Effects of correlated and synchronized stochastic inputs to leaky integrator neuronal model

We present an analysis of neuronal model behaviour with correlated synaptic inputs including the cases that correlated inputs are equivalent to exactly synchronized inputs and correlated inputs are not equivalent to exactly synchronized inputs. For the former case, it is found that the fully (synaptically) correlated inputs assumption (see Section 1 for definition), which is used in most, if not all, theoretical and experimental work in the past few years, results in a waste of resources and might be an unrealistic assumption; with an exactly balanced excitatory and inhibitory, and synaptically correlated input, the integrate-and-fire model simply behaves as a synchrony detector in certain parameter regions; the well-known diffusion model, upon which most theoretical work is based, fails to approximate the model with synaptically correlated Poisson inputs. A novel way to approximate synaptically correlated Poisson inputs is then presented;an optimization principle on neuronal models with partially (synaptically) correlated inputs is proposed, which enables us to predict microscopic structures in neuronal systems. For the latter case,with tightly synchronized inputs (see Section 1 for definition), the model behaviour depends on its integration time of input signals and could exhibit bursting discharge.for loosely synchronized inputs, we found that correlated inputs are equivalent to the post-spike voltage reset mechanism proposed in the literature.     

A minimal mechanistic model for temporal signal processing in the lateral geniculate nucleus

The receptive fields of cells in the lateral geniculate nucleus (LGN) are shaped by their diverse set of impinging inputs: feedforward synaptic inputs stemming from retina, and feedback inputs stemming from the visual cortex and the thalamic reticular nucleus. To probe the possible roles of these feedforward and feedback inputs in shaping the temporal receptive-field structure of LGN relay cells, we here present and investigate a minimal mechanistic firing-rate model tailored to elucidate their disparate features. The model for LGN relay ON cells includes feedforward excitation and inhibition (via interneurons) from retinal ON cells and excitatory and inhibitory (via thalamic reticular nucleus cells and interneurons) feedback from cortical ON and OFF cells. From a general firing-rate model formulated in terms of Volterra integral equations, we derive a single delay differential equation with absolute delay governing the dynamics of the system. A freely available and easy-to-use GUI-based MATLAB version of this minimal mechanistic LGN circuit model is provided. We particularly investigate the LGN relay-cell impulse response and find through thorough explorations of the model’s parameter space that both purely feedforward models and feedback models with feedforward excitation only, can account quantitatively for previously reported experimental results. We find, however, that the purely feedforward model predicts two impulse response measures, the time to first peak and the biphasic index (measuring the relative weight of the rebound phase) to be anticorrelated. In contrast, the models with feedback predict different correlations between these two measures. This suggests an experimental test assessing the relative importance of feedforward and feedback connections in shaping the impulse response of LGN relay cells.     

On-line identification of sensory systems using pseudorandom binary noise perturbations.

A technique of on-line identification of linear system characteristics from sensory systems with spike train or analog voltage outputs was developed and applied to the semicircular canal. A pseudorandom binary white noise input was cross-correlated with the system's output to produce estimates of linear system unit impulse responses (UIRs), which were then corrected for response errors of the input transducers. The effects of variability in the system response characteristics and sensitivity were studied by employing the technique with known linear analog circuits. First-order unit afferent responses from the guitarfish horizontal semicircular canal were cross-correlated with white noise rotational acceleration inputs to produce non-parametric UIR models. In addition, the UIRs were fitted by nonlinear regression to truncated exponential series to produce parametric models in the form of low-order linear system equations. The experimental responses to the white noise input were then compared with those predicted from the UIR models linear convolution, and the differences were expressed as a percent mean-square-error (%MSE). The average difference found from a population of 62 semicircular canal afferents was relatively low mean and standard deviation of 10.2 +/- 5.9 SD%MSE, respectively. This suggests that relatively accurate inferences can be made concerning the physiology of the semicircular canal from the linear characteristics of afferent responses.     

