scholarly journals Computation in a Single Neuron: Hodgkin and Huxley Revisited

2003 ◽  
Vol 15 (8) ◽  
pp. 1715-1749 ◽  
Author(s):  
Blaise Agüera y Arcas ◽  
Adrienne L. Fairhall ◽  
William Bialek
Keyword(s):  
Mutual Information ◽  
Feature Detection ◽  
Dimensional Space ◽  
Model Neuron ◽  
Realistic Model ◽  
High Time Resolution ◽  
Correlation Technique ◽  
Two Dimensional ◽  
Dimensional Approximation ◽  
Low Dimensional

A spiking neuron “computes” by transforming a complex dynamical input into a train of action potentials, or spikes. The computation performed by the neuron can be formulated as dimensional reduction, or feature detection, followed by a nonlinear decision function over the low-dimensional space. Generalizations of the reverse correlation technique with white noise input provide a numerical strategy for extracting the relevant low-dimensional features from experimental data, and information theory can be used to evaluate the quality of the low-dimensional approximation. We apply these methods to analyze the simplest biophysically realistic model neuron, the Hodgkin-Huxley (HH) model, using this system to illustrate the general methodological issues. We focus on the features in the stimulus that trigger a spike, explicitly eliminating the effects of interactions between spikes. One can approximate this triggering “feature space” as a two-dimensional linear subspace in the high-dimensional space of input histories, capturing in this way a substantial fraction of the mutual information between inputs and spike time. We find that an even better approximation, however, is to describe the relevant subspace as two dimensional but curved; in this way, we can capture 90% of the mutual information even at high time resolution. Our analysis provides a new understanding of the computational properties of the HH model. While it is common to approximate neural behavior as “integrate and fire,” the HH model is not an integrator nor is it well described by a single threshold.

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.

scholarly journals Predictability and Information Theory. Part II: Imperfect Forecasts

2005 ◽  
Vol 62 (9) ◽  
pp. 3368-3381 ◽  
Author(s):  
Timothy DelSole
Keyword(s):  
Information Theory ◽  
Mutual Information ◽  
Dimensional Space ◽  
Nonlinear Behavior ◽  
Potential Predictability ◽  
Low Dimensional ◽  
Lower Dimensional Space ◽  
True System ◽  
Normally Distributed

Abstract This paper presents a framework for quantifying predictability based on the behavior of imperfect forecasts. The critical quantity in this framework is not the forecast distribution, as used in many other predictability studies, but the conditional distribution of the state given the forecasts, called the regression forecast distribution. The average predictability of the regression forecast distribution is given by a quantity called the mutual information. Standard inequalities in information theory show that this quantity is bounded above by the average predictability of the true system and by the average predictability of the forecast system. These bounds clarify the role of potential predictability, of which many incorrect statements can be found in the literature. Mutual information has further attractive properties: it is invariant with respect to nonlinear transformations of the data, cannot be improved by manipulating the forecast, and reduces to familiar measures of correlation skill when the forecast and verification are joint normally distributed. The concept of potential predictable components is shown to define a lower-dimensional space that captures the full predictability of the regression forecast without loss of generality. The predictability of stationary, Gaussian, Markov systems is examined in detail. Some simple numerical examples suggest that imperfect forecasts are not always useful for joint normally distributed systems since greater predictability often can be obtained directly from observations. Rather, the usefulness of imperfect forecasts appears to lie in the fact that they can identify potential predictable components and capture nonstationary and/or nonlinear behavior, which are difficult to capture by low-dimensional, empirical models estimated from short historical records.

scholarly journals Planform evolution in convection–an embedded centre manifold

1992 ◽  
Vol 34 (2) ◽  
pp. 174-198 ◽  
Author(s):  
A. J. Roberts
Keyword(s):  
Dynamical Systems ◽  
Wide Class ◽  
Dimensional Manifold ◽  
Two Dimensional ◽  
Centre Manifold ◽  
Higher Dimensional ◽  
Dimensional Approximation ◽  
Low Dimensional ◽  
Geometric Picture ◽  
Better Than

