Accurate Path Integration in Continuous Attractor Network Models of Grid Cells

Grid cells in the rat entorhinal cortex display strikingly regular firing responses to the animal''s position in 2-D space and have been hypothesized to form the neural substrate for dead-reckoning. However, errors accumulate rapidly when velocity inputs are integrated in existing models of grid cell activity. To produce grid-cell-like responses, these models would require frequent resets triggered by external sensory cues. Such inadequacies, shared by various models, cast doubt on the dead-reckoning potential of the grid cell system. Here we focus on the question of accurate path integration, specifically in continuous attractor models of grid cell activity. We show, in contrast to previous models, that continuous attractor models can generate regular triangular grid responses, based on inputs that encode only the rat''s velocity and heading direction. We consider the role of the network boundary in the integration performance of the network and show that both periodic and aperiodic networks are capable of accurate path integration, despite important differences in their attractor manifolds. We quantify the rate at which errors in the velocity integration accumulate as a function of network size and intrinsic noise within the network. With a plausible range of parameters and the inclusion of spike variability, our model networks can accurately integrate velocity inputs over a maximum of ∼10–100 meters and ∼1–10 minutes. These findings form a proof-of-concept that continuous attractor dynamics may underlie velocity integration in the dorsolateral medial entorhinal cortex. The simulations also generate pertinent upper bounds on the accuracy of integration that may be achieved by continuous attractor dynamics in the grid cell network. We suggest experiments to test the continuous attractor model and differentiate it from models in which single cells establish their responses independently of each other.     

Evaluation of the Oscillatory Interference Model of Grid Cell Firing through Analysis and Measured Period Variance of Some Biological Oscillators

Models of the hexagonally arrayed spatial activity pattern of grid cell firing in the literature generally fall into two main categories: continuous attractor models or oscillatory interference models. Burak and Fiete (2009, PLoS Comput Biol) recently examined noise in two continuous attractor models, but did not consider oscillatory interference models in detail. Here we analyze an oscillatory interference model to examine the effects of noise on its stability and spatial firing properties. We show analytically that the square of the drift in encoded position due to noise is proportional to time and inversely proportional to the number of oscillators. We also show there is a relatively fixed breakdown point, independent of many parameters of the model, past which noise overwhelms the spatial signal. Based on this result, we show that a pair of oscillators are expected to maintain a stable grid for approximately t = 5µ ³ /(4πσ) ² seconds where µ is the mean period of an oscillator in seconds and σ² its variance in seconds². We apply this criterion to recordings of individual persistent spiking neurons in postsubiculum (dorsal presubiculum) and layers III and V of entorhinal cortex, to subthreshold membrane potential oscillation recordings in layer II stellate cells of medial entorhinal cortex and to values from the literature regarding medial septum theta bursting cells. All oscillators examined have expected stability times far below those seen in experimental recordings of grid cells, suggesting the examined biological oscillators are unfit as a substrate for current implementations of oscillatory interference models. However, oscillatory interference models can tolerate small amounts of noise, suggesting the utility of circuit level effects which might reduce oscillator variability. Further implications for grid cell models are discussed.     

pH Effect of the Sphingomyelin Membrane Interfacial Tension

The effect of pH on the interfacial tension of a sphingomyelin membrane in aqueous solution has been studied. Three models describing H⁺ and OH⁻ ion adsorption on the bilayer lipid surface are presented. In models I and II, the membrane surface is continuous, with uniformly distributed functional groups as centers of H⁺ and OH⁻ ion adsorption. In model III, the membrane surface is composed of lipid molecules, with and without adsorbed H⁺ and OH⁻ ions. The contribution of each individual lipid molecule to the overall interfacial tension of the bilayer was assumed to be additive in models I and II. In model III, the Gibbs isotherm was used to describe adsorption of H⁺ and OH⁻ ions at the bilayer surface. Theoretical equations are derived to describe the interfacial tension as a function of pH for all three models. Maximum interfacial tension was observed experimentally at the isoelectric point.     