Applications of a parametric model for informative censoring

In this article, we explore the use of a parametric model (for analyzing survival data) which is defined to allow sensitivity analysis for the presence of informative censoring. The dependence between the failure and the censoring processes is expressed through a parameter delta and a general bias function B(t, theta). We calculate the expectation of the potential bias due to informative censoring, which is an overall measure of how misleading our results might be if censoring is actually nonignorable. Bounds are also calculated for quantities of interest, e.g., parameter of the distribution of the failure process, which do not depend on the choice of the bias function for fixed delta. An application that relates to systematic lupus erythematosus data illustrates how additional information can result in reducing the uncertainty on estimates of the location parameter. Sensitivity analysis on a relative risk parameter is also explored.     

Rate of excitation propagation in a reduced Hodgkins-Huxley model. I. Rapid relaxation of the sodium current]

The question of calculating excitation propagation velocity is analyzed on the basis of the Hodgkin-Huxley model. The activation of the sodium current is assumed to be rapid as compared to the rate of potential variation. Because of slow variation of potassium activation and sodium inactivation the dynamics of these processes is assumed to be of negligible effect in the region of impulse velocity formation. By means of pieace-wise linear approximation of thus obtained voltage-current characteristics the characteristics the analytical solution of the problem was found. In two limiting cases this solution coincides with the solutions of Kolmogorov and Scott. The dependence of impulse velocity on parameters is analyzed and illustrated graphically.     

Dynamics of chromaticity horizontal cells in the freshwater turtle retina

A model of the cone-horizontal cell circuit is presented based on morphological evidence recently found in the Reeves' turtle: a luminosity horizontal cell (LHC) that receives inputs from red-, green-, and blue-sensitive cones in the ratio of 15:3:1, a triphasic horizontal cell (THC) that receives inputs from one class of red-sensitive and from blue-sensitive cones in the ratio of 2:1; and a biphasic chromaticity horizontal cell (BHC) that receives inputs from green-sensitive cones as well as from a special class of red-sensitive (i.e. the broad spectrum) and from blue-sensitive cones in the ratio of 3:2:1. A study of the simulated impulse responses strongly suggests that the basic response patterns of the BHC and THC can be readily explained by a simple wiring diagram consisting of direct hyper-polarizing inputs from the appropriate cones and a depolarizing input from the LHC which acts as a voltage inverter. A negative feedback circuit from the LHC to the cone pedicles is included and its negative feedback gain increases as the mean illuminance level (Io) increases. The negative feedback circuit, which promotes adaptation in the cones to changing Io's, is not necessary for opponent polarization in the BHC or THC, but does explain variabilities of impulse responses.     

Nonequilibrium voltage fluctuations in biological membranes. II. Voltage and current noise generated by ion carriers, channels and electrogenic pumps

As applications of the general theoretical framework of charge transport in biological membranes and related voltage and current noise, a number of model calculations are presented for ion carriers, rigid channels, channels with conformational substates and electrogenic pumps. The results are discussed with special reference to the problem of threshold values for sensory transduction processes and their limitations by voltage fluctuations. Furthermore, starting from the special results of model calculations, an attempt is made to determine more general aspects of electric fluctuations generated by charge-transport processes in biological membranes: different frequency dependences of voltage and current noise, and dependence of noise intensities with increasing distance from the equilibrium state.     

Scaling phloem transport: water potential equilibrium and osmoregulatory flow

In this work, the common assumption that phloem sap is in water potential equilibrium with the surrounding apoplast was examined. With a dimensionless model of phloem translocation that scales with just two dimensionless parameters (R?and F?), a ‘map’ of phloem behaviour as a function of these parameters was produced, which shows that the water potential equilibrium assumption (R?F? >> 1) is valid for essentially all realistic values of the relevant scales. When in water potential equilibrium, a further parameter reduction is possible that limits model dependence to a single parameter (F?), which describes the ratio of the solution's osmotic strength to its axial pressure drop. Due to the locally autonomous nature of individual sieve element/companion cell complexes, it is argued that long‐distance integrative control is most efficient when F? is large (that is, when the pressure drop is relatively small), permitting the sieve tube to regulate solute loading in response to global changes in turgor. This mode of transport has been called ‘osmoregulatory flow.’ Limitations on the pressure drop within the transport phloem could require that sieve tubes be shorter than the long axis of the plant, and thus arranged in series and hydraulically isolated from one another.     

