Combinatorial vector fields and the valley structure of fitness landscapes

Adaptive (downhill) walks are a computationally convenient way of analyzing the geometric structure of fitness landscapes. Their inherently stochastic nature has limited their mathematical analysis, however. Here we develop a framework that interprets adaptive walks as deterministic trajectories in combinatorial vector fields and in return associate these combinatorial vector fields with weights that measure their steepness across the landscape. We show that the combinatorial vector fields and their weights have a product structure that is governed by the neutrality of the landscape. This product structure makes practical computations feasible. The framework presented here also provides an alternative, and mathematically more convenient, way of defining notions of valleys, saddle points, and barriers in landscape. As an application, we propose a refined approximation for transition rates between macrostates that are associated with the valleys of the landscape.     

Ecology of bacterial communities in the schistosomiasis vector snailBiomphalaria glabrata

The internal colony-forming bacterial flora of the schistosome intermediate host snailBiomphalaria glabrata (Say) has been characterized in ca. 500 individual snails from Puerto Rico, Guadeloupe, and St. Lucia, and from laboratory aquaria. Freshly captured wild snails harbor 2–40×10⁶ CFU·g^–1, and healthy aquarium snails harbor 4–16×10⁷ CFU·g^–1, whereas moribund individuals have 4–10 times as many bacteria as healthy individuals from the same habitats.Pseudomonas spp. are the most common predominant bacteria in normal snails, whereasAcinetobacter, Aeromonas, andMoraxella spp. predominate in moribund snails. External bacterial populations in water appear to have little effect on the composition and size of the flora in any snail. In addition to normal (healthy) and moribund snails, a third group of snails has been distinguished on the basis of internal bacterial density and predominating genera. These high-density snails may have undergone stresses and may harbor opportunistic pathogens. The microfloras of wild and laboratory-reared snails can be altered and stimulated to increase in density by crowding the snails or treating them with antibiotics.     

Analysis and interpretation of quadratic models of receptive fields

In this protocol, we present a procedure to analyze and visualize models of neuronal input-output functions that have a quadratic, a linear and a constant term, to determine their overall behavior. The suggested interpretations are close to those given by physiological studies of neurons, making the proposed methods particularly suitable for the analysis of receptive fields resulting from physiological measurements or model simulations.     

Building better models of visual cortical receptive fields

Scientists usually study the receptive fields of visual cortical neurons by measuring responses to "optimal stimuli." In this issue of Neuron, Rust and colleagues have taken a promising alternative approach: build a receptive field model based on the cell responses to a stimulus subset and then use the model to predict responses to other stimuli.     

Influence of head models on neuromagnetic fields and inverse source localizations

Background  

The magnetoencephalograms (MEGs) are mainly due to the source currents. However, there is a significant contribution to MEGs from the volume currents. The structure of the anatomical surfaces, e.g., gray and white matter, could severely influence the flow of volume currents in a head model. This, in turn, will also influence the MEGs and the inverse source localizations. This was examined in detail with three different human head models.     

Comparison of models for subtractive and shunting lateral-inhibition in receptor-neuron fields