The effect of dispersal on single-species nonautonomous dispersal models with delays

In this paper, single-species nonautonomous dispersal models with delays are considered. An interesting result on the effect of dispersal for persistence and extinction is obtained. That is, if the species is persistent in a patch then it is also persistent in all other patches; if the species is permanent in a patch then it is also permanent in all other patches; if the species is extinct in a patch then it is also extinct in all other patches. Furthermore, some new sufficient conditions for the permanence and extinction of the species in a patch are established. The existence of positive periodic solutions is obtained in the periodic case by employing Teng and Chen's results on the existence of positive periodic solutions for functional differential equations. Received: 26 June 2000 / Revised version: 6 October 2000 / Published online: 10 April 2001     

Continuous integrated antibody precipitation with two-stage tangential flow microfiltration enables constant mass flow

Continuous precipitation is a new unit operation for the continuous capture of antibodies. The capture step is based on continuous precipitation with PEG6000 and Zn⁺⁺ in a tubular reactor integrated with a two-stage continuous tangential flow filtration unit. The precipitate cannot be separated with centrifugation, because a highly compressed sediment results in poor resolubilization. We developed a new two-stage tangential flow microfiltration method, where part of the concentrated retentate of the first stage was directly fed to the second stage, together with the wash buffer. Thus, the precipitate was concentrated and washed in a continuous process. We obtained 97% antibody purity, a 95% process yield during continuous operation, and a fivefold reduction in pre-existing high-molecular-weight impurities. For other unit operations, surge tanks are often required, due to interruptions in the product mass flow out of the unit operation (e.g., the bind/elute mode in periodic counter-current chromatography). Our setup required no surge tanks; thus, it provided a truly continuous antibody capture operation with uninterrupted product mass flow. Continuous virus inactivation and other flow-through unit operations can be readily integrated downstream of the capture step to create truly continuous, integrated, downstream antibody processing without the need for hold tanks.     

Dynamical effects of nonlocal interactions in discrete-time growth-dispersal models with logistic-type nonlinearities

The paper is devoted to the study of discrete time and continuous space models with nonlocal resource competition and periodic boundary conditions. We consider generalizations of logistic and Ricker's equations as intraspecific resource competition models with symmetric nonlocal dispersal and interaction terms. Both interaction and dispersal are modeled using convolution integrals, each of which has a parameter describing the range of nonlocality. It is shown that the spatially homogeneous equilibrium of these models becomes unstable for some kernel functions and parameter values by performing a linear stability analysis. To be able to further analyze the behavior of solutions to the models near the stability boundary, weakly nonlinear analysis, a well-known method for continuous time systems, is employed. We obtain Stuart–Landau type equations and give their parameters in terms of Fourier transforms of the kernels. This analysis allows us to study the change in amplitudes of the solutions with respect to ranges of nonlocalities of two symmetric kernel functions. Our calculations indicate that supercritical bifurcations occur near stability boundary for uniform kernel functions. We also verify these results numerically for both models.     

具时滞的n斑块捕食-食饵扩散系统的正周期解

讨论了具时滞和功能反应的n斑块捕食—食饵扩散系统,利用新的方法,得到了该系统正周期解存在性的判别准则．     

Species richness, area and climate correlates

Aim Species richness–area theory predicts that more species should be found if one samples a larger area. To avoid biases from comparing species richness in areas of very different sizes, area is often controlled by counting the numbers of co‐occupying species in near‐equal area grid cells. The assumption is that variation in grid cell size accrued from working in a three‐dimensional world is negligible. Here we provide a first test of this idea. We measure the surface area of c. 50 × 50 km and c. 220 × 220 km grid cells across western Europe. We then ask how variation in the area of grid cells affects: (1) the selection of climate variables entering a species richness model; and (2) the accuracy of models in predicting species richness in unsampled grid cells. Location Western Europe. Methods Models are developed for European plant, breeding bird, mammal and herptile species richness using seven climate variables. Generalized additive models are used to relate species richness, climate and area. Results We found that variation in the grid cell area was large (50 × 50 km: 8–3311 km²; 220 × 220: 193–55,100 km²), but this did not affect the selection of variables in the models. Similarly, the predictive accuracy was affected only marginally by exclusion of area within models developed at the c. 50 × 50 km grid cells, although predictive accuracy suffered greater reductions when area was not included as a covariate in models developed for c. 220 × 220 km grid cells. Main conclusions Our results support the assumption that variation in near‐equal area cells may be of second‐order importance for models explaining or predicting species richness in relation to climate, although there is a possibility that drops in accuracy might increase with grid cell size. The results are, however, contingent on this particular data set, grain and extent of the analyses, and more empirical work is required.     

