Fig. 2: From axon geometry to along-axon diffusivity.

a Relative cross-sectional variations α for representative SBEM-segmented myelinated axons (sham and TBI), the synthetic axon, and their power spectral densities Γ_η(q). The finite plateau Γ₀ = Γ_η(q)∣_q→0 > 0 signifies the short-range disorder (finite correlation length) in the cross-sections. b Monte Carlo simulated D(t) ensemble-averaged over N_axon = 50 randomly synthesized, N_axon = 43 randomly sampled SBEM myelinated sham and N_axon = 57 TBI axons, with colors corresponding to (a). c D(t) for all three cases scales asymptotically linearly with \(1/\sqrt{t}\), validating the functional form of Eq. (1). d Coarse-graining over the increasing diffusion length ℓ(t) makes an axon appear increasingly more uniform, suppressing shape fluctuations Γ_η(q) with q ≳ 1/ℓ(t), such that only the q → 0 plateau Γ₀ “survives” for long t and governs the diffusive dynamics (1). To illustrate the effect, an axon segment is Gaussian-filtered with the standard deviation \(\ell ({t}_{i})/\sqrt{2}\) for ℓ(t_i) = 0, 5, 10, 20 μm. The coarse-graining of the axon segment along its length is color-coded for increasing diffusion times t_i according to the color bar. e The exact tortuosity limit (2) is validated for both synthetic and SBEM individual axons. Axons with larger cross-sectional variations var α have higher tortuosity. The center represents the mean, and horizontal error bars reflect errors in estimating D_∞ from Eq. (1) (see Methods). f The predicted amplitude c_D of the t-dependent contribution to D(t), Eq. (3), validated against its MC counterpart estimated from Eq. (1), for individual synthetic and TBI axons (colors as in (a)). The coefficient c_D is larger for axons with greater cross-sectional variations. The filled circles and error bars reflect means and errors in estimating c_D from MC-simulated D(t) (horizontal) and estimating the plateau Γ₀ from Γ_η(q) (vertical), as shown by dashed lines in the power spectral densities of (a) (see Methods). The number of samples is indicated in (b). Source data are provided as a Source Data file.