2.1 Voltage decay in Axons with Passive Membranes

The simplest axon case to consider is that in which its membrane electrical properties consist of a capacitance (1ufd/cm^2) in parallel with a resistance. Such a membrane is called "passive" to distinguish it from those with active channels. Because the steady-state current throught such membranes is proportional to the transmembrane voltage, they are also referred to as "linear".

The current and voltage distributions in such axons are described by "cable equations" because they are anagolous to those which describe underseas electrical cables. An injection of current at any point in a passive axon results in a local potential change. As this voltage signal spreads along a passive cable (axon with blocked ionic channels), it changes shape and is rapidly attenuated. You can examine how the diameter and membrane resistivity affect this rapid decay of signal and you can also change the membrane capacitance: this is just "thought" experiment because the capacitance of lipid membrane is rather constant at 1 ufd/cm^2 and rather insensitive to lipid composition. From these experiments you cna come to appreciate the neecessity for the presence of voltage-sensitive ionic channels to generate action potentials whose shape and amplitude are maintained with high integrity along an of uniform diameter as it propagates shown in the next section (2.2 Propagation in a uniform squid axon).

A classic exposition of this subject may be found in 

Electric current flow in excitable cells. by
J. J. B. Jack, D. Noble, and R. W. Tsien
Oxford Press. 1975, 1985
Below is shown the time course of voltage rise and fall at various points along a passive cable following a current step injected at the "0" end. Note the decay in amplitude and change in the time course of voltage at various points along this passive cable.

If the location of the current injecting electrode is moved from the left end of the the axon (0.01) to other locations, changes occur in: In the figure below is a composite display of the voltage profiles for two locations of the current injecting electrode. [This composite made by pressing right mouse button in the graph window and selecting "keep lines". The number of plots/mS was reduced to prevent too much overlap].

Run the Passive Cable Experiments

to see these interesting patterns. I suggest that you also: If you reduce the membrane conductane all the way to zero (completely "blocked"), you will see a very interesting pattern.