## 4 Discussion  
DW-fMRI performed in this preliminary study shows that the \(\gamma\) parameter (i.e., the fractional order of the space derivative in the fractional diffusion representation) response to a brain activation is opposite to that provided by ADC at high b-values and matches well to the BOLD activation, being more pronounced than that observed using conventional ADC methods (Figure 2). However, to obtain the ADC results, we used b = 2000 s/mm\({}^{2}\) and b = 80 s/mm\({}^{2}\) instead of the optimized b-values to eliminate any T2-BOLD and perfusion residue (i.e., b = 1800 s/mm\({}^{2}\) and b = 1000 s/mm\({}^{2}\)) [30]. Consequently, the ADC fMRI result could be affected by residual effects of T\({}_{2}\)-BOLD, which attenuate the ADC signal decrease during activation. The increase in \(\gamma\) during activation, which is opposite to the ADC decrease, can be explained by considering \(\gamma\)'s dependence on diffusion multi-compartmentalization and local \(\Delta\chi\). At first glance, the similarity of the time course shown for \(\gamma\) and the BOLD response would suggest that the \(\Delta\chi\) dependence largely predominates, and therefore, \(\gamma\)'s response is due to the vascular contribution of capillaries and small veins. Although the changes may likely be vascular, they appear more localized than those of BOLD, probably due to minimal intravascular effects. However, it is worth noting that \(\gamma\) increases not only when \(\Delta\chi\) decreases, but also when the multi-compartmentalization decreases. In particular, the presumed cell swelling occurring during functional activation involves a decrease in the multiplicity of extracellular diffusion compartments. To better elucidate the biophysical mechanism underpinning the \(\gamma\) parameter and its relationship with the physiological phenomena occurring during activation, we performed a 2D simulation to obtain \(\gamma\)'s behavior as a function of cell packing in extravascular tissue, \(\phi\). In Figure 3, the graph of \(\gamma\) vs. \(\phi\) shows that when increasing the cell packing (i.e., decreasing the diffusion multi-compartmentation), \(\gamma\) increases. In parallel, \(\gamma\) increases when \(\Delta\chi\) decreases, a phenomenon that occurs during activation. Considering that the diffusion signal that we acquire in vivo derives from extracellular water, our results confirm that ADC decreases during activation, probably due to cellular swelling, and increases a little with the decrease in DX. In fact, \(\gamma\) depends, as ADC, on \(\Delta\chi\) and cellular swelling, but in a different way. As already highlighted in vitro on phantoms [43; 44], in ex vivo excised tissues [45; 52] and in vivo in the human brain [40; 46], \(\gamma\) increases significantly with a decrease in \(\Delta\chi\), more than ADC does [44], and \(\gamma\), in contrast to ADC, increases with cellular swelling. In Figure 4, we report a simple scheme to highlight the main dependencies of the T\({}_{2}\)*, ADC, and \(\gamma\) parameters on the main physiological and microstructural variations that occur during neuronal activation. Therefore, the greater signal intensity observed in the functional \(\gamma\) maps shown in Figure 2c is because the two main variations that occur during activation cause a coherent sum of \(\gamma\) responses. This feature of \(\gamma\) supports it being a potential candidate for performing innovative diffusion fMRI experiments. To summarize, \(\rm G_{i}\) induces an ADC increase during activation that may mask the ADC-decreasing effect due to extravascular cell swelling [31; 32; 33]. On the other hand, \(\rm G_{i}\) induces an increase in \(\gamma\) (higher than the ADC one) [46], which sums up to its additional increase due to extravascular cell swelling. This ensures the potential sensitivity of \(\gamma\) to extracellular space changes due to cell swelling, independently of the \(\rm G_{i}\) strength.