Passive properties of cells
Modelling the cell as a single compartment is limited because neurons have extended structures including dendrites and axons. To create more realistic model, we will model the neuron as a cable. The cable contains many cylindrical compartments coupled to each other with resistors.
The diameter of the cylinder is d, its length is called h so the cross-section has an area as an equation below.
The resistance between the compartments, the resistance along a cable, is proportional to its length and inversely proportional to the cross-sectional area.
where ri is the intracellular resistivity.
Moreover, the membrane resistance of a single cylinder is
where the area is the the outside area of the cylinder.
The capacitance of the cylinder is
from the specific capacitance is defined as
The function of the voltage of the position in the cable
Suppose the voltage at the position x will have the voltage V(x,t) then its neighbouring compartments will be V(x+h, t) and V(x-h, t) which is shown in the following diagram.
Using the Kirchhoff’s law, we will get the equation below.
Finally, we will get the equation below.
We then are be able to take the limit of small h, which the cable is splitted into very small elements.
From the derivative equation which is defined as
The second derivative equation is also defined as
So, we will get the passive cable equation.
Steady state solution to cable equation
Consider the steady state of the cable equation, dV/dt will be zero. Suppose a current injected at x = 0 is represent as a delta function, it is zero except at zero which it will be infinite, and its total area is one.