Follow up on how how we get the structures:
If you have ever played around with the Barkley model or reaction-diffusion simulations before, you are likely used to carefully drawing a line (a "ribbon") or a circle to cleanly ignite a smooth, predictable wave.
Seeing structured, twisting filaments emerge out of pure, messy static feels like magic, but it is actually a beautifully violent process of mathematical survival and self-organization.
Here is exactly how the math forces random noise to spontaneously generate organized scroll filaments:
1. The Immediate "Die-Off" (Survival of the Fittest Cells)
When you initialize with pure random noise, every single cell is assigned a random value for the activator (u) and the inhibitor (v). At time step zero, the grid is a chaotic soup:
• Some cells have high activator (u) but also high inhibitor (v). Because the inhibitor blocks the activator, these cells immediately fizzle out and die.
• Some cells have low activator (u) and just stay at rest.
However, purely by statistical chance, there will be tiny, localized clusters of cells where the activator (u) is high, but the inhibitor (v) happens to be low. These rare clusters are the only ones that survive the first few frames. They instantly fire, releasing chemical waves that begin to spread outward into the resting cells around them.
2. Spontaneous Wave-Splitting (The Clog Effect)
Because the initial background is a random mess, a newly born wave cannot expand smoothly.
Imagine a tiny wave front trying to move forward. As it travels, it immediately bumps into patches of cells that still have high leftover inhibitor values (v) from the random initialization.
• The parts of the wave front that hit these "inhibitor blocks" are stopped dead in their tracks.
• The parts of the wave front that hit resting cells keep moving forward.
This brutally tears the wave front apart, shredding a single expanding bubble into thousands of jagged, broken wave fragments.
3. The Natural Curling of Free Edges
In the physics of excitable media, a wave front hates having a broken, open edge.
When a wave is ripped apart, the open edge finds itself right next to a region of completely unexcited cells. Because of diffusion, the wave front naturally spills sideways into that unexcited space, trying to heal itself. This sideways leaking causes the broken edge to curl inward.
• In 2D, this curling edge creates a spiral.
• In 3D, that torn wave edge is a 2D sheet, and its entire broken boundary rolls up like a scroll—creating a scroll wave.
4. Condensing into the Filament Core
Within a fraction of a second, the millions of random chaotic pixels have either completely died off, or they have rolled up into these tiny localized scroll waves.
The center axis of each newly rolled scroll wave is where the filament is born. Because the initial noise was completely disorganized, these filaments don't form as nice, straight lines; they form as a tangled, messy "jungle" of microscopic loops and fragments.
As the simulation progresses, these tiny fragments collide. The rules of the Barkley model dictate that when filaments collide, they don't just pass through each other—they cross-connect, merge, and organize into the macroscopic, macro-scale swirling tubes you see by second 15 of the video.
Essentially, the random noise acts as a brutal filter. It destroys 99% of the chaos, leaves behind thousands of tiny torn waves, and the math forces every single one of those tears to curl into a filament.