A model-based approach for the analysis of neuronal information transmission in multi-input and -output systems

We present a new method to characterize multi-input and output neuronal systems using information theory. To obtain a lower bound of transinformation we take three steps: (1) Estimation of the deterministic response to isolate components carrying stimulus information. The deviation of the original response from the deterministic estimate is defined as noise. (2) Coordinate transformation using PCA yields an uncorrelated representation. (3) Partial transinformation values are calculated independently either by Shannon's formula assuming normality or based on density estimation for arbitrary distributions. We investigate the performance of the algorithms using simulated data and discuss suitable parameter settings. The approach allows to evaluate the degree to which stimulus features are encoded. Its potential is illustrated by analyses of neuronal activity in cat primary visual cortex evoked by electrical retina stimulation.     

Information transmission by isolated frog muscle spindle

Transmission of sensory information was calculated for the isolated frog muscle spindle receptor, using Shannon's information measure. Sinusoidal movements, random noise stretches, and sinusoids with superimposed auxiliary noise were applied as stimuli. In addition, the static prestretch level of the intrafusal muscle bundle was adjusted between resting length (L0) and L0 + 600 micron, so that the analysis of the information transmission properties covered the entire dynamic range of the sensory receptor organ. Sinusoidal stretches below 2 Hz evoked smoothly modulated cycle histograms, which were approximately linearly related to the stimulating sinewave. The transinformation rates under these conditions were generally low (5-17 bit X s-1), regardless of the amplitude of the applied movement. Increasing prestretch enhanced the modulation depth of the cycle histograms considerably, but increased the transinformation rates by less than 10 bit X s-1. By contrast, sinusoids above 2 Hz evoked clearly nonlinear cycle histograms, because each action potential was firmly phase-locked to a small segment of the stretch cycle. Under these conditions the transinformation rates grew larger with increasing stimulus frequency and approached 130 bit X s-1 at 60 Hz. Small amplitude sinusoidal stretches, however, evoked considerable transinformation rates in the high frequency region only then, when the spindle receptor was extended to higher prestretch levels. Random stretches evoked transinformation rates between 5 and 30 bit X s-1 depending on both the prestretch level and the intensity of the noise stimulus. The linear response components carried only about 25% of the transinformation rates transmitted by both the linear and nonlinear response components. Auxiliary noise stimuli greatly improved the information transmission of sinusoidal stretches. For example, a pure sinusoid evoked 5 bit X s-1. Adding a noise signal with equal energy to the sinusoidal movement elicited 20 bit X s-1. This facilitation effect of auxiliary noise was restricted to low frequency sinusoidal stimuli. The present results are discussed with respect to the information transmission properties of various sensory systems evaluated by either the same or different information processing procedure as that used in the present study. The functional significance of high transinformation rates sent by the muscle spindle to the central nervous system is discussed with respect to motor control.     

Marginalized models for moderate to long series of longitudinal binary response data

Marginalized models (Heagerty, 1999, Biometrics 55, 688-698) permit likelihood-based inference when interest lies in marginal regression models for longitudinal binary response data. Two such models are the marginalized transition and marginalized latent variable models. The former captures within-subject serial dependence among repeated measurements with transition model terms while the latter assumes exchangeable or nondiminishing response dependence using random intercepts. In this article, we extend the class of marginalized models by proposing a single unifying model that describes both serial and long-range dependence. This model will be particularly useful in longitudinal analyses with a moderate to large number of repeated measurements per subject, where both serial and exchangeable forms of response correlation can be identified. We describe maximum likelihood and Bayesian approaches toward parameter estimation and inference, and we study the large sample operating characteristics under two types of dependence model misspecification. Data from the Madras Longitudinal Schizophrenia Study (Thara et al., 1994, Acta Psychiatrica Scandinavica 90, 329-336) are analyzed.     

Measures of transinformation for multiple input/single output neuronal systems

This paper presents a method for the calculation of the information transfer, or transinformation, in multiple input/single output neuronal systems. Our approach is an extension of an approach introduced by Eckhorn and Poepel in 1974. These authors computed the transinformation in single input/single output neuronal channels by regarding the spike (or stimulus) trains involved as finite Markov chains. The expressions for multiple input systems presented here are derived in close analogy to formulae in linear systems theory which show explicity the correlations between the different input channels. A number of equivalent forms for the transinformation are discussed.     