AbstractThe new motion of embedding a centre manifold in some higher-dimensional manifold leads to a practical approach to the rational low-dimensional approximation of a wide class of dynamical systems; it also provides a simple geometric picture for these approximations. In particular, I consider the problem of finding an approximate, but accurate, description of the evolution of a two-dimensional planform of convection. Inspired by a simple example, the straightforward adiabatic iteration is proposed to estimate an embedding manifold and arguments are presented for its effectiveness. Upon applying the procedure to a model convective planform problem I find that the resulting approximations perform remarkably well–much better than the traditional Swift-Hohenberg approximation for planform evolution.

scholarly journals Detecting Causality in Multivariate Time Series via Non-Uniform Embedding

Entropy ◽  
2019 ◽  
Vol 21 (12) ◽  
pp. 1233 ◽  
Author(s):  
Ziyu Jia ◽  
Youfang Lin ◽  
Zehui Jiao ◽  
Yan Ma ◽  
Jing Wang
Keyword(s):  
Time Series ◽  
Mutual Information ◽  
Chaotic Systems ◽  
Multivariate Time Series ◽  
Conditional Mutual Information ◽  
Uniform Embedding ◽  
Actual Application ◽  
Dimensional Approximation ◽  
Coupling Strengths ◽  
Low Dimensional

Causal analysis based on non-uniform embedding schemes is an important way to detect the underlying interactions between dynamic systems. However, there are still some obstacles to estimating high-dimensional conditional mutual information and forming optimal mixed embedding vector in traditional non-uniform embedding schemes. In this study, we present a new non-uniform embedding method framed in information theory to detect causality for multivariate time series, named LM-PMIME, which integrates the low-dimensional approximation of conditional mutual information and the mixed search strategy for the construction of the mixed embedding vector. We apply the proposed method to simulations of linear stochastic, nonlinear stochastic, and chaotic systems, demonstrating its superiority over partial conditional mutual information from mixed embedding (PMIME) method. Moreover, the proposed method works well for multivariate time series with weak coupling strengths, especially for chaotic systems. In the actual application, we show its applicability to epilepsy multichannel electrocorticographic recordings.

Electronic properties of quasi two-dimensional transition metals chalcogenides with low-dimensional magnetism

Doklady BGUIR ◽  
2020 ◽  
Vol 18 (7) ◽  
pp. 87-95
Author(s):  
M. S. Baranava ◽  
P. A. Praskurava
Keyword(s):  
Transition Metal ◽  
Transition Metals ◽  
Exchange Interaction ◽  
Electronic Properties ◽  
Metal Atom ◽  
Transition Metal Atom ◽  
Two Dimensional ◽  
Transition Metal Atoms ◽  
Ground Magnetic ◽  
Low Dimensional

The search for fundamental physical laws which lead to stable high-temperature ferromagnetism is an urgent task. In addition to the already synthesized two-dimensional materials, there remains a wide list of possible structures, the stability of which is predicted theoretically. The article suggests the results of studying the electronic properties of MAX3 (M = Cr, Fe, A = Ge, Si, X = S, Se, Te) transition metals based compounds with nanostructured magnetism. The research was carried out using quantum mechanical simulation in specialized VASP software and calculations within the Heisenberg model. The ground magnetic states of twodimensional MAX3 and the corresponding energy band structures are determined. We found that among the systems under study, CrGeTe3 is a semiconductor nanosized ferromagnet. In addition, one is a semiconductor with a bandgap of 0.35 eV. Other materials are antiferromagnetic. The magnetic moment in MAX3 is localized on the transition metal atoms: in particular, the main one on the d-orbital of the transition metal atom (and only a small part on the p-orbital of the chalcogen). For CrGeTe3, the exchange interaction integral is calculated. The mechanisms of the formation of magnetic order was established. According to the obtained exchange interaction integrals, a strong ferromagnetic order is formed in the semiconductor plane. The distribution of the projection density of electronic states indicates hybridization between the d-orbital of the transition metal atom and the p-orbital of the chalcogen. The study revealed that the exchange interaction by the mechanism of superexchange is more probabilistic.

