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Jose Louton
Jose Louton

A Brief On Tensor Analysis

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A Brief On Tensor Analysis

The first two books treat a large amount of subjects in mathematics, including tensor calculus, geometry etc. The aim is to provide a bridge between mathematics and physics. In Munkres's book, you will find a nice exposition about tensor products of vector spaces, which is used in the study of multivariate integrals. Greub's book is a more abstract account on the subject (and, in my opinion, more advanced), but a very nice reference too. Maybe Winitzki's book is more appropriate for you, since the book is a linear algebra-type of book, so it has proofs for theorems and some nice tools for direct applications too. Roman's book also treats the case of tensor products of vector spaces.

Diffusion tensor magnetic resonance imaging (DT-MRI) is a powerful quantitative technique with the ability to detect in vivo microscopic characteristics and abnormalities of brain tissue. It has been successfully applied to a number of neurological conditions, such as stroke, multiple sclerosis and brain tumors, providing information otherwise inaccessible on the pathological substrates. DT-MRI has also been used to study patients with cognitive decline, mainly those with Alzheimer's disease. Several image-analysis approaches have been employed, including region of interest, histogram, voxel-based analyses and DT-MRI-based tractography. Specific patterns of spatial distribution of tissue damage and correlations with neuropsychological measures have been reported. This review focuses on the use of DT-MRI to investigate dementias. The main clinical results and the different methods of image analysis will be overviewed and discussed.

The notion of a tensor captures three great ideas: equivariance, multilinearity, separability. But trying to be three things at once makes the notion difficult to understand. We will explain tensors in an accessible and elementary way through the lens of linear algebra and numerical linear algebra, elucidated with examples from computational and applied mathematics.

Our investigation of the preferred mode was guided by a three-part hypothesis. First, we hypothesized that empirical population responses may often have a clear preferred mode. Second, we hypothesized that the preferred mode likely differs between brain areas. To address these hypotheses, we assessed the preferred mode for three neural datasets recorded from primary visual cortex (V1) and four neural datasets recorded from M1. V1 datasets were strongly neuron-preferred, while M1 datasets were strongly condition-preferred. Third, we hypothesized that the preferred mode might be informative regarding the origin of population responses. We concentrated on models of M1, and found that existing models based on tuning for external variables were neuron-preferred, in opposition to the M1 data. However, existing models with strong internal dynamics were condition-preferred, in agreement with the data. Model success or failure depended not on parameter choice or fit quality, but on model class. We conclude that tensor structure is informative regarding the predominant origin of time-varying activity, and can be used to test specific hypotheses. In the present case, the tensor structure of M1 datasets is consistent with only a subset of existing models.

(a) Responses of four example neurons for a V1 dataset recorded via an implanted electrode array during presentation of movies of natural scenes. Each colored trace plots the trial-averaged firing rate for one condition (one of 25 movies). For visualization, traces are colored red to blue based on the firing rate early in the stimulus. (b) Responses of four example neurons for an M1 dataset recorded via two implanted electrode arrays during a delayed-reach task (monkey J). Example neurons were chosen to illustrate the variety of observed responses. Each colored trace plots the trial-averaged firing rate for one condition; i.e., one of 72 straight and curved reach trajectories. For visua


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