Dependence Calibration in Conditional Copulas: A Nonparametric Approach

Summary The study of dependence between random variables is a mainstay in statistics. In many cases, the strength of dependence between two or more random variables varies according to the values of a measured covariate. We propose inference for this type of variation using a conditional copula model where the copula function belongs to a parametric copula family and the copula parameter varies with the covariate. In order to estimate the functional relationship between the copula parameter and the covariate, we propose a nonparametric approach based on local likelihood. Of importance is also the choice of the copula family that best represents a given set of data. The proposed framework naturally leads to a novel copula selection method based on cross‐validated prediction errors. We derive the asymptotic bias and variance of the resulting local polynomial estimator, and outline how to construct pointwise confidence intervals. The finite‐sample performance of our method is investigated using simulation studies and is illustrated using a subset of the Matched Multiple Birth data.     

A parametric approach to numerical modeling of TKR contact forces

In vivo knee contact forces are difficult to determine using numerical methods because there are more unknown forces than equilibrium equations available. We developed parametric methods for computing contact forces across the knee joint during the stance phase of level walking. Three-dimensional contact forces were calculated at two points of contact between the tibia and the femur, one on the lateral aspect of the tibial plateau, and one on the medial side. Muscle activations were parametrically varied over their physiologic range resulting in a solution space of contact forces. The obtained solution space was reasonably small and the resulting force pattern compared well to a previous model from the literature for kinematics and external kinetics from the same patient. Peak forces of the parametric model and the previous model were similar for the first half of the stance phase, but differed for the second half. The previous model did not take into account the transverse external moment about the knee and could not calculate muscle activation levels. Ultimately, the parametric model will result in more accurate contact force inputs for total knee simulators, as current inputs are not generally based on kinematics and kinetics inputs from TKR patients.     

A nonlinear numerical model of percutaneous drug absorption.

A nonlinear mathematical model developed by Chandrasekaran et al. is examined to monitor pharmacokinetic profiles in percutaneous drug absorption and is addressed to several associated problems that could occur in the data analysis of in vitro experiments. The formulation of the model gives rise to a nonlinear partial differential equation (PDE) of parabolic type, and a family of finite-difference methods is developed for the numerical solution of the associated initial/boundary-value problem. The value given to a parameter in this family determines the stability properties of the resulting method and whether the solution is obtained explicitly or implicitly. In the case of implicit members of the family it is seen that the solution of the nonlinear PDE is obtained by solving a linear algebraic system, the coefficient matrix of which is tridiagonal. The behaviors of two methods of the family are examined in a series of numerical experiments. Numerical differentiation and integration procedures are combined to monitor the cumulative amount of drug eliminated into the receptor cell per unit area as time increases. It is found that the use of the equation for the simple membrane model to estimate the permeability coefficient and lag time is warranted even if the system should be described by the dual-sorption model, provided cumulative amount versus time data collected for a sufficiently long time are used. However, being different from the behavior in the simple membrane model, the lag time, which can be estimated in this way, is dose-dependent and decreases with increasing donor cell concentration. On the other hand, the permeability coefficient in the dual-sorption model remains constant irrespective of the donor cell concentrations as in the simple membrane model.     

