Cable Theory Parameters and Units

The membrane capacitance, intracellular resistance, and membrane resistance of the axon or dendrite of a neuron can be expressed in more than one way using different units. What follows is a guide for working with these quantities.

Membrane capacitance

• Membrane capacitance per unit length, $c_m$
  • Units: $\mu\text{F}/\text{cm}$
  • Depends on axon/dendrite diameter

• Membrane capacitance per unit area, $C_m$
  • Units: $\mu\text{F}/\text{cm}^2$
  • Independent of axon/dendrite diameter
  • This quantity is given directly in the cable simulations.

• The quantities $c_m$ and $C_m$ are related through the circumference of the axon/dendrite: $$c_m = C_m \pi d$$

Intracellular resistance

• Intracellular resistance per unit length, $r_i$
  • Units: $\text{k}\Omega/\text{cm}$
  • Depends on axon/dendrite diameter

• Intracellular resistivity, $R_i$
  • Units: $\text{k}\Omega\cdot\text{cm}$ (Note that in the simulation, the units given are: $\Omega\cdot\text{cm}$; you will need to keep this in mind during your calculations.)
    • Since the total intracellular resistance of a segment of an axon/dendrite decreases with diameter and increases with length, this quantity can be divided by cross-sectional area and multiplied by length to get the total intracelluar resistance between two points on an axon/dendrite. This means the quantity must have units $\text{k}\Omega\cdot\text{cm}$.
  • Independent of axon/dendrite diameter
  • This quantity is given directly in the cable simulations.

• The quantities $r_i$ and $R_i$ are related through the cross-sectional area of the axon/dendrite: $$r_i = \frac{R_i}{\pi d^2 / 4}$$

Membrane resistance

• Membrane resistance of a unit length, $r_m$
  • Units: $\text{k}\Omega\cdot\text{cm}$
    • The membrane resistance of a unit length of the axon/dendrite is defined as the reciprocal of the membrane conductance per unit length, giving it units $1/(\text{mS}/\text{cm}) = \text{cm}/\text{mS} = \text{k}\Omega\cdot\text{cm}$.
  • Depends on axon/dendrite diameter

• Membrane resistance of a unit area, $R_m$
  • Units: $\text{k}\Omega\cdot\text{cm}^2$
    • The membrane resistance of a unit area of the membrane is defined as the reciprocal of the membrane conductance per unit area, $g_m$, giving it units $1/(\text{mS}/\text{cm}^2) = \text{cm}^2/\text{mS} = \text{k}\Omega\cdot\text{cm}^2$.
  • Independent of axon/dendrite diameter
  • This quantity is given indirectly in the cable simulations by the membrane conductances.

• The quantities $r_m$ and $R_m$ are related through the circumference of the axon/dendrite: $$r_m = \frac{R_m}{\pi d}$$

Input resistance

• Input resistance, $r_\text{input}$
  • Units: $\text{k}\Omega$
  • Depends on axon/dendrite diameter
  • This quantity is related to both the membrane resistance and the internal resistance: $$r_\text{input} = \sqrt{\frac{r_m r_i}{4}} = \sqrt{\frac{(R_m / \pi d) (R_i / (\pi d^2 / 4))}{4}} = \frac{1}{\pi} \sqrt{\frac{R_m R_i}{d^3}} = \frac{1}{\pi} \sqrt{\frac{R_i}{g_m d^3}}$$

Time and length constants

• Time constant, $\tau$
  • Units: $\text{ms}$
  • Independent of axon/dendrite diameter
  • This quantity is related to both the membrane resistance and capacitance: $$\tau = r_m c_m = (R_m / \pi d) (C_m \pi d) = R_m C_m$$

• Length constant, $\lambda$
  • Units: $\text{cm}$
  • Depends on axon/dendrite diameter
  • This quantity is related to both the membrane resistance and the internal resistance: $$\lambda = \sqrt{\frac{r_m}{r_i}} = \sqrt{\frac{R_m / \pi d}{R_i / (\pi d^2 / 4)}} = \sqrt{\frac{d R_m}{4 R_i}} = \sqrt{\frac{d}{4 g_m R_i}}$$