SPIRE—a software tool for bicontinuous phase recognition: application for plastid cubic membranes

Bicontinuous membranes in cell organelles epitomize nature’s ability to create complex functional nanostructures. Like their synthetic counterparts, these membranes are characterized by continuous membrane sheets draped onto topologically complex saddle-shaped surfaces with a periodic network-like structure. Their structure sizes, (around 50–500 nm), and fluid nature make transmission electron microscopy (TEM) the analysis method of choice to decipher their nanostructural features. Here we present a tool, Surface Projection Image Recognition Environment (SPIRE), to identify bicontinuous structures from TEM sections through interactive identification by comparison to mathematical “nodal surface” models. The prolamellar body (PLB) of plant etioplasts is a bicontinuous membrane structure with a key physiological role in chloroplast biogenesis. However, the determination of its spatial structural features has been held back by the lack of tools enabling the identification and quantitative analysis of symmetric membrane conformations. Using our SPIRE tool, we achieved a robust identification of the bicontinuous diamond surface as the dominant PLB geometry in angiosperm etioplasts in contrast to earlier long-standing assertions in the literature. Our data also provide insights into membrane storage capacities of PLBs with different volume proportions and hint at the limited role of a plastid ribosome localization directly inside the PLB grid for its proper functioning. This represents an important step in understanding their as yet elusive structure–function relationship.

The SPIRE tool facilitates recognition of bicontinuous phases on transmission electron micrographs, which enables in-depth spatial prolamellar body structure analysis from 2D micrographs.     

Computational models of grid cells

Grid cells are space-modulated neurons with periodic firing fields. In moving animals, the multiple firing fields of an individual grid cell form a triangular pattern tiling the entire space available to the animal. Collectively, grid cells are thought to provide a context-independent metric representation of the local environment. Since the discovery of grid cells in 2005, a number of models have been proposed to explain the formation of spatially repetitive firing patterns as well as the conversion of these signals to place signals one synapse downstream in the hippocampus. The present article reviews the most recent developments in our understanding of how grid patterns are generated, maintained, and transformed, with particular emphasis on second-generation computational models that have emerged during the past 2-3 years in response to criticism and new data.     

SIR-SVS epidemic models with continuous and impulsive vaccination strategies

Vaccination is important for the control of some infectious diseases. This paper considers two SIR-SVS epidemic models with vaccination, where it is assumed that the vaccination for the newborns is continuous in the two models, and that the vaccination for the susceptible individuals is continuous and impulsive, respectively. The basic reproduction numbers of two models, determining whether the disease dies out or persists eventually, are all obtained. For the model with continuous vaccination for the susceptibles, the global stability is proved by using the Lyapunov function. Especially for the endemic equilibrium, to prove the negative definiteness of the derivative of the Lyapunov function for all the feasible values of parameters, it is expressed in three different forms for all the feasible values of parameters. For the model with pulse vaccination for the susceptibles, the global stability of the disease free periodic solution is proved by the comparison theorem of impulsive differential equations. At last, the effect of vaccination strategies on the control of the disease transmission is discussed, and two types of vaccination strategies for the susceptible individuals are also compared.     

