This is a diametral cross-section through a nerve guidance conduit (NGC): instead of a 3D cylinder, the canvas shows the exact rectangular slice you would see if you cut the conduit lengthwise through its central axis. That cut turns the cylindrical lumen wall into two flat, parallel walls at ±R (a geometrically exact reduction, not an approximation), and the constrained problem becomes genuinely two-dimensional: a longitudinal coordinate z (proximal → distal stump) and a lateral coordinate y (offset from the conduit's central axis).
Each growth cone follows the same biased persistent random walk as the full 3D model, just embedded directly in this 2D plane:
n(t+dt) = normalize( p·n(t) + (1-p)·ξ + β·ĝ_chem + γ·ĝ_guide )
x(t+dt) = x(t) + v·n(t+dt)·dt
- n = (nz, ny) — 2D unit heading; p — persistence; ξ — random unit vector on the circle (filopodial sampling noise).
- ĝchem — points from the cone toward the distal NGF source at (z=L, y=0): it pulls forward and re-centers the cone toward the conduit axis.
- ĝguide = (1,0) — contact guidance from the aligned internal microchannels (drawn as horizontal fibers): a purely forward pull with no centering, so it straightens paths without correcting drift.
- Cones that cross y = ±R reflect exactly off the flat wall (no approximation needed — the cross-section really is a rectangle); cones that fail to reach z = L within a time budget stall and re-sprout from the proximal stump.
Because the two cues act differently — chemotaxis centers and advances, guidance only advances — you can watch a completely guidance-dominated conduit ferry cones straight down the middle even without any NGF gradient, while a chemotaxis-only conduit still tends to succeed (the centering pull acts as its own weak guidance), matching the tradeoff described for the full 3D model.