Microscopic description of voltage effects on ion-driven cotransport systems

A microscopic model for the analysis of voltage effects on ion-driven cotransport systems is described. The model is based on the notion that the voltage dependence of a given rate constant is directly related to the amount of charge which is translocated in the corresponding reaction step. Charge translocation may result from the movement of an ion along the transport pathway, from the displacement of charged ligand groups of the ion-binding site, or from reorientation of polar residues of the protein in the course of a conformational transition. The voltage dependence of overall transport rate is described by a set of dimensionless coefficients reflecting the dielectric distances over which charge is displaced in the elementary reaction steps. The dielectric coefficients may be evaluated from the shape of the experimental flux-voltage curve if sufficient information on the rate constants of the reaction cycle is available. Examples of flux-voltage curves which are obtained by numerical simulation of the transport model are given for a number of limiting cases.     

Structural stability,alternate descriptions and information

The inactivation properties of a model of the nerve membrane are examined. The inactivation kinetics are closely first order and may be characterized by Hodgkin-Huxley (H-H) parameters h_∞ and τ_h which depend on potential in agreement with experiments. Some differences from the H-H equations are identified. The forms predicted for τ_h variation with hyper-polarization and change of external [K⁺] agree with available data. While the inactivation time delay predicted by the model is too small to be detected experimentally, there are grounds for expecting that it may be larger in other tissues, as observed in Myxicola giant axons. The variation of the delay with test potential is predicted to be exponential. Although the model is coupled in the sense defined by Hoyt, it gives rise to an inactivation shift of negligible magnitude. However, introducing a simple variability in one physical parameter leads to the observed form of both the peak transient current voltage relation and the inactivation shift. Inactivation shift thus does not unambiguously indicate coupling; that it results from parametric heterogeneity may be a better hypothesis, and is readily testable. The inactivation shift dependence on current ratio, from experimental data, can be used to correct for the effects of parametric heterogeneity and obtain the value of a previously predicted fundamental parameter of excitable membranes. It is suggested that the effects of parametric heterogeneity must be considered in interpreting experiments and designing models for excitable systems.     

Calculation of NMR relaxation, covolume, and scattering-related properties of bead models using the SOLPRO computer program

The hydrodynamic properties of macromolecules and bioparticles, represented by bead models, can be calculated using methods implemented in the computer routine HYDRO. Recently, a new computer routine, SOLPRO, has been presented for the calculation of various SOLution PROperties. These include (1) time-dependent electro-optic and spectroscopic properties related to rotational diffusion, (2) non-dynamic properties like scattering curves, and (3) dimensionless quantities that combine two or more solution properties in a form which depends on the shape of the macromolecule but not on its size. In the present work we describe the inclusion of more of those types of properties in a new version of SOLPRO. Particularly, we describe the calculation of relaxation rates in nuclear magnetic resonance (NMR). For dipolar coupling, given the direction of the dipole the program calculates values of the spectral density, from which the NMR relaxation times can be obtained. We also consider scattering-related properties, namely the distribution of distances for the bead model, which is directly related to the angular dependence of scattered intensity, and the particle's longest distance. We have devised and programmed a procedure to calculate the covolume of the bead model, related to the second virial coefficient and, in general, to the concentration dependence of solution properties. Various shape-dependent dimensionless quantities involving the covolume are calculated. In this paper we also discuss some aspects, namely bead overlapping and hydration, that are not explicitely included in SOLPRO, but should be considered by the user. Received: 25 May 1998 / Revised version: 30 July 1998 / Accepted: 30 July 1998     

Sensitivity analysis for informative censoring in parametric survival models

Most statistical methods for censored survival data assume there is no dependence between the lifetime and censoring mechanisms, an assumption which is often doubtful in practice. In this paper we study a parametric model which allows for dependence in terms of a parameter delta and a bias function B(t, theta). We propose a sensitivity analysis on the estimate of the parameter of interest for small values of delta. This parameter measures the dependence between the lifetime and the censoring mechanisms. Its size can be interpreted in terms of a correlation coefficient between the two mechanisms. A medical example suggests that even a small degree of dependence between the failure and censoring processes can have a noticeable effect on the analysis.     

