Analysis of the Hodgkin Huxley equations

Aims: to explore how the Hodgkin-Huxley simulations respond to maintained current and to pulses of current.

Getting the simulation worksheet: Make a copy of the worksheet in a clean directory (e.g. h:\hh). Using Netscape, you can download it from http://biolpc9.york.ac.uk/hh/hh_pos.xls

If you are on the Biology teaching network, with a windows 3.1 machine you should open the file f:\public\apps\ex5\hh_prac\hh_pos.xls and save a copy in the clean directory. If you have a machine wanting to run windows95 you will need to open it from j:\public\apps\ex5\hh_prac\hh_pos.xls.

The following features of the file are of interest before you start.:

  • 1) Most of the columns are named, so that references are not in the simple A:1, B:2... style but include the names of the columns

    2) Many parameters (e.g. gKmax, dt) are supplied as fixed named variables. You will have to change these, and the way to do it is to select Insert | Name w Define and then type an equals sign followed by the new value in the box marked Refers to.

    3) Note that the sheet automatically recalculates the answers when you change any value, so that the easy way to save your data is to copy the picture and, in Word for Windows, choose Edit | paste special and then Picture. If you just choose Edit | Paste you will get a linked picture which will always try to track the changes in your file. Otherwise you need to keep lots of saved copies of your work.

    4) Note that (unless you do something about it) the charts will be plotted with the Na current going upwards and K current going downwards; this is the opposite of the figures in the books and lectures.

  • The sheet implements Simpson's rule for the solution of the equations. This assumes that at every time point you know V and dv/dt and uses a constant small time interval (called dt in the worksheet) to calculate the change in voltage that would be expected to occur in the short time. If dv/dt changes, then you get an error which may accumulate. Thus this is not the most accurate method to solve the equations - it depends on a good choice of the value of the short time interval.

    Handy hints:

    You may like to add a column time in column A. Type 0 on A2 and =A2+dt in square A3. Fill Down. Then you can check the duration (in msec) of the action potential.

    To plot a graph of one column: click on the column header box (e.g. for a voltage plot the box which says K) and then use the chart wizard icon to plot a line graph.

    To add a column to an existing chart: Select the column header, click on Edit | copy and then double-click on the chart and choose Edit | paste ( or Edit | Paste Special )

    To extend the sheet of formulae downwards: Use Ctrl+End to get to the last occupied square. Now choose Edit | Goto and type the new limit (e.g. A2000) in the space marked Reference. Press shift as you click on OK. Now use Edit | Fill w down to propagate the formulae across the selected area!

    Questions to examine:

    1) What happens when dt is increased by a factor of (a) 2 (b) 5 (c) 10? Don't forget to reset dt to 0.02.

    2) What happens when the stimulus current (iset) is decreased and increased? and when iset is negative (e.g. -80, -50)?

    3) Plot a graph of the Na and K currents. When is their peak? What is their sum? How does it relate to the maximum voltage? What happens if you reduce gNamax substantially?

    4) How do gNa and gK relate to INa and IK ?

    5) Extend the stimulus to long periods of time What happens if you try a second stimulus after the first one? Why cannot you evoke a second action potential?

    6) What happens in your long duration spreadsheet if you have a long tonic stimulus? Can you get repetitive firing?

    Reminder: The Hodgkin-Huxley Equations are as follows:

    Total current flow:

     Ohms Law applied:
    Na and K channels opening, closing (and inactivating)
    The rates at which the channels open and close are voltage dependent