# Why Is Action Potential Propagation Not Described by Telegrapher's But Cable Equation?

In modelling the propagation of action potential in an axon, why is the partial differential equation the cable equation rather than the telegrapher's equation? The difference between the two is that the former does not have inductance while the latter has. Does an axon not have inductance whether it is myelinated or unmyelinated? This is related to my previous question. There the only answer I obtained attributes this choice not to experimental evidence but the deficiency on the part of biologists in mathematical sophistication in dealing with the complication of electromagnetism arising from including inductance.

## 1 Answer

I am the individual who answered your earlier question. The answer remains much the same. The Cable Equation is inappropriate. It was developed originally as the Heat Equation of Thomson, While Thomson disowned its application to the axon as a cable, Herman continued to promote the solution within the Biology community. The problem is the Cable Equation requires as an initial condition that the stimulation of the cable continues until the signal reaches the terminus of the axon by conduction (diffusion).

When exploring the myelinated axon, the Telegrapher's equation must be used. The formal name for this equation is the General Wave Equation, GWE, of Maxwell. It involves an initial condition that is much more lenient. The stimulus must only be applied to the axon for long enough to ensure the desired full waveform is represented. Thus the GWE involves propagation and not conduction (or diffusion). See section 7.4 of Chapter 7 of "Processes in Biological Hearing," PBH, on my website.

Cole & Baker, "Longitudinal Impedance of the Squid Giant Axon", Journal of General Physiology, (1941) vol 24(6) pp 771-783 present their initial measurements of inductance within the large axon of a small squid. This axon generated "swim waveforms" and not "Action Potentials" as currently defined. The inductance is much smaller in a non-myelinated axon than in a myelinated axon. It remains calculable in either case.

I have reviewed the two papers by Lieberstein & Mahrous you cited for my benefit. The authors are apparently two very good mathematicians without much knowledge of biophysics or the biophysics literature. They quote a diffusion velocity from Hodgkin & Huxley (1952) that is several orders of magnitude slower than the phase velocity along an axon that is very well documented by Cole and more recently by Smith et al. I have uploaded a more complete analysis in Section 9.1.1.4.3 of the above citation to chapter 9 of my work, 9SignalTransmission.pdf.

These two papers should not be relied upon until reviewing Section 9.1.1.4.3