Phenomenological Incorporation of Nonlinear Dendritic Integration Using Integrate-and-Fire Neuronal Frameworks

It has been discovered recently in experiments that the dendritic integration of excitatory glutamatergic inputs and inhibitory GABAergic inputs in hippocampus CA1 pyramidal neurons obeys a simple arithmetic rule as , where , and are the respective voltage values of the summed somatic potential, the excitatory postsynaptic potential (EPSP) and the inhibitory postsynaptic potential measured at the time when the EPSP reaches its peak value. Moreover, the shunting coefficient in this rule only depends on the spatial location but not the amplitude of the excitatory or inhibitory input on the dendrite. In this work, we address the theoretical issue of how much the above dendritic integration rule can be accounted for using subthreshold membrane potential dynamics in the soma as characterized by the conductance-based integrate-and-fire (I&F) model. Then, we propose a simple I&F neuron model that incorporates the spatial dependence of the shunting coefficient by a phenomenological parametrization. Our analytical and numerical results show that this dendritic-integration-rule-based I&F (DIF) model is able to capture many experimental observations and it also yields predictions that can be used to verify the validity of the DIF model experimentally. In addition, the DIF model incorporates the dendritic integration effects dynamically and is applicable to more general situations than those in experiments in which excitatory and inhibitory inputs occur simultaneously in time. Finally, we generalize the DIF neuronal model to incorporate multiple inputs and obtain a similar dendritic integration rule that is consistent with the results obtained by using a realistic neuronal model with multiple compartments. This generalized DIF model can potentially be used to study network dynamics that may involve effects arising from dendritic integrations.     

Scaling for Dynamical Systems in Biology

Asymptotic methods can greatly simplify the analysis of all but the simplest mathematical models and should therefore be commonplace in such biological areas as ecology and epidemiology. One essential difficulty that limits their use is that they can only be applied to a suitably scaled dimensionless version of the original dimensional model. Many books discuss nondimensionalization, but with little attention given to the problem of choosing the right scales and dimensionless parameters. In this paper, we illustrate the value of using asymptotics on a properly scaled dimensionless model, develop a set of guidelines that can be used to make good scaling choices, and offer advice for teaching these topics in differential equations or mathematical biology courses.     

Estimating survival and association in a semicompeting risks model

Summary .   In many follow-up studies, patients are subject to concurrent events. In this article, we consider semicompeting risks data as defined by Fine, Jiang, and Chappell (2001, Biometrika 88 , 907–919) where one event is censored by the other but not vice versa. The proposed model involves marginal survival functions for the two events and a parametric family of copulas for their dependency. This article suggests a general method for estimating the dependence parameter when the dependency is modeled with an Archimedean copula. It uses the copula-graphic estimator of Zheng and Klein (1995, Biometrika 82 , 127–138) for estimating the survival function of the nonterminal event, subject to dependent censoring. Asymptotic properties of these estimators are derived. Simulations show that the new methods work well with finite samples. The copula-graphic estimator is shown to be more accurate than the estimator proposed by Fine et al. (2001) ; its performances are similar to those of the self-consistent estimator of Jiang, Fine, Kosorok, and Chappell (2005, Scandinavian Journal of Statistics 33, 1–20). The analysis of a data set, emphasizing the estimation of characteristics of the observable region, is presented as an illustration.     

The existence of non-axisymmetric bilayer vesicle shapes predicted by the bilayer couple model

The existence of non-axisymmetric shapes with minimal bending energy is proved by means of a mathematical model. A parametric model is used; the shapes considered have an elliptical top view whilst their front view contour is described using Cassim ovals. Taking into account the bilayer couple model, the minimization of the membrane bending energy is performed at a constant membrane area A, a constant enclosed volume V and a constant difference between the two membrane leaflet areas A. It is shown that for certain sets of A, V and A the non-axisymmetric shapes calculated with the use of the parametric model have lower energy than the corresponding axisymmetric shapes obtained by the exact solution of the general variational problem. As an exact solution of the general variational problem for non-axisymmetric shapes would yield even lower energy, this indicates the existence of non-axisymmetric shapes with minimal bending energy in a region of the V/4A phase diagram.