  Brownian motion of water molecules provides an essential length scale, the
diffusion length, commensurate with cell dimensions in biological tissues.
Measuring the diffusion coefficient as a function of diffusion time makes in
vivo diffusion MRI uniquely sensitive to the cellular features about three
orders of magnitude below imaging resolution. However, there is a longstanding
debate, regarding which contribution --- intra- or extra-cellular --- is more
relevant in the overall time-dependence of the diffusion metrics. Here we
resolve this debate in the human brain white matter. By varying not just the
diffusion time, but also the gradient pulse duration of a standard diffusion
pulse sequence, we identify a functional form of the measured time-dependent
diffusion coefficient transverse to white matter tracts in 5 healthy
volunteers. This specific functional form is shown to originate from the
extra-axonal space, and provides estimates of the fiber packing correlation
length for axons in a bundle. Our results offer a metric for the outer axonal
diameter, a promising candidate marker for demyelination in neurodegenerative
diseases. From the methodological perspective, our analysis demonstrates how
competing models, which describe different physics yet interpolate standard
measurements equally well, can be distinguished based on their prediction for
an independent "orthogonal" measurement.
