Cable Theory - Deriving The Cable Equation

Deriving The Cable Equation

r_m and c_m, as introduced above, are measured per membrane-length unit (per meter (m)). Thus r_m is measured in ohm-meters (Ω·m) and c_m in farads per meter (F/m). This is in contrast to R_m (in Ω·m²) and C_m (in F/m²), which represent the specific resistance and capacitance respectively of one unit area of membrane (in m2). Thus, if the radius, a, of the axon is known, then its circumference, 2πa, its r_m, and its c_m values can be calculated as follows:

(1)

(2)

These relationships make sense intuitively, because the greater the circumference of the axon, the greater the area for charge to escape through its membrane, and therefore the lower the membrane resistance (we divide R_m by 2πa); and the more membrane available to store charge (we multiply C_m by 2πa). Similarly, the specific resistance, R_l, of the axoplasm allows us to calculate the longitudinal intracellular resistance per unit length, r_l, (in Ω·m−1) by the equation:

(3)

Again, a sensible equation, because the greater the cross sectional area of the axon, πa², the greater the number of paths for the current to flow through its axoplasm, and the lower the axoplasmic resistance.

To better understand how the cable equation is derived, let's first simplify our theoretical neuron even further and pretend it has a perfectly sealed membrane (r_m=∞) with no loss of current to the outside, and no capacitance (c_m = 0). A current injected into the fiber at position x = 0 would move along the inside of the fiber unchanged. Moving away from the point of injection and by using Ohm's law (V = IR) we can calculate the voltage change as:

(4)

If we let Δx go towards zero and have infinitely small increments of x we can write (4) as:

(5)

or

(6)

Bringing r_m back into the picture is like making holes in a garden hose. The more holes, the faster the water will escape from the hose, and the less water will travel all the way from the beginning of the hose to the end. Similarly, in an axon, some of the current traveling longitudinally through the axoplasm will escape through the membrane.

If i_m is the current escaping through the membrane per length unit, m, then the total current escaping along y units must be y·i_m. Thus the change of current in the axoplasm, Δi_l, at distance, Δx, from position x=0 can be written as:

(7)

or, using continuous ,infinitesimally small increments:

(8)

can be expressed with yet another formula, by including the capacitance. The capacitance will cause a flow of charge (current) towards the membrane on the side of the cytoplasm. This current is usually referred to as displacement current (here denoted .) The flow will only take place as long as the membrane's storage capacity has not been reached. can then be expressed as:

(9)

where is the membrane's capacitance and is the change in voltage over time. The current that passes the membrane can be expressed as:

(10)

and because the following equation for can be derived if no additional current is added from an electrode:

(11)

where represents the change per unit length of the longitudinal current.

By combining equations (6) and (11) we get a first version of a cable equation:

(12)

which is a second-order partial differential equation (PDE.)

By a simple rearrangement of equation (12) (see later) it is possible to make two important terms appear, namely the length constant (sometimes referred to as the space constant) denoted and the time constant denoted . The following sections focus on these terms.