Source
Frederic Folz, Lukas Wettmann, Giovanna Morigi and Karsten Kruse (2018), “The sound of an axon’s growth”, Physical Review E 99, 050401 (2019), LIT-034, read in full with the SM in NOTE-067.
What was actually shown
One stage cannot read frequency (after Eq. 2). For a response f_κ(c) = cⁿ/(κⁿ + cⁿ) and any β > 0, (1/T)∫₀^T f(c(t))dt = (β/T)∫₀^(T/β) f(c(βt))dt. A periodic input that only changes period gives the same mean response. In the Heaviside limit the mean is set by the fraction of the period above threshold (Eq. 3).
Two stages can (Eq. 4, Fig. 2). After one low-pass, saturating stage the waveform of the intermediate signal changes with period, and a second threshold stage turns that change into a change in mean: ⟨c_R⟩ increases with T (shown for n = 4 and n = 50).
The loop (Eqs. 5–9). Kinesin carries a signal I from soma to growth cone; dynein carries O back, suppressing I’s release, with delay τ = L/v. The loop oscillates only above a minimum length, from J̄ sin(γτ√(J̄² − 1)) = 1 (Eq. 7), and its frequency falls with length, ω ≈ √3 v/L for long axons (Eq. 9), matching full simulations (Fig. 3b).
Length control (Eq. 10, Fig. 4). L̇ = v_g c_I − v_s c_R: I drives protrusion and the readout R drives contraction. From L = 0 the length settles with oscillation under 1% of the mean. In the model, lowering motor influx J₀ below a critical value lengthens the axon, as does lowering dynein coupling, which matches the experimental finding that fewer motors give longer axons. Gaussian delay noise (σ = 0.1τ₀) and lossy transport leave the behaviour intact (SM Figs. S2–S3).
What could have come out otherwise. The cascade could have failed to make the mean depend on period, and the coupled system could have oscillated or run away instead of settling. Neither happened for the parameter set used.
What this does not say
- That axons measure length this way. The paper proposes no experiment that would tell spectral readout from gradient mechanisms, and its claim that gradients cannot work beyond a few micrometres is uncited. The authors note that sub-µm length oscillations would likely be masked by fluctuations in a real axon.
- That the set point is robust. One parameter set; sweeps cover only J_R, n, J₀ and J_O. Equal kinesin and dynein speeds are assumed and membrane tension is neglected. The model is lumped, and the authors say a spatial description of contractility is “most likely” needed.
- That 125 µm means anything. Lengths are in units of L_min = vτ_min, and v is not given anywhere.
- That fewer motors always lengthen the axon. Only below a critical J₀; above it length rises with J₀, and the paper does not say which side real axons are on.
- That the steady state is a laser. The gain–loss balance and phase-locking are an analogy (p. 4); phase-locking is not computed.
- More than the stated scope of period independence. The proof covers signals that differ by a rescaling of time with the same waveform; the whole mechanism depends on the waveform changing.
Connections
- It builds on the delayed-feedback oscillator of Karamched and Bressloff (2015) and the motor-oscillation proposal of Rishal et al. (2012), neither held, and supplies the transduction they lacked.