Summary Two types of neuronal lateral inhibition in one-dimensional fields of receptors and neurons are considered. The first type, which has been demonstrated in the eye of Limulus, is called subtractive inhibition (SI): it assumes that neuronal activity depends on the difference between the total excitation and inhibition. The second type is called shunting inhibition (SHI): it assumes that inhibitory influences cause a shunting of a portion of the excitation-produced depolarizing current. Consideration of the shunting model is dictated by its considerable physiological plausibility. The actions of SI and SHI, examined for a variety of coupling conditions and time-stationary positive inputs, are shown to be markedly different. The results indicate that SI is most suited for obtaining (1) a linearity between input and output, (2) a contrasting effect that does not depend on the presence of input discontinuities, and (3) contrasting whose degree is independent of input amplitude. SI is especially useful if coupling coefficients can be varied to accommodate the various input form functions or if, for fixed coupling coefficients, the class of input form functions is limited. On the other hand SHI appears most suited for obtaining (1) a nonlinear input-output relation, (2) a relative contrasting only of discontinuities, and (3) a dependence of the contrasting upon input amplitude.List of Main Symbols a coupling coefficient for neighboring units, also called coupling amplitude - V _j output of receptor number j - i _j generator current of neuron number j - g inhibitory function for subtractive inhibition - h inhibitory function for shunting inhibition - v ₂/v ₁ [applies to two-unit case] - N _k neuron number k - I _k total source current produced by excitatory influences on N _k - G _k conductance for source current not shunted (with shunting inhibition) - i portion of source current shunted as a result of inhibition - m number of inhibitory influences [in Eq. (1)] - G _kj conductance of inhibitory shunt path j for neuron N _k - q number of receptors - n number of neurons - R _j receptor number j - x distance - y(x) input stimulus to receptors - y _j =y(x _j ) input stimulus to receptor R _j - v _j v_j for v _j 0, zero otherwise - a _kj G _kj /v _j , inhibitory coupling coefficient for forward shunting inhibition [refer to Eq. (2)] - b _kj excitatory coupling coefficient for contribution to source current of neuron N _k by receptor R _j [refer to Eq. (3)] - i _j i _j for i _j 0, zero otherwise - c _kj G _kj /i _j , inhibitory coupling coefficient for backward shunting inhibition [refer to Eq. (4)] - â _kj inhibitory coupling coefficient for forward subtractive inhibition [refer to Eq. (5)] - _kj inhibitory coupling coefficient for backward subtractive inhibition [refer to Eq. (6)] - y(x _j )=Af(x _j ) sensory input function - A input amplitude - f(x _j ) sensory input form function, also called a sensory image - i(x _j ) generator current output of neuron Nj which is located at x=x _j - y (y ₁, y ₂, ..., y _n), a column vector - i (i ₁, i ₂, ..., i _n), a column vector, also called generator current configuration - a an n by n matrix having a _kj as the term in the k-th row, j-th column - U the unit matrix - d ¦k-j¦, separation between neurons N _k and N _j - a a _kj for d=1, called coupling amplitude - SI subtractive inhibition - SHI shunting inhibition - FSI forward subtractive inhibition - BSI backward subtractive inhibition - FSHI forward shunting inhibition - BSHI backward shunting inhibition - s i/i ₅₁ = (s ₁, s ₂, ..., s _n), normalized generator current vector, also called normalized generator current configuration - s _j i _j/i ₅₁, normalized generator current of neuron N _j - f(x) continuous input form function of which f(x _j ) is a sampled version - ^p f(x)/x ^p p-th order derivative of f(x)     

State vector models of the cell cycle. III: Continuous time cell cycle models

In this paper the elements of the matrix of the Hahn cell-cycle model are identified with the infinitesimal transition probabilities of a Markov process, and as a limiting process a differential equation analogue is derived. The probability density function of the discrete time model is derived and used to obtain the density function for transit times of the continuous time model. It is shown that the mean transit time remains constant and that the variances of the discrete and continuous time models are the same to the order of the time increment. Finally, it is shown how to derive the Takahashi model from the continuous time Hahn model.     

Malaria in Africa: vector species' niche models and relative risk maps

A central theoretical goal of epidemiology is the construction of spatial models of disease prevalence and risk, including maps for the potential spread of infectious disease. We provide three continent-wide maps representing the relative risk of malaria in Africa based on ecological niche models of vector species and risk analysis at a spatial resolution of 1 arc-minute (9 185 275 cells of approximately 4 sq km). Using a maximum entropy method we construct niche models for 10 malaria vector species based on species occurrence records since 1980, 19 climatic variables, altitude, and land cover data (in 14 classes). For seven vectors (Anopheles coustani, A. funestus, A. melas, A. merus, A. moucheti, A. nili, and A. paludis) these are the first published niche models. We predict that Central Africa has poor habitat for both A. arabiensis and A. gambiae, and that A. quadriannulatus and A. arabiensis have restricted habitats in Southern Africa as claimed by field experts in criticism of previous models. The results of the niche models are incorporated into three relative risk models which assume different ecological interactions between vector species. The "additive" model assumes no interaction; the "minimax" model assumes maximum relative risk due to any vector in a cell; and the "competitive exclusion" model assumes the relative risk that arises from the most suitable vector for a cell. All models include variable anthrophilicity of vectors and spatial variation in human population density. Relative risk maps are produced from these models. All models predict that human population density is the critical factor determining malaria risk. Our method of constructing relative risk maps is equally general. We discuss the limits of the relative risk maps reported here, and the additional data that are required for their improvement. The protocol developed here can be used for any other vector-borne disease.     

