## 1 Introduction  
The brain  
## 1 Introduction  
Functional MRI (fMRI) is crucial for studying brain function [1]. It allows researchers to map neural activity in response to different cognitive and sensory tasks, providing valuable insights into brain connectivity and functional organization. On the other hand, diffusion MRI [2,3] is essential for investigating the brain's microstructure by tracking the movement of biological water molecules. It enables the reconstruction of neural pathways and the assessment of structural integrity to study neurodevelop, aging, and neurological disorders. A key limitation of fMRI, which is based on the blood-oxygen-level-dependent (BOLD) effect, is its limited spatial specificity. This arises because the BOLD signal is related to the neurovascular coupling, which leads to macrovascular changes that are not strictly confined to the regions with increased neural activity [4, 5, 6, 7]. The origin of the BOLD fMRI effect lies in the different magnetic properties of oxy-hemoglobin, which is approximately isomagnetic relative to tissue, and deoxy-hemoglobin, which is slightly paramagnetic relative to tissue [8]. The magnetic susceptibility difference (\(\Delta\chi\)) decreases during activation due to changes in the relative amounts of oxy- and deoxy-hemoglobin. This, in turn, leads to an increase in T\({}_{2}\) and T\({}_{2}\)*, resulting in an enhanced BOLD signal. The reduction in \(\Delta\chi\) during activation [9] is associated with a reduction in internal magnetic field inhomogeneities (called internal or background gradients) G\({}_{i}\). However, the brain parenchyma also contains some other microstructural changes associated with neuronal activation. For example, cell swelling and membrane expansion involving neural soma, axons, and glial cells have been observed in vitro using invasive techniques [10,11]. Considering the potential of diffusion techniques to infer microstructural changes in vivo [12, 13, 14], diffusion-weighted imaging (DWI) techniques have also been applied in the field of fMRI, the so-called DW-fMRI. Early studies conducted at intermediate-to-low b-values revealed that the apparent diffusion coefficient (ADC) time course closely mirrors the vascular BOLD signal, alongside an observed increase in water ADC during activation. This phenomenon can be attributed to the inverse dependence of ADC on Gi [15, 16, 17]. Moreover, DWI at b-values lower than 500 s/mm\({}^{2}\) has been used to eliminate the unwanted (mostly perfusion) signal from arteries and large vessels [18,19], which can be spatially displaced from the site of neural activity [20]. Other studies have proposed the hypothesis that DW-fMRI signals at high b-values may be coupled to neuronal activity more directly than the regional blood flow, which is exploited in conventional fMRI approaches [21,22]. Notably, instead of the usual increase in ADC values during brain activation, using b = 1400 s/mm\({}^{2}\), [PERSON] and [PERSON] [22] found a synchronized decrease in ADC that was attributed to cell swelling induced by enhanced neuronal activity, together with the suppression of the intravascular signal due to high b-values. In subsequent studies using a bi-exponential diffusion model [23,24], the feeble/moderate decrease in ADC was interpreted as an effect of small dilation of the slow diffusion water compartment and a volume decrease in the fast diffusion water compartment in the extravascular space. While [PERSON] and co-workers in subsequent high-b-value experiments using drugs to block the neurovascular effect have reinforced the hypothesis of ADC decrease being associated with cellular swelling [25,26], other groups suggested that a large component of the DW-fMRI signal at high b-values is vascular rather than neuronal [27,28]. Moreover, in simultaneous calcium fluorescence imaging and DW-fMRI, no evidence was found that diffusion signals directly correlate with normal spontaneous neuronal activity [29].