Can Two Opposite Running Action Potential Cross Each Other without Annihilation in One Axon

Combining the lecture notes on the interchannel and channel electric potential leads to a description of the propagation of the action potential $V$. Specifically, substituting in $I_{text{ion}}=I_{text{ext}}$ of equation (6) of cable theory by equation (1) of Hodgkin-Huxley model gives the full propagation equation. A linearization of the Hodgkin-Huxley model leads to a system of linear partial differential equations. I will write out the full formulation and the linear approximation later. That allows complete superposition of action potential. The original non-linear system of partial differential equations under a set of functions/parameterizations at least close to the linear version should permit two opposing wave to cross each other though with interference. So it is not true that all parameterizations of the Hodgkin-Huxley model prohibit two-wave-crossing.

It would of course be intriguing to investigate the existence of a particular form/parameterization where the model does extinguish two opposing waves with its initial condition being two pulses close to a dirac delta function at each end of the axon.

Here is a paper M. Argentina, P. Coullet and V. Krinsky, Head-on Collisions of Waves in an Excitable FitzHugh-Nagumo System: a Transition from Wave Annihilation to Classical Wave Behavior a special case of the Hodgkin-Huxley model. Just as I expected, the waves do not always annihilate each other, but rather may cross, annihilate or even coalesce or even coalesce then re-emit, all depending on the particular values of the parameters of the model. Another paper Oleg Aslanidi, O. A. Mornev, Can Colliding Nerve Pulses Be Reflected? on the numerical simulation of a full-blown Hodgkin-Huxley model reporting crossing of colliding waves. This paper Follman et al, Dynamics of Signal Propagation and Collision in Axons reports under their particular set of parameterization of Hodgkin-Huxley model, two colliding waves annihilate. This paper by Alfredo Gonzalez-Perez et al demonstrates Penetration of Action Potentials During Collision in the Median and Lateral Giant Axons of Invertebrates based on the soliton theory of nervous signal propation as described in On soliton propagation in biomembranes and nerves by Thomas Heimburg and Andrew D. Jackson.

The wikipedia page on autowave has some interesting information including references for propagation and collision of nonlinear waves in excitable media.