A real-time malleable molecular surface

We describe a method for generating a molecular surface using a parametric patch representation. Unlike previous methods, this algorithm generates a parametric patch surface which is smooth and G¹ continuous and manipulable in real-time. Crucial to our approach is the creation of a net of approximately equilateral triangles from which we generate the control points used as the basis for describing the surface. We present in detail the method used for generating the triangular net and accompanying control points, along with examples of the resulting surfaces.     

A novel method for the observation of membrane transporter dynamics.

A new method is proposed for measuring the dynamic properties of a membrane transporter by means of steady-state fluxes. Any voltage-sensitive transporter will give a flow of substrate in the presence of a steady-state periodic membrane potential. The periodic steady-state flow, averaged over one period, is a flux that can be measured by traditional steady-state techniques, such as the radioactive tracer method. The average flux, solely due to the periodic field, is described by a set of Lorentzian functions that depend on the applied periodic field amplitude and frequency. The normal mode amplitudes and frequencies of these Lorentzians are model-independent parameters of the transport mechanism. Measurement of the average flux as a function of the applied periodic frequency permits determination of system relaxation times as the reciprocals of the midpoints of the Lorentzian curves, which in turn can be used to estimate individual rate constants of specific models. It was found by simulation of a six-state model of the electrogenic Na+/glucose cotransporter, using published estimates of the model rate constants, that the periodic field effects can be large and rich with measurable details that can be used to study the mechanism thoroughly. The new method serves in this case to complement and expand on the information obtainable by means of the voltage clamp method. It was also found by means of simulations of a nonelectrogenic six-state cotransporter model that experimentally measurable effects are expected and that results can be used to distinguished among alternative kinetic models as well as to estimate individual rate constants.(ABSTRACT TRUNCATED AT 250 WORDS)     

A New Fluorescence Method for the Continuous Determination of Surface Lipid Oxidation in Lipoproteins and Plasma

We report on a new method for the determination of lipid oxidation in lipoproteins and plasma. The biological lipid system is preloaded with a fluorescent analog of phosphatidylcholine containing diphenylhexatriene (DPH) propionic acid covalently linked to the sn-2 position. When externally added, the respective phospholipid label (DPHPC) localizes to the surface monolayer of a lipoprotein. Under oxidative conditions (e.g. in the presence of Cu²⁺ ions) the fluorophore undergoes decomposition, resulting in a continuous decrease of fluorescence intensity which reflects the oxidation of a chemically defined phospholipid molecule with well defined localization. When incorporated into LDL particles, the kinetics of the decrease in DPHPC fluorescence intensity upon exposure to Cu²⁺ is very similar to that of conjugated diene accumulation. Furthermore, our assay can be applied to follow the oxidation of lipids in diluted serum and may also be developed into a suitable test system for clinical studies of susceptibility of plasma lipids to oxidation.     

A differentiable,periodic function for pulsatile cardiac output based on heart rate and stroke volume

Many mathematical models of human hemodynamics, particularly those which describe pressure and flow pulses throughout the circulatory system, require as specified input a modeling function which describes cardiac output in terms of volume per unit time. To be realistic, this cardiac output function should capture, to the greatest extent possible, all relevant features observed in measured physical data. For model analysis purposes, it is also highly desirable to have a model function that is continuous, differentiable, and periodic. This paper addresses both classes of needs by developing such a function. Physically, the present function provides an accurate model for flow into the ascending aorta. It is completely specified by a minimal number of standard input parameters associated with left ventricle dynamics, including heart rate, mean cardiac output, and an estimation of the peak-to-mean flow ratio. Analytically, it can be expressed as a product of two continuous, differentiable and periodic factors. Further, the Fourier expansion of this model function is shown to be a finite Fourier series, and explicit closed-form expressions are given for the non-zero coefficients in this series.     

