If at node N1, an AP is generated and then the passive spread initiates another AP at node N2, is there 2 APs existing at the same time?
In fact, the second AP is ever initiated before the first has ended. It seems it contradicts the theory?

AFAIK the theory doesn't predict that the first AP will end before the second is initiated. In fact, according to cable theory, when the first has ended then there is no depolarization to propagate passively.

At any one time there are action potentials at more than one location along the axon, but each action potential is at a different point in its time course. At three successive locations with an AP, one AP has just passed its peak, another AP is at its peak, and the third AP has not reached its peak.

The above description is only approximate because APs at different points in the axon do not have identical time courses. In the internode, the AP changes shape, but retains enough shape that an onset, peak and end can be reasonably defined.

At any one time there are action potentials at more than one location along the axon, but each action potential is at a different point in its time course. At three successive locations with an AP, one AP has just passed its peak, another AP is at its peak, and the third AP has not reached its peak.

The above description is only approximate because APs at different points in the axon do not have identical time courses. In the internode, the AP changes shape, but retains enough shape that an onset, peak and end can be reasonably defined.

DaleSpam,
You're funny. An equation without parameters and conditions... Just funny!
Is it decay like this since you insist on the light speed propagation?
or decay + delay?
it cant' be delay only.

I had assumed that you would be familiar with the cable equation since it is the basic equation of the cable theory that we have been discussing, and I wrote it using standard notation. So I didn't think it was necessary to explain the terms any more than I would explain the terms of "f=ma" in the physics section. I apologize for my unwarranted assumption and I will gladly describe it in detail now.

The cable equation is:
[tex]\lambda ^2 \frac{\partial ^2v}{\partial x^2}=-i
r+v+\tau \frac{\partial v}{\partial t}[/tex]
where v is the transmembrane voltage, x is the distance along the axon or dendrite, i is the current through the membrane, r is the transmembrane resistance, and t is time. [tex]\lambda[/tex] is known as the space constant and [tex]\tau[/tex] is known as the time constant. They are free parameters that depend on the electrical properties of the specific neuron with typical values from tens to hundreds of micrometers for the space constant and from tens to hundreds of microseconds for the time constant.

I have a fundamental question for you: if you do not even recognize the cable equation in standard notation then on what basis are you objecting to cable theory?

Does a cable is able to expand or shrink?
Does a cable exhibit heat changes in both directions?
Does a cable swells?
Does a cable uses ions?

The cable theory is unable to integrate this fate.
BTW, the simulator produces curves. These curves are produced upon models. They must fit facts.
Once again, it is easy to test your model but you obstinately refuse to give any value.
If you're true then the curves will show the predictions of the cable theory and I'll shut my mouth...

I assume by these questions you are talking about industrial cables such as the underseas telegraph cables that originally inspired the cable equation rather than biological cables where the cable equation is more commonly used now. Even with industrial cables the answer is yes to all of these.

Industrial cables expand (swell) and shrink, particularly in response to changes in temperature (in both directions). In addition industrial cables are designed to a specific size, the cable equation is generally important in determining the optimal size. During the design process a wide variation of size can be considered (much wider than would occur thermally). The cable equation can even handle changes in size and temperature even that are not uniform simply by including the appropriate terms (e.g. tau becomes a function of x). That covers your first three points.

Your last point, about using ions, is interesting. I assume that you think there is something fundamentally different between electrical currents where the charge carriers are free electrons in a metal and ones where the charge carriers are free ions in an electrolyte. Interestingly, the original cable equations were for a metal cable immersed in sea-water, so both electronic and ionic currents were considered. The purely ionic currents in neurons are actually simpler than the mix of ionic and electronic currents in industrial cables.

There is a wealth of experimental evidence going back several decades supporting neuronal cable theory. The models do fit the facts as you well know from your studies.

Really?
A cable that exudes water when a message travels?
A cable where ions are going in and out of the cable at the same speed but having different atomic size?
A cable that is hot when the message arrives and makes cold when he goes?
A cable where the diameter is enlarged when the message runs?

Does the cable theory describe all these points?

And we have demonstrated that your cable theory miss, at least, 2 points: https://www.physicsforums.com/showthread.php?t=258168 Where is the component implied by the missing capacitor?
Where is the internal axonal resistance?

That is why I brought that one!
https://www.physicsforums.com/showthread.php?t=260444 [Broken]

Sodium gates are at nodes.
Potassium gates are at paranodes (under myelin).
Myelin is impermeable to sodium and potassium.
The two kind of gates are functional in sequence.