/(e-(V+0.035)/0.010 - 1), a(-0.035) = 103

b(V) = 4000e-(V+0.060)/0.018

·    Plot the curve for m¥ and tm for Vm = -80 to 80 with steps 10 mV.

·    Plot a curve of the state of such a particle for a period of  0.05 s at Vm = -80 to 80 with steps 20 mV.

Implement a Na-h-particle as:

a(V) = 70e-(V+30)/0.020

b(V) = 1000/(e-V/0.010 + 1)

·    Plot the curve for h¥ and th for Vm = -80 to 80 with steps 10 mV.

·    Plot a curve of the state of one such particle for a period of  0.05 s at Vm = -80 to 80 with steps 20 mV.

Compose a sodium channel from three Na-m and one Na-h particle (a m3h sodium channel).

·    Verify its voltage dependence by plotting a curve of the state of the channel for a period of  0.05 s at Vm = -80 to 80 with steps 20 mV.

6.  A compartment with added voltage dependent sodium channels

The program Membcomp.java implements a spine head sized membrane compartment with a number of stochastic voltage dependent sodium channels added. It runs 0.05 s with the same sodium permeability change as above. Several program parameters can be given on the command line and the program outputs to 'standard out' various state variables at each timestep. To save program output, redirect 'standard out' to a file as shown below. You should use this program to investigate some properties of this system.

Initialize your directory like this:

Ø       source /info/biomod01/labbar/java/init.tcsh

or init.bash if you use bash.

The program is used as follows:

java lab1.Membcomp [nNa] [P1] [radius] [> outfile]

where the three command line arguments are optional and correspond respectively  to: nNa = number of channels, P1 = Na leak permeability (m/s, see P1 in the table above), radius = compartment radius (mm).

The program output has the following format:

 time (ms) Vm (mV) Im (pA) INa (pA) PNa (m/s*1e18) PNaV (m/s*1e18)

where PNaV  is the permeability of the voltage dependent Na-channels and PNa the permeability of the other, non-voltage dependent, Na-channels.

Start with 100 sodium channels (default). Adjust the sodium leak permeability and number of channels so that you get a sodium spike during which the top potential reaches up to about 20 mV given the conditions above, the potential with nNa = 0 not exceeding -45 mV.

·    What is the threshold permeability?

·    Draw a set of plots of Vm over time for some different permeability increments, subthreshold as well as superthreshold.

·    Decrement the number of channels (nNa) by dividing by two down to some value around 5. Draw plots of Vm over time for each condition. Is the maximum of Vm linear or nonlinear with nNa?



Now you are done, thank you!