Use of optical spectroscopic N15-analyses to trace a specific nitrogen application in tobacco plants

Summary The pathway of a specific nitrogen application to tobacco plants was traced with N¹⁵. The spectroscopic N¹⁵-analysis was found to be a reliable, fast, and convenient method. Total-N concentration always increases from bottom to top leaves. From a specific application, nitrogen will preferentially move into growing tissues, but even there a substantial fraction of N originates from previous applications. As time proceeds, the label decreases in new growth even without an additional N-supply. Noticeable amounts of labeled N will also move into old, even senescent leaves which actually show a net loss of N, and even in such leaves, N will be incorporated into proteins at a fast rate. It is concluded that there is a continuous influx and efflux of N and a continuous turnover in all living parts of the plants including old, yellow leaves.The author wishes to thank Prof. Dr. G. Michael, Director, Institute of Plant Nutrition, Stuttgart-Hohenheim for helpful discussions and for generous support of the project.     

Highly Sensitive Plasmonic Sensor Based on Fano Resonance from Silver Nanoparticle Heterodimer Array on a Thin Silver Film

We investigate the optical spectrum of a multilayer metallic slab using multiple-scattering formalism. A thin silver film is attached to a periodic array of heterodimers consisting of two vertically spaced silver nanoparticles of different radii. Depending on the radius of nanoparticles, heterodimer array presents a simple nanoscale geometry which gives rise to remarkable plasmonic properties of multipolar resonances. Due to the coherent interference of the localized nanoparticle plasmons (discrete mode) and surface plasmon polaritons of metallic film (continuous mode), the reflection spectrum represents a sharp asymmetric Fano resonance dip, which is strongly sensitive to the refractive index of the surrounding embedded dielectric host. The physical features contribute to a highly efficient plasmonic sensor for refractive index sensing with sensitivity of ～1.5?×?10^?3 RIU/nm.     

Static and dynamic behavior of a class of unstructured models of continuous bioreactors with growth associated product

The static and dynamic behavior of a class of unstructured models of continuous bioprocesses, for which the product is growth associated, are analyzed using elementary concepts of singularity theory and continuation techniques. The class consists of models for which both the rates of utilization of limiting substrate and product formation are linearly proportional to the specific cell growth rate. The kinetic expressions are allowed to assume general forms of substrate and nonbiomass product. The steady-state analysis allows the derivation of analytical results and the construction of a useful picture in the models' parameter space delineating the different static behavior these models can predict, including unique steady states and bistability. The analysis of the dynamic behavior allows the derivation of general analytical conditions for the occurrence of periodic behavior in the models. It is also shown that the subclass of these models for which the specific cell growth rate expression is monotonic with respect to the nonbiomass product is unable to predict a stable oscillatory behavior regardless of the expression of the growth rate. These results illustrate the fundamental weakness of this class of unstructured models in predicting transient behavior in continuous cultures. The effect of kinetic and operating parameters on the stability characteristics of these models is also investigated.     

A simple and highly sensitive method for sequence determination of 32P-labeled oligonucleotides

A new procedure is described for the sequence determination of oligonucleotides produced by digestion of RNA with pancreatic RNase A. The oligonucleotide is treated with spleen exonuclease and all intermediates are resolved by thin-layer chromatography on polyethyleneimine plates. On the basis of the increase in mobility it can be decided for each successive step whether a Gp- or an Ap-residue has been removed by reference to a calibration grid. The method is very simple and can easily be applied to a large number of samples. An amount of ³²P-radioactivity corresponding to 40 dpm/nucleotide is sufficient for analysis.     

The identification of the periodic behaviour in an epidemic model

In this paper we study a method for the identification of the unknown parameter of the periodic function and also the first component of the state vector, in a mathematical model which describes the evolution of some diseases with an oro-fecal transmission.To solve the identification problem we use a numerical method to integrate the differential equations system, which reproduces the stability properties of the above mentioned continuous system.The numerical methods which we propose can be applied also to a spatial semi discretization of the reaction-diffusion model which is a diffusive generalization of the system that we consider in this paper.Finally, through an analysis on both the continuous and the discrete system we also obtain a necessary condition on the experimental data in order that a periodic trajectory of the system exists.Work supported by: Progetto Finalizzato Controllo Malattie da Infezione-CNR and by Ministero Pubblica Istruzione