Eigenvalues and eigenvectorsIn linear algebra, an eigenvector (ˈaɪgənˌvɛktər) or characteristic vector of a linear transformation is a nonzero vector that changes at most by a constant factor when that linear transformation is applied to it. The corresponding eigenvalue, often represented by , is the multiplying factor. Geometrically, a transformation matrix rotates, stretches, or shears the vectors it acts upon. The eigenvectors for a linear transformation matrix are the set of vectors that are only stretched, with no rotation or shear.
Elastic fiberElastic fibers (or yellow fibers) are an essential component of the extracellular matrix composed of bundles of proteins (elastin) which are produced by a number of different cell types including fibroblasts, endothelial, smooth muscle, and airway epithelial cells. These fibers are able to stretch many times their length, and snap back to their original length when relaxed without loss of energy. Elastic fibers include elastin, elaunin and oxytalan.
Soma (biology)In cellular neuroscience, the soma (pl. somata or somas; ), perikaryon (pl. perikarya), neurocyton, or cell body is the bulbous, non-process portion of a neuron or other brain cell type, containing the cell nucleus. Although it is often used to refer to neurons, it can also refer to other cell types as well, including astrocytes, oligodendrocytes, and microglia. There are many different specialized types of neurons, and their sizes vary from as small as about 5 micrometres to over 10 millimetres for some of the smallest and largest neurons of invertebrates, respectively.
Magnetic resonance imagingMagnetic resonance imaging (MRI) is a medical imaging technique used in radiology to form pictures of the anatomy and the physiological processes of the body. MRI scanners use strong magnetic fields, magnetic field gradients, and radio waves to generate images of the organs in the body. MRI does not involve X-rays or the use of ionizing radiation, which distinguishes it from computed tomography (CT) and positron emission tomography (PET) scans.
Eigendecomposition of a matrixIn linear algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. When the matrix being factorized is a normal or real symmetric matrix, the decomposition is called "spectral decomposition", derived from the spectral theorem. Eigenvalue, eigenvector and eigenspace A (nonzero) vector v of dimension N is an eigenvector of a square N × N matrix A if it satisfies a linear equation of the form for some scalar λ.
Brain cellBrain cells make up the functional tissue of the brain. The rest of the brain tissue is structural or connective called the stroma which includes blood vessels. The two main types of cells in the brain are neurons, also known as nerve cells, and glial cells also known as neuroglia. Neurons are the excitable cells of the brain that function by communicating with other neurons and interneurons (via synapses), in neural circuits and larger brain networks.
Generalized eigenvectorIn linear algebra, a generalized eigenvector of an matrix is a vector which satisfies certain criteria which are more relaxed than those for an (ordinary) eigenvector. Let be an -dimensional vector space and let be the matrix representation of a linear map from to with respect to some ordered basis. There may not always exist a full set of linearly independent eigenvectors of that form a complete basis for . That is, the matrix may not be diagonalizable.
Edge-preserving smoothingEdge-preserving smoothing or edge-preserving filtering is an technique that smooths away noise or textures while retaining sharp edges. Examples are the median, bilateral, guided, anisotropic diffusion, and Kuwahara filters. In many applications, e.g., medical or satellite imaging, the edges are key features and thus must be preserved sharp and undistorted in smoothing/denoising. Edge-preserving filters are designed to automatically limit the smoothing at “edges” in images measured, e.g., by high gradient magnitudes.
Orthogonal matrixIn linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is where QT is the transpose of Q and I is the identity matrix. This leads to the equivalent characterization: a matrix Q is orthogonal if its transpose is equal to its inverse: where Q−1 is the inverse of Q. An orthogonal matrix Q is necessarily invertible (with inverse Q−1 = QT), unitary (Q−1 = Q∗), where Q∗ is the Hermitian adjoint (conjugate transpose) of Q, and therefore normal (Q∗Q = QQ∗) over the real numbers.
Fick's laws of diffusionFick's laws of diffusion describe diffusion and were first posited by Adolf Fick in 1855 on the basis of largely experimental results. They can be used to solve for the diffusion coefficient, D. Fick's first law can be used to derive his second law which in turn is identical to the diffusion equation. A diffusion process that obeys Fick's laws is called normal or Fickian diffusion; otherwise, it is called anomalous diffusion or non-Fickian diffusion.
Orthogonal diagonalizationIn linear algebra, an orthogonal diagonalization of a symmetric matrix is a diagonalization by means of an orthogonal change of coordinates. The following is an orthogonal diagonalization algorithm that diagonalizes a quadratic form q(x) on Rn by means of an orthogonal change of coordinates X = PY. Step 1: find the symmetric matrix A which represents q and find its characteristic polynomial Step 2: find the eigenvalues of A which are the roots of . Step 3: for each eigenvalue of A from step 2, find an orthogonal basis of its eigenspace.
NeurofilamentNeurofilaments (NF) are classed as type IV intermediate filaments found in the cytoplasm of neurons. They are protein polymers measuring 10 nm in diameter and many micrometers in length. Together with microtubules (~25 nm) and microfilaments (7 nm), they form the neuronal cytoskeleton. They are believed to function primarily to provide structural support for axons and to regulate axon diameter, which influences nerve conduction velocity.
