non linear systems - Why do spiral waves annihilate each other when 2 wavefronts collide?

I was reading about Fitzhugh-Nagumo model. And in a 2D space the simulations a Reaction-Diffusion process associated with FitzHugh system look like this. But intuitively I could not satisfy myself about the annihilation of wave when 2 wavefronts meet each other. Why do they annihilate each other?

One example I heard in practical world is controlling forest fire using opposing wavefront of a controlled fire.

Answer

Let’s consider this incarnation of the FitzHugh–Nagumo model:

$$\begin{matrix} \dot{V} & = & V-V^3/3 - W + I \\ \dot{W} & = & 0.08(V+0.7 - 0.8W) \end{matrix}$$

This describes an excitable system. In the following I will explain what this means, how this relates to the above equation and use a small patch of forest as an example.

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  A small stimulus (via $I$) can trigger a huge response (excitation). In the equation, the extent of excitation is represented by the variable $V$. The mechanism of excitation is somewhat difficult to see from the equations and comes from the non linear structure $V-V^3/3$ which causes the nullclines to be aligned such that a small excitation from the fixed point requires the system to perform a large excursion in phase space to return to the fixed point. In the forest example, the stimulus corresponds to throwing a match into our patch of forest.

  
  


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  Excitation triggers an inhibitory process that counteracts the excitation. The unit becomes refractory. In the equations this inhibitory process is represented by the variable $W$. Furthermore, you can see that a high $V$ (excitation) begets a high $W$ (inhibition), as $\dot{W}$ increases with $V$. Furthermore a high $W$ (inhibition) begets a lower $V$ (excitation), as $\dot{V}$ decreases with $W$. In the forest example the inhibitory process corresponds to ashes, which do not burn.

  
  


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  The inhibitory process self-terminates after a while. In the equations, this is implemented via $\dot{W}$ decreasing with $W$, i.e., without further influence, the variable $W$ drifts towards $0$. In the forest example, this corresponds to the forest regrowing from the ashes.

  
  

In an excitable medium such as a 2D reaction–diffusion system, each point in space is represented by an excitable system and these systems are coupled via a next-neighbour coupling. This coupling is usually diffusive, i.e., $I \propto ΔV$, but other couplings have a similar effect. In our forest example, an excitable medium corresponds to a whole forest, consisting of many patches and the coupling represents sparks of fire going from one burning patch to its neighbours.

In an excitable medium, excitations spread in a wave-like manner, the same way forest fires do: Whenever a single patch is excited, it is likely to also excite its neighbours, which then excite their neighbours and so on – the forest fire spreads in a wave-like manner. However, each excited unit becomes refractory shortly afterwards and thus a front of excitation is followed by a front of inhibition.

Now, if two wavefronts collide, consider the single front of excitation formed by this collision. As both wavefronts are followed by fronts of inhibition, it is now enclosed by inhibitory regions. Thus the excitation cannot spread anywhere and thus once this single front becomes inhibitory, the excitation dies and the wavefronts have annihilated.

Fighting fire with fire works the same way: Each forest-fire front leaves behind nothing but ashes. If this front runs into another front, it is enclosed by ashes and can thus not spread any further.