A final word on compartmental modeling

In attempting to model the behavior of whole neurons, detailed microscopy and staining are required to produce a full three-dimensional picture of the cell. This picture is then broken down into a series of cylinders with varying diameters and lengths. These are then used to create a mathematical and computational description of the cell. In order to write equations for the compartments, we must determine the axial and the transmembrane resistances as well as the membrane capacitance. For a given cylindrical compartment, the membrane resistance is related to the area of the compartment as is the capacitance. Given a specific resistivity in Ohm-cm², R_M the total resistance is just this value divided by the area, R_m=R_M/A whare A is the area. Similarly, if there are active channels with conductances in millisiemens per square centimeter, the total conductance is the specific conductance times the area. (Note that the resistance is divided by the area and the conductance multiplied - more area means more conductance and less resistance.) The capacitance for the compartment is also proportional to the area.

 

Figure 1: Two different cylinders coupled by the average axial resistance

However, when we connect two compartments together, how do we define the axial resistance between them if they have different lengths and/or diameters? Obviously, if the two cylinders are identical, then you can use either of the axial resistances as they are both the same. If they are not the same, then there are several different possibilities:

• If both are cylinders, average the two resistances and use the average.
• If one is a cylinder and the other is a sphere, use the cylindrical value
• Order the compartments and use the resistance corresponding to the lower or higher number in the ordering

We will always use the first 2 algorithms. GENESIS lets you choose which algorithm to use. I am not sure how NEURON resolves this.

EXAMPLE

Suppose that the two compartments have dimensions (length X diameter) $100\ \mu m \times 5\ \mu m$ and $200\ \mu m \times 2\ \mu m$ respectively. Then

$$A_1 = 500 \pi = 1571 \mu m^2$$

$$A_2 = 400 \pi = 1257 \mu m^2$$

$$Ri_1 = R_I 100/(\pi 2.5^2) = 5.1 R_I$$

$$Ri_2 = R_I 200/(\pi 1^2) = 63.66 R_I$$

R_i = 34.4 R_I

Thus, the two equations are

$$C_M A_1 \frac{dV_1}{dt} = (V_M - V_1)/(R_M/A_1) + (V_2-V_1)/R_i$$

$$C_M A_2 \frac{dV_2}{dt} = (V_M - V_2)/(R_M/A_1) + (V_1-V_2)/R_i$$

Dividing these by the area, we get

$$C_M \frac{dV_1}{dt} = (V_M - V_1)/R_M + (V_2-V_1)/(A_1 R_i)$$

$$C_M \frac{dV_2}{dt} = (V_M - V_2)/R_M + (V_1-V_2)/(A_2 R_i)$$

In other words the two equations for the voltages look identical except for the coupling strength between them. (If the specific resistances and potentials and channel densities were different for the two cells, the noncoupling coefficients would also be different.) The key point is that the effect of a big compartment on a little compartment is asymmetric. In this example, since compartment 1 is larger than compartment 2, the influence of compartment 2 on compartment 1 is less than that of compartment 1 on compartment 2 since the former coupling is divided by a bigger number. Often in modeling a 2 compartment system with compartment 1 bigger than compartment 2, it is convenient to write the couplings as:

$$\hbox{coupling}_{2\to1} = G (V_2-V_1)$$

$$\hbox{coupling}_{1\to2} = G (V_2-V_1)/(1-p)$$

where G is a fixed conductance and p=1-A₂/A₁ is an asymmetry parameter. p=0 if the compartments have the same area. As p gets closer to 1, the ratio of the small to the big goes to zero. Big dendrites have strong effects on little somas. This is why when the soma spikes, there is little propagation back up the dendrite unless the dendrites themselves have active channels.

 

Figure 2: A 4 compartment model representing a pyramidal cell

Homework

Write the equations for a compartmental model of a neuron with 4 compartments that are arranged as follows with dimensions given in the figure (length and diameter in microns). Compartments 1 and 2 are apical dendrites with $R_M=10000\Omega/cm^2$, $C_M=1\mu F/cm^2$, $R_a=100 \Omega cm.$ Compartment 3 is a basal dendrite with the same properties and compartment 4 is the spherical soma that has a transient sodium conductance and delayed rectifier as well as a leak. Assume that it has a transient sodium channel with a density of 100 mS/cm² a persistent potassium channel with density of 80mS/cm², a leak conductance .1mS/cm² and V_Na=50mV,V_K=-100mV,V_l=-67mV. (Note that if one uses millisiemens and microfarads as the dimensions, then the times will be in milliseconds which is convenient.) (Also note that a leak conductance of 0.1mS/cm² corresponds to a membrane resistance of $10000\Omega cm^2.$) You do not have to simulate this (yet); I am mainly interested in seeing that you understand how to put the compartments together with the dynamics. The resting potential is -67mV.