scholarly journals Cortical Entropy Values Correlate with Brain Scale-Free Dynamics

2017 ◽  
Author(s):  
Arturo Tozzi ◽  
James F. Peters ◽  
Mehmet Niyazi Çankaya
Keyword(s):  
Brain Activity ◽  
Brain Area ◽  
Three Dimensional ◽  
Renyi Entropy ◽  
Rényi Entropy ◽  
Two Dimensional ◽  
Informational Entropy ◽  
Scale Free ◽  
Dimensional Object ◽  
Available Information

ABSTRACTA two-dimensional shadow may encompass more information than its corresponding three-dimensional object. If we rotate the object, we achieve a pool of observed shadows from different angulations, gradients, shapes and variable length contours that make it possible for us to increase our available information. Starting from this simple observation, we show how informational entropies might turn out to be useful in the evaluation of scale-free dynamics in the brain. Indeed, brain activity exhibits a scale-free distribution, which appears as a straight line when plotted in a log power versus log frequency plot. A variation in the scale-free exponent and in the line scaling slope may occur during different functional neurophysiological states. Here we show that modifications in scaling slope are associated with variations in Rényi entropy, a generalization of Shannon informational entropy. From a three-dimensional object’s perspective, by changing its orientation (standing for the cortical scale-free exponent), we detect different two-dimensional shadows from different perception angles (standing for Rènyi entropy in different brain areas). We perform simulations showing how, starting from known values of Rènyi entropy (easily detectable in brain fMRIs or EEG traces), it is feasible to calculate the scaling slope in a given moment and a given brain area. Because changes in scale-free cortical dynamics modify brain activity, suggests the possibility of novel insights in mind reading and description of the forces required for transcranial stimulation.

scholarly journals Excited state Rényi entropy and subsystem distance in two-dimensional non-compact bosonic theory. Part I. Single-particle states

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Jiaju Zhang ◽  
M.A. Rajabpour
Keyword(s):  
Excited States ◽  
Single Particle ◽  
Renyi Entropy ◽  
Rényi Entropy ◽  
Large Momentum ◽  
Particle State ◽  
Two Dimensional ◽  
Universal Form ◽  
Conformal Field ◽  
Harmonic Chain

Abstract We investigate the Rényi entropy of the excited states produced by the current and its derivatives in the two-dimensional free massless non-compact bosonic theory, which is a two-dimensional conformal field theory. We also study the subsystem Schatten distance between these states. The two-dimensional free massless non-compact bosonic theory is the continuum limit of the finite periodic gapless harmonic chains with the local interactions. We identify the excited states produced by current and its derivatives in the massless bosonic theory as the single-particle excited states in the gapless harmonic chain. We calculate analytically the second Rényi entropy and the second Schatten distance in the massless bosonic theory. We then use the wave functions of the excited states and calculate the second Rényi entropy and the second Schatten distance in the gapless limit of the harmonic chain, which match perfectly with the analytical results in the massless bosonic theory. We verify that in the large momentum limit the single-particle state Rényi entropy takes a universal form. We also show that in the limit of large momenta and large momentum difference the subsystem Schatten distance takes a universal form but it is replaced by a new corrected form when the momentum difference is small. Finally we also comment on the mutual Rényi entropy of two disjoint intervals in the excited states of the two-dimensional free non-compact bosonic theory.

Statistical Approach to Shape from Shading: Reconstruction of Three-Dimensional Face Surfaces from Single Two-Dimensional Images

1996 ◽  
Vol 8 (6) ◽  
pp. 1321-1340 ◽  
Author(s):  
Joseph J. Atick ◽  
Paul A. Griffin ◽  
A. Norman Redlich
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Practical Interest ◽  
Principal Component ◽  
Shape From Shading ◽  
Shape Space ◽  
Two Dimensional ◽  
Two Dimensional Image ◽  
Low Dimensional ◽  
Dimensional Object

The human visual system is proficient in perceiving three-dimensional shape from the shading patterns in a two-dimensional image. How it does this is not well understood and continues to be a question of fundamental and practical interest. In this paper we present a new quantitative approach to shape-from-shading that may provide some answers. We suggest that the brain, through evolution or prior experience, has discovered that objects can be classified into lower-dimensional object-classes as to their shape. Extraction of shape from shading is then equivalent to the much simpler problem of parameter estimation in a low-dimensional space. We carry out this proposal for an important class of three-dimensional (3D) objects: human heads. From an ensemble of several hundred laser-scanned 3D heads, we use principal component analysis to derive a low-dimensional parameterization of head shape space. An algorithm for solving shape-from-shading using this representation is presented. It works well even on real images where it is able to recover the 3D surface for a given person, maintaining facial detail and identity, from a single 2D image of his face. This algorithm has applications in face recognition and animation.

Three Dimensional Fourier Analysis of the Ribonucleoprotein Particle (Ribosome) Helix of Entamoeba Invadens

1970 ◽  
Vol 28 ◽  
pp. 266-267 ◽  
Author(s):  
James A. Lake ◽  
Henry S. Slayter
Keyword(s):  
Fourier Analysis ◽  
Three Dimensional ◽  
Spherical Particles ◽  
Two Dimensional ◽  
Ribonucleoprotein Particle ◽  
Single View ◽  
Ordered Arrays ◽  
Dimensional Object ◽  
Entamoeba Invadens ◽  
Lines Of Evidence

Cysts of Entamoeba Invadens contain large ordered arrays of closely packed helices which absorb strongly in the ultraviolet. The helices consist of small, approximately spherical particles about 250Å in diameter. Several lines of evidence have indicated that they may be ribosomes. We shall refer to these particles as ribosomes in this paper.DeRosier and Klug (1) have demonstrated that it is possible to reconstruct a three dimensional object from two dimensional projected images, i.e. micrographs, provided that sufficient views, of individual molecules are available. A single view (micrograph) of one ribosomal helix provides many views of individual ribosomes.

Sign in / Sign up

Export Citation Format

Share Document