Algorithmic choice of coordinates for injections into the brain: encoding a neuroanatomical atlas on a grid

Given an atlas of the brain and a number of injections to be performed in order to map out the connections between parts of the brain, we propose an algorithm to compute the coordinates of the injections. The algorithm is designed to sample the brain in the most homogeneous way compatible with the separation of brain regions. It can be applied to other species for which a neuroanatomical atlas is available. The computation is tested on the annotation at a resolution of 25 microns corresponding to the Allen Reference Atlas, which is hierarchical and consists of 209 regions. The resulting injection coordinates are being used for the injection protocol of the Mouse Brain Architecture project. Due to its large size and layered structure, the cerebral cortex is treated in a separate algorithm, which is more adapted to its geometry.Comment: 13 pages, LaTe

Marker Genes for Anatomical Regions in the Brain: Insights from the Allen Gene Expression Atlas

Quantitative criteria are proposed to identify genes (and sets of genes) whose expression marks a specific brain region (or a set of brain regions). Gene-expression energies, obtained for thousands of mouse genes by numerization of in-situ hybridization images in the Allen Gene Expression Atlas, are used to test these methods in the mouse brain. Individual genes are ranked using integrals of their expression energies across brain regions. The ranking is generalized to sets of genes and the problem of optimal markers of a classical region receives a linear-algebraic solution. Moreover, the goodness of the fitting of the expression profile of a gene to the profile of a brain region is closely related to the co-expression of genes. The geometric interpretation of this fact leads to a quantitative criterion to detect markers of pairs of brain regions. Local properties of the gene-expression profiles are also used to detect genes that separate a given grain region from its environment.Comment: 26 pages, LaTe

A General Class of Collatz Sequence and Ruin Problem

In this paper we show the probabilistic convergence of the original Collatz (3n + 1) (or Hotpo) sequence to unity. A generalized form of the Collatz sequence (GCS) is proposed subsequently. Unlike Hotpo, an instance of a GCS can converge to integers other than unity. A GCS can be generated using the concept of an abstract machine performing arithmetic operations on different numerical bases. Original Collatz sequence is then proved to be a special case of GCS on base 2. The stopping time of GCS sequences is shown to possess remarkable statistical behavior. We conjecture that the Collatz convergence elicits existence of attractor points in digital chaos generated by arithmetic operations on numbers. We also model Collatz convergence as a classical ruin problem on the digits of a number in a base in which the abstract machine is computing and establish its statistical behavior. Finally an average bound on the stopping time of the sequence is established that grows linearly with the number of digits

On The Dynamical Nature Of Computation

Dynamical Systems theory generally deals with fixed point iterations of continuous functions. Computation by Turing machine although is a fixed point iteration but is not continuous. This specific category of fixed point iterations can only be studied using their orbits. Therefore the standard notion of chaos is not immediately applicable. However, when a suitable definition is used, it is found that the notion of chaos and fractal sets exists even in computation. It is found that a non terminating Computation will be almost surely chaotic, and autonomous learning will almost surely identify fractal only sets.Comment: arXiv admin note: substantial text overlap with arXiv:1407.7417, arXiv:1111.494

'Almost Sure' Chaotic Properties of Machine Learning Methods

It has been demonstrated earlier that universal computation is 'almost surely' chaotic. Machine learning is a form of computational fixed point iteration, iterating over the computable function space. We showcase some properties of this iteration, and establish in general that the iteration is 'almost surely' of chaotic nature. This theory explains the observation in the counter intuitive properties of deep learning methods. This paper demonstrates that these properties are going to be universal to any learning method.Comment: 10 pages : to be submitted to Theoretical Computer Science. arXiv admin note: text overlap with arXiv:1111.494

A Pseudo Random Number Generator from Chaos

A random number generator is proposed based on a theorem about existence of chaos in fixed point iteration of x= cot2(x). Digital computer simulation of this function iteration exhibits random behavior. A method is proposed to extract random bytes from this simulation. Diehard and NIST test suite for randomness detection is run on this bytes, and it is found to pass all the tests in the suite. Thus, this method qualifies even for cryptographic quality random number generation

Invariance And Inner Fractals In Polynomial And Transcendental Fractals

A lot of formal and informal recreational study took place in the fields of Meromorphic Maps, since Mandelbrot popularized the map z <- z^2 + c. An immediate generalization of the Mandelbrot z <-z^n + c also known as the Multibrot family were also studied. In the current paper, general truncated polynomial maps of the form z =2} a_px^p +c are studied. Two fundamental properties of these polynomial maps are hereby presented. One of them is the existence of shape preserving transformations on fractal images, and another one is the existence of embedded Multibrot fractals inside a polynomial fractal. Any transform expression with transcendental terms also shows embedded Multibrot fractals, due to Taylor series expansion possible on the transcendental functions. We present a method by which existence of embedded fractals can be predicted. A gallery of images is presented alongside to showcase the findings.Comment: 22 pages, 22 figure

Parallel Computation Is ESS

There are enormous amount of examples of Computation in nature, exemplified across multiple species in biology. One crucial aim for these computations across all life forms their ability to learn and thereby increase the chance of their survival. In the current paper a formal definition of autonomous learning is proposed. From that definition we establish a Turing Machine model for learning, where rule tables can be added or deleted, but can not be modified. Sequential and parallel implementations of this model are discussed. It is found that for general purpose learning based on this model, the implementations capable of parallel execution would be evolutionarily stable. This is proposed to be of the reasons why in Nature parallelism in computation is found in abundance.Comment: Submitted to Theoretical Computer Science - Elsevie

Computational neuroanatomy and gene expression: optimal sets of marker genes for brain regions

The three-dimensional data-driven Allen Gene Expression Atlas of the adult mouse brain consists of numerized in-situ hybridization data for thousands of genes, co-registered to the Allen Reference Atlas. We propose quantitative criteria to rank genes as markers of a brain region, based on the localization of the gene expression and on its functional fitting to the shape of the region. These criteria lead to natural generalizations to sets of genes. We find sets of genes weighted with coefficients of both signs with almost perfect localization in all major regions of the left hemisphere of the brain, except the pallidum. Generalization of the fitting criterion with positivity constraint provides a lesser improvement of the markers, but requires sparser sets of genes.Comment: 6 pages, 5 figure

A study of two new generalized negative KdV type equations

We give a simple geometric interpretation of the mapping of the negative KdV equation as proposed by Qiao and Li {arXiv:1101.1605 [math-ph], Europhys. Lett.,94 (2011) 50003} and the Fuchssteiner equation using geometry of projective connection on S1 or stabilizer set of the Virasoro orbit. We propose a similar connection between with the higher-order negative KdV equations of Fuchssteiner type described as and respectively. We study the Painleve and symmetry analyses of these newly found equations and show that they yield soliton solutions.Comment: 14 pages. Constructive suggestions and criticism are most welcom