Symmetric Jacobians

This article is about polynomial maps with a certain symmetry and/or antisymmetry in their Jacobians, and whether the Jacobian Conjecture is satisfied for such maps, or whether it is sufficient to prove the Jacobian Conjecture for such maps. For instance, we show that it suffices to prove the Jacobian conjecture for polynomial maps x + H over C such that JH satisfies all symmetries of the square, where H is homogeneous of arbitrary degree d >= 3.Comment: 18 pages, minor corrections, grayscale eepic boxes have been replaced by colorful tikz boxe

Quantitative assessment of prefrontal cortex in humans relative to nonhuman primates

Significance A longstanding controversy in neuroscience pertains to differences in human prefrontal cortex (PFC) compared with other primate species; specifically, is human PFC disproportionately large? Distinctively human behavioral capacities related to higher cognition and affect presumably arose from evolutionary modifications since humans and great apes diverged from a common ancestor about 6–8 Mya. Accurate determination of regional differences in the amount of cortical gray and subcortical white matter content in humans, great apes, and Old World monkeys can further our understanding of the link between structure and function of the human brain. Using tissue volume analyses, we show a disproportionately large amount of gray and white matter corresponding to PFC in humans compared with nonhuman primates.</jats:p

Mapping human cortical areas in vivo based on myelin content as revealed by t1- and t2-weighted MRI

Non-invasively mapping the layout of cortical areas in humans is a continuing challenge for neuroscience. We present a new method of mapping cortical areas based on myelin content as revealed by T1-weighted (T1w) and T2-weighted (T2w) MRI. The method is generalizable across different 3T scanners and pulse sequences. We use the ratio of T1w/T2w image intensities to eliminate the MR-related image intensity bias and enhance the contrast to noise ratio for myelin. Data from each subject was mapped to the cortical surface and aligned across individuals using surface-based registration. The spatial gradient of the group average myelin map provides an observer-independent measure of sharp transitions in myelin content across the surface—i.e. putative cortical areal borders. We found excellent agreement between the gradients of the myelin maps and the gradients of published probabilistic cytoarchitectonically defined cortical areas that were registered to the same surface-based atlas. For other cortical regions, we used published anatomical and functional information to make putative identifications of dozens of cortical areas or candidate areas. In general, primary and early unimodal association cortices are heavily myelinated and higher, multi-modal, association cortices are more lightly myelinated, but there are notable exceptions in the literature that are confirmed by our results. The overall pattern in the myelin maps also has important correlations with the developmental onset of subcortical white matter myelination, evolutionary cortical areal expansion in humans compared to macaques, postnatal cortical expansion in humans, and maps of neuronal density in non-human primates

Neurite imaging reveals microstructural variations in human cerebral cortical gray matter

We present distinct patterns of neurite distribution in the human cerebral cortex using diffusion magnetic resonance imaging (MRI). We analyzed both high-resolution structural (T1w and T2w images) and diffusion MRI data in 505 subjects from the Human Connectome Project. Neurite distributions were evaluated using the neurite orientation dispersion and density imaging (NODDI) model, optimized for gray matter, and mapped onto the cortical surface using a method weighted towards the cortical mid-thickness to reduce partial volume effects. The estimated neurite density was high in both somatosensory and motor areas, early visual and auditory areas, and middle temporal area (MT), showing a strikingly similar distribution to myelin maps estimated from the T1w/T2w ratio. The estimated neurite orientation dispersion was particularly high in early sensory areas, which are known for dense tangential fibers and are classified as granular cortex by classical anatomists. Spatial gradients of these cortical neurite properties revealed transitions that colocalize with some areal boundaries in a recent multi-modal parcellation of the human cerebral cortex, providing mutually supportive evidence. Our findings indicate that analyzing the cortical gray matter neurite morphology using diffusion MRI and NODDI provides valuable information regarding cortical microstructure that is related to but complementary to myeloarchitecture

Anatomical variability, multi-modal coordinate systems, and precision targeting in the marmoset brain

Localising accurate brain regions needs careful evaluation in each experimental species due to their individual variability. However, the function and connectivity of brain areas is commonly studied using a single-subject cranial landmark-based stereotactic atlas in animal neuroscience. Here, we address this issue in a small primate, the common marmoset, which is increasingly widely used in systems neuroscience. We developed a non-invasive multi-modal neuroimaging-based targeting pipeline, which accounts for intersubject anatomical variability in cranial and cortical landmarks in marmosets. This methodology allowed creation of multi-modal templates (MarmosetRIKEN20) including head CT and brain MR images, embedded in coordinate systems of anterior and posterior commissures (AC-PC) and CIFTI grayordinates. We found that the horizontal plane of the stereotactic coordinate was significantly rotated in pitch relative to the AC-PC coordinate system (10 degrees, frontal downwards), and had a significant bias and uncertainty due to positioning procedures. We also found that many common cranial and brain landmarks (e.g., bregma, intraparietal sulcus) vary in location across subjects and are substantial relative to average marmoset cortical area dimensions. Combining the neuroimaging-based targeting pipeline with robot-guided surgery enabled proof-of-concept targeting of deep brain structures with an accuracy of 0.2 mm. Altogether, our findings demonstrate substantial intersubject variability in marmoset brain and cranial landmarks, implying that subject-specific neuroimaging-based localization is needed for precision targeting in marmosets. The population-based templates and atlases in grayordinates, created for the first time in marmoset monkeys, should help bridging between macroscale and microscale analyses