Bicomplex quantum mechanics: I. The generalized Schr\"odinger equation

We introduce the set of bicomplex numbers $\mathbb{T}$ which is a commutative ring with zero divisors defined by $\mathbb{T}=\{w_0+w_1 \bold{i_1}+w_2\bold{i_2}+w_3 \bold{j}| w_0,w_1,w_2,w_3 \in \mathbb{R}\}$ where $\bold{i^{\text 2}_1}=-1, \bold{i^{\text 2}_2}=-1, \bold{j}^2=1,\ \bold{i_1}\bold{i_2}=\bold{j}=\bold{i_2}\bold{i_1}$. We present the conjugates and the moduli associated with the bicomplex numbers. Then we study the bicomplex Schr\"odinger equation and found the continuity equations. The discrete symmetries of the system of equations describing the bicomplex Schr\"odinger equation are obtained. Finally, we study the bicomplex Born formulas under the discrete symetries. We obtain the standard Born's formula for the class of bicomplex wave functions having a null hyperbolic angle

BOLD and perfusion changes during epileptic generalised spike wave activity

It is unclear whether neurovascular coupling is maintained during epileptic discharges. Knowing this is important to allow appropriate inferences from functional imaging studies of epileptic activity. Recent blood oxygen level-dependent (BOLD) functional MRI (fMRI) studies have demonstrated negative BOLD responses (NBR) in frontal, parietal and posterior cingulate cortices during generalised spike wave activity (GSW). We hypothesized that GSW-related NBR commonly reflect decreased cerebral blood flow (CBF). We measured BOLD and cerebral blood flow responses using simultaneous EEG with BOLD and arterial spin label (ASL) fMRI at 3 T. Four patients with epilepsy were studied; two with idiopathic generalized epilepsy (IGE) and two with secondary generalized epilepsy (SGE). We found GSW-related NBR in frontal, parietal and posterior cingulate cortices. We measured the coupling between BOLD and CBF changes during GSW and normal background EEG and found a positive correlation between the simultaneously measured BOLD and CBF throughout the imaged volume. Frontal and thalamic activation were seen in two patients with SGE, concordant with the electro-clinical features of their epilepsy. There was striking reproducibility of the GSW-associated BOLD response in subjects previously studied at 1.5 T. Our results show a preserved relationship between BOLD and CBF changes during rest and GSW activity consistent with normal neurovascular coupling in patients with generalized epilepsy and in particular during GSW activity. Cortical activations appear to reflect areas of discharge generation whilst deactivations reflect changes in conscious resting state activity

Investigation of the neurovascular coupling in positive and negative BOLD responses in human brain at 7T

Decreases in stimulus-dependent blood oxygenation level dependent (BOLD) signal and their underlying neurovascular origins have recently gained considerable interest. In this study a multi-echo, BOLD-corrected vascular space occupancy (VASO) functional magnetic resonance imaging (fMRI) technique was used to investigate neurovascular responses during stimuli that elicit positive and negative BOLD responses in human brain at 7 T. Stimulus-induced BOLD, cerebral blood volume (CBV), and cerebral blood flow (CBF) changes were measured and analyzed in ‘arterial’ and ‘venous’ blood compartments in macro- and microvasculature. We found that the overall interplay of mean CBV, CBF and BOLD responses is similar for tasks inducing positive and negative BOLD responses. Some aspects of the neurovascular coupling however, such as the temporal response, cortical depth dependence, and the weighting between ‘arterial’ and ‘venous’ contributions, are significantly different for the different task conditions. Namely, while for excitatory tasks the BOLD response peaks at the cortical surface, and the CBV change is similar in cortex and pial vasculature, inhibitory tasks are associated with a maximum negative BOLD response in deeper layers, with CBV showing strong constriction of surface arteries and a faster return to baseline. The different interplays of CBV, CBF and BOLD during excitatory and inhibitory responses suggests different underlying hemodynamic mechanisms