The purpose of this work is to identify the relevant microstructural features for the human brain gray matter. For that, we estimate the diffusivity and kurtosis time-dependence in 25 gray matter sub-regions of 10 healthy subjects, and compare the effects of the structural disorder and exchange (Karger model). The estimated power-law dynamical exponent θ≈1/2 is consistent with the structural disorder picture in a 1-dimensional micro-geometry of randomly positioned restrictions along neurites. In contrast, Karger model yields exchange time much shorter than values in previous studies, and below our shortest diffusion time.