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

Can two opposite travelling action potential cross each other annihilation in an axon?

My answer would be affirmative. If the propagation mechanism is linear as described by https://en.wikipedia.org/wiki/Cable_theory or even https://en.wikipedia.org/wiki/Telegrapher%27s_equations with approximately constant coefficients, the waves are then linearly additive and each signal should propagate unperturbed. The refractory period on each end of the axon can not possibly prevent the other end from being excited at the same time since the signal has not arrived yet.

This question is inspired by the question Can a single axon propagate multiple simultaneous action potentials?

• When the waves collide they should extinguish each other since the membrane each encounters is depolarized (because of the other's propagation). Commented Sep 17, 2016 at 6:57
• @mgkrebbs: How does depolarization lead to extinction of both waves? The equations contradict this claim since the electric (action) potential is additive.
– Hans
Commented Sep 17, 2016 at 9:05
• @Hans the waves would extinguish each other because both will lead to depolarization of the membrane. The AP cannot proceed further. AFAIK, the amplitude of the AP would be the same for a certain type of neuron. If they are different then the dynamics may be different. Using a model you may be able to predict the behaviour but I am not sure if anyone has checked this experimentally. Commented Sep 18, 2016 at 8:17
• @WYSIWYG: Your first sentences are stated without reason, just like the statement from mgkrebbs. It seems you are making the same analogy for AP to domino as Superbest does in his answer judging from your concern with the amplitude of the opposing AP's. However, if you switch to the wave analogy then the two should cross each other without interference. It sounds very much like speculation with different analogies leading to completely different conclusions with one no more convincing than the other. But please do derive the proposition if you think otherwise and are certain of its validity.
– Hans
Commented Sep 19, 2016 at 3:24
• @Hans No it is not like a usual wave. You have to understand the physiology behind that. AP leads to membrane depolarization. An already depolarized section of the membrane cannot conduct AP. Of course it is not as simple as two waves clashing and annihilating each other. That's why I said that a simulation would be useful (there may be non-intuitive behaviour as these are non-linear systems). I would do the derivations if I had the time. I am a bit occupied and can tell you only what I know right now. Commented Sep 19, 2016 at 4:38

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.

• This answer is theoretical, and it doesn't convince me that the model is a perfect representation of reality. It would be greatly improved if you linked to experimental results showing this phenomenon. Commented Sep 27, 2016 at 10:41
• @MarchHo: The theoretical answer shows the model allows for wave crossing and annihilation, and that the depolarization is NOT sufficient to prevent wave crossing. No model is a perfect representation of reality. This model and numerical simulation precisely refutes the hypothesis "recently polarized membrane" would prevent wave crossing --- as insufficient.
– Hans
Commented Sep 27, 2016 at 17:35
• @MarchHo Although I wouldn't personally accept the answer, it's definitely valuable to know that some models permit the phenomenon. It's an answer in case experimental data does not exist. Commented Oct 31, 2016 at 4:12