On the Classification of Fractal and Non-Fractal Objects in Two-Dimensional Space

2020 ◽  
Vol 14 (6) ◽  
pp. 1232-1239
Author(s):  
P. M. Pustovoit ◽  
E. G. Yashina ◽  
K. A. Pshenichnyi ◽  
S. V. Grigoriev
Keyword(s):  
Dimensional Space ◽  
Two Dimensional

Political Cleavages and Political Parties

2018 ◽  
Author(s):  
Russell J. Dalton
Keyword(s):  
Political Parties ◽  
Dimensional Space ◽  
New Left ◽  
Party Systems ◽  
European Parliament ◽  
Two Dimensional ◽  
Far Right ◽  
Policy Positions ◽  
Cultural Policies ◽  
Policy Choices

This chapter uses the cleavage positions of Candidates to the European Parliament (CEPs) to as representative of their parties’ political positions. Three surveys of CEPs track the evolution of party supply in European party systems. In 1979 parties were primarily aligned along a Left–Right economic cleavage. Gradually new left and Green parties began to compete in elections and crystallized and represented liberal cultural policies. In recent decades new far-right parties arose to represent culturally conservative positions. The cross-cutting cultural cleavage has also prompted many of the established parties to alter their policy positions. In most multiparty systems, political parties now compete in a fully populated two-dimensional space. This increases the supply of policy choices for the voters. The analyses are based on the Candidates to the European Parliament Studies in 1979, 1994, and 2009.

Wake Flow of Single and Multiple Yawed Cylinders

2004 ◽  
Vol 126 (5) ◽  
pp. 861-870 ◽  
Author(s):  
A. Thakur ◽  
X. Liu ◽  
J. S. Marshall
Keyword(s):  
Computational Study ◽  
Three Dimensional ◽  
Side Wall ◽  
Wake Flow ◽  
Upstream Region ◽  
Two Dimensional ◽  
Cylinder Wake ◽  
Dimensional Approximation ◽  
The Face ◽  
Drag And Lift

An experimental and computational study is performed of the wake flow behind a single yawed cylinder and a pair of parallel yawed cylinders placed in tandem. The experiments are performed for a yawed cylinder and a pair of yawed cylinders towed in a tank. Laser-induced fluorescence is used for flow visualization and particle-image velocimetry is used for quantitative velocity and vorticity measurement. Computations are performed using a second-order accurate block-structured finite-volume method with periodic boundary conditions along the cylinder axis. Results are applied to assess the applicability of a quasi-two-dimensional approximation, which assumes that the flow field is the same for any slice of the flow over the cylinder cross section. For a single cylinder, it is found that the cylinder wake vortices approach a quasi-two-dimensional state away from the cylinder upstream end for all cases examined (in which the cylinder yaw angle covers the range 0⩽ϕ⩽60°). Within the upstream region, the vortex orientation is found to be influenced by the tank side-wall boundary condition relative to the cylinder. For the case of two parallel yawed cylinders, vortices shed from the upstream cylinder are found to remain nearly quasi-two-dimensional as they are advected back and reach within about a cylinder diameter from the face of the downstream cylinder. As the vortices advect closer to the cylinder, the vortex cores become highly deformed and wrap around the downstream cylinder face. Three-dimensional perturbations of the upstream vortices are amplified as the vortices impact upon the downstream cylinder, such that during the final stages of vortex impact the quasi-two-dimensional nature of the flow breaks down and the vorticity field for the impacting vortices acquire significant three-dimensional perturbations. Quasi-two-dimensional and fully three-dimensional computational results are compared to assess the accuracy of the quasi-two-dimensional approximation in prediction of drag and lift coefficients of the cylinders.

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