Current densities induced in swine and rat models by power-frequency electric fields

Measurements have been made of vector current densities induced by vertical, uniform, 60-Hz electric fields in the torsos of homogeneous models of swine and rats. The observed data were a strong function of the five grounding configurations invested: all four feet grounded, only front feet grounded, only rear feet grounded, left front and right rear feet grounded, and right front and left rear feet grounded. In the first configuration and with an exposure field strength of 10 kV/m, average total current densities induced in the torsos of pigs and rats were 34 nA/cm2 and 20 nA/cm2, respectively. The corresponding value for human exposure is about 250 nA/cm2, 7.3 and 12.5 times larger than for swine and rats, respectively. Current densities measured at 60 Hz can be linearly extrapolated to frequencies in a range extending from at least 1 Hz to 1 MHz. Human and animal current-density data can provide an improved rationale for extrapolating biological data across species. In addition, these data can be used to validate the predictions of numerical models.     

RNA stability under different combinations of amber force fields and solvation models

The proper matching of force field and solvent is critical to obtain correct result in molecular dynamics simulation of bio-molecules. This problem has been intensively investigated for protein but not for RNA yet. In this paper, we use standard molecular dynamics and replica exchange molecular dynamics to take a series of tests on the RNA stability under different combinations of Amber force field parameters (ff98, ff99 and ff99bsc0) and the general Born implicit solvent models (igb1, igb2 and igb5). It is found that only ff98 and ff99bsc0 with igb1 can keep the native conformations of RNA hairpin and duplex. Our results suggest that ff98 plus igb1 may be reasonable choice for molecular dynamics simulation of RNA dynamics.     

Electric fields induced in rat and human models by 60-Hz magnetic fields: comparison of calculated and measured values.

The calculated distribution of electric fields induced in homogeneous human and rat models by a 60-Hz magnetic field is compared with values measured in instrumented mannequins. The calculated values agree well with measured values.     

From basis functions to basis fields: vector field approximation from sparse data

Recent investigations (Poggio and Girosi 1990b) have pointed out the equivalence between a wide class of learning problems and the reconstruction of a real-valued function from a sparse set of data. However, in order to process sensory information and to generate purposeful actions living organisms must deal not only with real-valued functions but also with vector-valued mappings. Examples of such vector-valued mappings range from the optical flow fields associated with visual motion to the fields of mechanical forces produced by neuromuscular activation. In this paper, I discuss the issue of vector-field processing from a broad computational perspective. A variety of vector patterns can be efficiently represented by a combination of linearly independent vector fields that I call basis fields. Basis fields offer in some cases a better alternative to treating each component of a vector as an independent scalar entity. In spite of its apparent simplicity, such a component-based representation is bound to change with any change of coordinates. In contrast, vector-valued primitives such as basis fields generate vector field representations that are invariant under coordinate transformations.     

Current densities measured in human models exposed to 60-Hz electric fields

This paper gives current densities measured in homogeneous grounded human models exposed to vertical, 60-Hz electric fields. The methods used for these measurements were validated by measuring the current densities induced in a grounded hemisphere and in a grounded prolate hemispheroid; agreement between measurement and theory was good. For an unperturbed field strength of 10 kV/m, current densities measured in the human chest were in the range 125-300 nA/cm2. A strong horizontal current-density enhancement was observed in the axillae, with peak values of about 400 nA/cm2. The vertical current density in the arms, when held downward, was in the opposite direction to that in the chest. Current densities in the abdomen, pelvis, and legs were a strong function of whether the body was grounded through one or both feet. With one foot grounded, the horizontal current density in the lower pelvic region, just above the crotch, was 770 nA/cm2. This value was the largest of those measured in the head, arms, or torso of the human model. Scaling factors derived from these data and similar data for animals will provide a quantitative basis for comparing animal and human exposure to 60-Hz electric fields. In addition, current-density data given in this paper can be directly extrapolated to higher frequencies, at least to 1 MHz. These extrapolated data may be useful to individuals and groups involved in the determination of safety standards for the lower radiofrequency region.