Number Of Effective Dimensions Research Articles

Low-dimensional models of single neurons: a review.

The classical Hodgkin-Huxley (HH) point-neuron model of action potential generation is four-dimensional. It consists of four ordinary differential equations describing the dynamics of the membrane potential and three gating variables associated to a transient sodium and a delayed-rectifier potassium ionic currents. Conductance-based models of HH type are higher-dimensional extensions of the classical HH model. They include a number of supplementary state variables associated with other ionic current types, and are able to describe additional phenomena such as subthreshold oscillations, mixed-mode oscillations (subthreshold oscillations interspersed with spikes), clustering and bursting. In this manuscript we discuss biophysically plausible and phenomenological reduced models that preserve the biophysical and/or dynamic description of models of HH type and the ability to produce complex phenomena, but the number of effective dimensions (state variables) is lower. We describe several representative models. We also describe systematic and heuristic methods of deriving reduced models from models of HH type.

Patterns of quantitative genetic variation in multiple dimensions

A fundamental question for both evolutionary biologists and breeders is the extent to which genetic correlations limit the ability of populations to respond to selection. Here I view this topic from three perspectives. First, I propose several nondimensional statistics to quantify the genetic variation present in a suite of traits and to describe the extent to which correlations limit their selection response. A review of five data sets suggests that the total variation differs substantially between populations. In all cases analyzed, however, the "effective number of dimensions" is less than two: more than half of the total genetic variation is explained by a single combination of traits. Second, I consider how patterns of variation affect the average evolutionary response to selection in a random direction. When genetic variation lies in a small number of dimensions but there are a large number of traits under selection, then the average selection response will be reduced substantially from its potential maximum. Third, I discuss how a low genetic correlation between male fitness and female fitness limits the ability of populations to adapt. Data from two recent studies of natural populations suggest this correlation can diminish or even erase any genetic benefit to mate choice. Together these results suggest that adaptation (in natural populations) and genetic improvement (in domesticated populations) may often be as much constrained by patterns of genetic correlation as by the overall amount of genetic variation.

Effective dimensionality for principal component analysis of time series expression data

Large-scale expression data are today measured for thousands of genes simultaneously. This development has been followed by an exploration of theoretical tools to get as much information out of these data as possible. Several groups have used principal component analysis (PCA) for this task. However, since this approach is data-driven, care must be taken in order not to analyze the noise instead of the data. As a strong warning towards uncritical use of the output from a PCA, we employ a newly developed procedure to judge the effective dimensionality of a specific data set. Although this data set is obtained during the development of rat central nervous system, our finding is a general property of noisy time series data. Based on knowledge of the noise-level for the data, we find that the effective number of dimensions that are meaningful to use in a PCA is much lower than what could be expected from the number of measurements. We attribute this fact both to effects of noise and the lack of independence of the expression levels. Finally, we explore the possibility to increase the dimensionality by performing more measurements within one time series, and conclude that this is not a fruitful approach.

Survey of two-time physics

Two-time physics (2T) is a general reformulation of one-time physics(1T)that displays previously unnoticed hidden symmetries in 1T dynamicalsystems and establishes previously unknown duality-type relations amongthem. This may play a role in displaying the symmetries and constructingthe dynamics of little understood systems, such as M-theory. 2T-physicsdescribes various 1T dynamical systems as different d-dimensional`holographic' views of the same 2T system in d + 2 dimensions. The`holography' is due to gauge symmetries that tend to reduce the number ofeffective dimensions. Different 1T evolutions (i.e. differentHamiltonians) emerge from the same 2T-theory when gauge fixing is donewith different embeddings of d dimensions inside d + 2 dimensions. Thus,in the 2T setting, the distinguished 1T which we call`time' is a gauge-dependent concept. The 2T-action also has a global SO(d,2) symmetry in flat spacetime, or a more general d + 2 symmetry incurved spacetime, under which all dimensions are on an equal footing. Thissymmetry is observable in many 1T-systems, but it remained unknownuntil discovered in the 2T formalism. The symmetry takes variousnonlinear (hidden) forms in the 1T-systems, and it is realized in the sameirreducible unitary representation (the same Casimir eigenvalues) in theirquantum Hilbert spaces. 2T-physics has mainly been developed in thecontext of particles, including spin and supersymmetry, but someadvances have also been made with strings and p-branes, and insights forM-theory have already emerged. In the case of particles, there exists ageneral worldline formulation with background fields, as well as a fieldtheory formulation, both described in terms of fields that depend on d + 2coordinates. All 1T particle interactions with Yang-Mills, gravitationaland other fields are included in the d + 2 reformulation. In particular, thestandard model of particle physics can be regarded as a gauge-fixed formof a 2T-theory in 4 + 2 dimensions. These facts already provide evidencefor a new type of higher-dimensional unification.

On the number of effective dimensions of exchange interactions in the triplet state of naphthalene

Recent spectra of the triplet exciton band of naphthalene, due to Doberer and Port (1984), thought to imply effective three-dimensional exchange coupling, are explained within the usual two-dimensional model. This model describes adequately both the substructure of the Davydov bands and their frequency shift, when account is taken of shallow traps present in the crystals.

Functional Method for the Off-Shell Dual Amplitudes

By adopting Corrigan and Fairlie's interpretation of the off·the·mass·shell states of Schwarz, we give a general framework to establish the dual amplitudes for off·shell particles in terms of purely functional-integral formalism. In our framework, the amplitudes with one·particle off-shell are derived, whose representation is of the Bardak~i-Ruegg type, only when the number of effective dimensions is 24 and the intercept of the leading trajectory is one. There is an involved problem around the effective dimensionality (in connection with the integration measure), but the origin of magic number 16 of Schwarz is undiscovered.