The fastest sort of signal transmitted in a neuron is the membrane potential. A patch of membrane acts like a resistor in parallel with a capacitor, between the inside and the outside of the cell:
In a real cell, the membrane potential is not the same everywhere, but varies from region to region on the cell. However, if we break the cell into hypothetical compartments -- little sphere- or cylinder-shaped sections -- finely enough, then we can imagine that within a compartment, the membrane potential is uniform. (Thus the term "isopotential compartment" which you may occasionally encounter.)
Now -- here's the neat bit -- each compartment is attached to its neighbors, as if joined by little resistors, allowing current to flow between them. Thus a potential change in one will spread to the others, according to the values of the resistors and the membrane capacitance (which, again, sets the charging rate).
This is the basics of what you need to simulate a "passive cable" -- so called because, so far, we've described an unresponsive neuron branch, which might as well be a transatlantic telegraph cable. Real neurons, of course, are a bit more complicated, as they add active membrane channels. These are like adding a variable resistor and battery in parallel with the membrane resistor and capacitor diagrammed above. The resistor is variable in that its value changes with the membrane potential: when V increases, some channels increase their resistance, and others decrease. The "battery" comes from the fact that each channel passes some specific ion or set of ions, each of which has its own electrical potential (due, you remember, to the concentration difference between the inside and outside of the cell). In effect, the channel engages or disengages the battery, according to the membrane potential (and sometimes other factors, such as the concentration of calcium, presense of neurotransmitter, etc.).
As one might imagine, this can lead to some interesting behavior: V affects active channels, which in turn affect V. This is in fact how action potentials propagate: an increasing V opens channels which increase V even further (these channels then close of their own accord). Without this regenerative system, a voltage change would never make it from your spine to your foot -- not even close.
Thus, compartmental models are closely analogous to real biological neurons. By appropriately specifying the geometry and connectivity of the compartments, we can recreate the morphology of a real neuron. If we then add the right passive and active channels in the right proportions, we should -- in principle -- be able to reproduce the behavior of the cell.
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