## 2 Materials and Methods  
### Theoretical Background  
It is known that the DW signal that is obtained using a PFG [13; 17] experiment is proportional to the Fourier transform (FT) of the motion propagator (MP) that describes the probability of spin motion. When the MP is Gaussian and Brownian, the PFG signal attenuation as a function of the b-values follows a mono-exponential decay. On the other hand, when the MP is non-Brownian, it is possible to define, in general, the average MP as the solution of the fractional diffusion equation (e.g., as introduced within the framework of the CTRW model [47]). This equation introduces two parameters, the fractional exponent of the time derivative, \(\alpha\), and the fractional exponent of the space derivative, \(\gamma\), which are the time and space derivative fractional orders. Specifically, \(\alpha\) is measured by varying the diffusion time \(\Delta\) in a PFG sequence [43; 48; 49; 53; 54; 55], and it has been recently shown to be a generalization of the Kurtosis representation [50]. Conversely, \(\gamma\) is measured by varying the gradient strength g in a PFG sequence [46; 48; 52; 56]. Of relevance for DW-fMRI applications, it has been reported both in vitro [44; 45] and in vivo in the human brain [40; 46] that \(\gamma\) maps show a useful image contrast to highlight tissue interfaces. The resulting image contrast of \(\gamma\) maps, deriving from water diffusion multi-compartmentalization, strictly reflects he local \(\Delta\chi\), and thus, it allows for improved discrimination of the interface between different compartments compared to T\({}_{2}\)* images. Leaving aside here any discussion about the real anomalous diffusion of extracellular water in the human brain, in the last 10 years, through various experiments conducted in vitro [52] and in vivo on the human brain [40; 46], we have highlighted that the DW signal representation of the anomalous superdiffusion may furnish a new useful image contrast in neuroimaging studies. We have recognized that, in a complex system such as the brain, the fractional diffusion representation related to the fractional exponent of the space derivative, \(\gamma\), can better describe water diffusion multi-compartmentalization, also thanks to its peculiar sensibility to magnetic field inhomogeneity. The magnetic field inhomogeneities may arise from the presence of heavy metal accumulation or the difference in local magnetic susceptibility between different adjacent compartments or tissues. In the human brain, extracellular water can diffuse with Brownian diffusion, non-Gaussian diffusion, or, in the context of anomalous diffusion, with subdiffusion. We do not find superdiffusion, which is quantified by the \(\gamma\) parameter. Superdiffusion is not real but mimicked by the nuclear spins of biological water, which, due to local inhomogeneities of the magnetic field and due to their indistinguishability, simulate superdiffusion dynamics. Due to the inhomogeneities, magnetization disappears in a voxel to \"appear\" in an adjacent (or even distant) one where there are magnetic field gradient conditions that are favorable to the refocusing of the magnetization components. For this reason, we say that \(\gamma\) quantifies pseudo-superdiffusion, and we use the fractional diffusion formalism to extract the \(\gamma\) parameter. Briefly, the fractional diffusion representation is based on the following relation [43; 47]:  
\[W(\mathbf{q},\Delta)\approx\exp\Bigl{[}-K_{2\gamma}|2\pi\mathbf{q}\mid^{2 \gamma}\Delta\Bigr{]} \tag{1}\]  
where is the FT of MP, \(K\) and \(K_{2\gamma}\) are generalized diffusion constants, whose units are (ms\({}^{-1}\)), \(q=\frac{1}{(2\pi)}\gamma g\delta\) is the wave vector, and \(0<\gamma<1\). To measure \(\gamma\), expression (1) has to be fitted to data obtained by PFG signal decay as a function of \(b=\gamma^{2}\delta^{2}g^{2}\Bigl{(}\Delta-\frac{\delta}{3}\Bigr{)}\), where \(\delta\) is the diffusion gradient duration, collected by changing the \(g\) strength at a constant value of \(\Delta\).