Making realistic neural compartments

The soma compartment that was simulated in the tutorial1.g script corresponds to the "generic neural compartment" diagram but without the variable ionic conductances G_k that we will add later. As it is a single isolated compartment, we didn't make use of the axial resistance R_a. The diagram reveals that the current I_inject flows through R_m to create a potential difference that is in series with E_m. The simulation results show that initially, V_m will equal E_m, and the steady state will be reached after a time given roughly by the time constant for charging the membrane capacitance, R_mC_m. With the values used, the time constant was 20.

A lot of the simplicity of the script stems from the fact that the numbers used in the simulation worked well with the default values of the graph axis scales and the default integration step size used by GENESIS to integrate the equation for the compartment V_m. In order to make a realistic soma compartment that we can then link to dendrite compartments and populate with ion channels, we will need to pick appropriate values for the passive cell parameters R_m, R_a, C_m, and the membrane resting potential E_m.

So far, we haven't said much about the units used to express the quantities R_m, R_a, C_m, V_m, etc. that appear in the neural compartment diagram and the differential equation for V_m.

Physicists and engineers like to use SI (MKS) units of ohms, farads, volts, and meters for describing resistance, capacitance, voltage, and length. Neurophysiologists are more likely to prefer kilohms, microfarads, millivolts, and either centimeters or micrometers. One can use any consistent set of units with GENESIS, but it is most common to use SI units.

The problem with using any of these units for resistance and capacitance is that R_m, C_m, and R_a will depend on the dimensions of the section of dendrite that is represented by the neural compartment. In order to specify parameters that are independent of the cell dimensions, specific units are used. For a cylindrical compartment, the membrane resistance is inversely proportional to the area of the cylinder, so we define a specific membrane resistance R_M, which has units of ohms·m². The membrane capacitance is proportional to the area, so it is expressed in terms of a specific membrane capacitance C_M, with units of farads/m². Compartments are connected to each other through their axial resistances R_a. The axial resistance of a cylindrical compartment is proportional to its length and inversely proportional to its cross-sectional area. Therefore, we define the specific axial resistance R_A to have units of ohms/m.

For a piece of dendrite or a compartment of length l and diameter d we then have

Note the membrane time constant R_m·C_m is also equal to R_M·C_M, so that it is independent of the dimensions of the membrane.

WARNING: Many treatments of the passive properties of neural tissue use the symbols R_m, R_a, and C_m for the specific resistances and capacitance, instead of this notation with R_M, R_A, and C_M. Also, many textbooks and journal papers define the resistance and capacitance in terms of that for a unit length of cable having a specified diameter, where

R_m = r_m/l, C_m = c_ml, R_a = r_al.

Although this notation is convenient and widely used, it obscures the fact that r_m and r_a depend on the dendrite diameter. In your reading, you should be aware of the units that are being used.

You can read more about passive properties of dendrites in the Digression on Cable Theory from the Introduction to Computational Neuroscience lectures.

Our goal is to build a cylindrical soma compartment that has the same physiological properties as those of the squid giant axon studied by Hodgkin and Huxley, and simulated in the GENESIS "squid" tutorial in genesis/Scripts/squid. So, we will use these values (in SI units) for the compartment parameters. However, we will make our soma smaller, with both the length and diameter equal to 30 micrometers.

We will also need to choose an appropriate time step for the numerical solution of the equation for V_m. With the values of R and CM that we will use (R_M = 0.33333 and C_M = 0.01), the membrane time constant will be 0.003333 seconds. We would then expect our integration time step to be a small fraction of this. In practice, it turns out that 50 microseconds (0.00005 sec) will be a good value.

You can (and should, at some point) read the section in the GENESIS Reference Manual on Clocks for further suggestions on choosing a time step. The documentation for the commands setclock and useclock gives the details of setting the time step.

Chapter 13 of the BoG leads the reader through the process of developing the script tutorial2.g. If you like, you can run this this script from the genesis/Scripts/tutorials directory. You should now examine tutorial2.g.

There are several features of the GENESIS script language that are introduced here:

• The use of a C-style multiline comment, extending from the second through fourth line.

• Declaration of variables to be used. In this script only floating point variables (float) are used, but "int" and "str" are also allowed. The declaration and use of variables is explained in the Reference Manual section on Variables.

• The line "setclock 0 {dt}" sets the global simulation clock (clock 0) to the specified time step, dt. Note the use of curly brackets around the variable name dt. Usually, the value of a variable or an expression must be evaluated by enclosing it in curly brackets in order to distinguish between the value represented by the character string (0.00005) and the actual string of characters "dt". This is particularly true when the expression is to be evaluated as an argument of a GENESIS command or script language function.

• The use of function declarations for makecompartment and make_Vmgraph, and their use in the "Main Script" section. GENESIS script language functions are described in the Reference Manual section on Functions.

• The use of a backslash to continue a line in the long setfield command in the makecompartment function.

• Setting the graph scales, and the use of "^" to denote the element last referenced (in this example, /data/voltage).

• The use of the "-time" option for the step command with the RUN button, to give the amount of time to run the simulation, rather than the number of steps. (See the documentation for step.)