A diffusible guidance cue forms a concentration field around its source. For a point source this falls off roughly exponentially with distance d:
C(d) = C₀ · e^(−d / λ)
∇C(d) = −(C(d) / λ) · d̂
The growth cone's filopodia sample this field on either side of its leading edge. Chemotactic turning obeys a Weber–Fechner (fractional) law: the response depends on the relative, not absolute, concentration difference sensed across the cone's width w:
Weber fraction: ΔC / C ≈ (∇C · n̂) · w / C
Turning rate: dθ/dt = k · tanh( g · ΔC/C ) + ξ(t)
where k is a base turning-rate constant, g is receptor gain (sensitivity), tanh() models receptor saturation at steep gradients, and ξ(t) is stochastic noise from actin/microtubule dynamics inside the growth cone. Netrin-1 binding DCC receptors pulls the cone up-gradient (attraction); Semaphorin-3A binding Neuropilin/Plexin pushes it down-gradient (repulsion) — this simulator sums both signed contributions before applying the turning law, exactly as a real growth cone integrates competing cues (Tessier-Lavigne & Goodman, 1996). This 2D version renders the same field top-down and additionally exposes the gradient's spatial decay length λ directly as a slider.
- Netrin-1 / Semaphorin-3A sliders — set each source's peak concentration C₀.
- Gradient steepness — sets 1/λ: a steeper gradient gives a stronger, more localized cue that fades faster with distance from its source.
- Sensitivity (gain) — scales g; higher gain turns more sharply per unit gradient, but tanh() caps the response so it can't turn faster than the cytoskeleton allows.
- Cytoskeletal noise — adds random filopodial jitter ξ(t), the same stochasticity that makes real axon paths wander rather than draw a perfect line.
Reach the glowing attractant target and the growth cone has found its synaptic partner — exactly how real axons navigate to the correct destination during nervous-system wiring.