Gain of the ventilatory exercise stimulus: definition and meaning

The ratio G = delta VE/delta VCO2 where delta VA is change in ventilation and delta VCO2 is change in CO2 production, is often used to quantitate the ventilatory response to exercise and is the overall system gain (G). However, the actual variable of interest often is the gain for the exercise stimulus (GEX). Exercise stimulus refers to a stimulus or group of stimuli other than the mean levels of arterial PO2 (PaCO2), PCO2 (PaCO2), and pH (pHa) that act to increase ventilation during exercise. GEX will be equal to G only if the response to exercise is precisely isocapnic, normoxic, and without metabolic acidosis. A mathematical model was used to examine the relationship between G and GEX when 1) the response to exercise is not strictly isocapnic and 2) when the resting PaCO2 is shifted away from its normal value. It was found that 1) when the exercise response was not strictly isocapnic, G was a poor estimate of GEX and 2) when resting PaCO2 was changed while GEX wa assumed to remain constant, G was a function of the resting PaCO2. However, this dependence of G on resting PaCO2 is a system property that was caused by the nonlinear properties of the gas exchange processes and was not a fundamental property of the controller. It is concluded that G may not always be a good estimate of GEX and may lead to incorrect conclusions concerning the nature of the exercise stimulus.     

On the Transmission of Information through Sensory Neurons

Information about muscle length is transmitted to the cerebellum from muscle spindle receptors through the dorsal spinocerebellar tract (DSCT). The “transinformation” about muscle length in single DSCT fibers was calculated from steady-state spike trains by two different methods, assuming that the decoding mechanisms use a frequency code. By the first method, the number of distinguishable muscle lengths (and thus the transiformation) was determined from the rate of convergence of the mean frequency of firing (with increasing number of intervals). The observation time necessary to estimate the mean frequency of the impulse train with a certain accuracy was independent of the stretch level, even though the number of intervals necessary to make this estimate was different at high and low levels of stretch. By the second method an input frequency-output frequency matrix was calculated. The transinformations and the rate of transinformation was then calculated from this matrix. There was an acceptable agreement in the estimates of transinformation by the two methods. The rates of transinformation are significantly increased by the particular time structure of the discharge patterns of the nerve cells. Consequently, the loss of information due to the synaptic coupling is appreciably reduced.     

Estimation of a Nonlinear Density-Dependence Parameter for Wild Turkey

Abstract: Although previous research and theory has suggested that wild turkey (Meleagris gallopavo) populations may be subject to some form of density dependence, there has been no effort to estimate and incorporate a density-dependence parameter into wild turkey population models. To estimate a functional relationship for density dependence in wild turkey, we analyzed a set of harvest-index time series from 11 state wildlife agencies. We tested for lagged correlations between annual harvest indices using partial autocorrelation analysis. We assessed the ability of the density-dependent theta-Ricker model to explain harvest indices over time relative to exponential or random walk growth models. We tested the homogeneity of the density-dependence parameter estimates (θ) from 3 different harvest indices (spring harvest no. reported harvest/effort, survey harvest/effort) and calculated a weighted average based on each estimate's variance and its estimated covariance with the other indices. To estimate the potential bias in parameter estimates from measurement error, we conducted a simulation study using the theta-Ricker with known values and lognormally distributed measurement error. Partial autocorrelation function analysis indicated that harvest indices were significantly correlated only with their value at the previous time step. The theta-Ricker model performed better than the exponential growth or random walk models for all 3 indices. Simulation of known parameters and measurement error indicated a strong positive upward bias in the density-dependent parameter estimate, with increasing measurement error. The average density-dependence estimate, corrected for measurement error ranged 0.25 ≤ θC ≤ 0.49, depending on the amount of measurement error and assumed spring harvest rate. We infer that density dependence is nonlinear in wild turkey, where growth rates are maximized at 39-42% of carrying capacity. The annual yield produced by density-dependent population growth will tend to be less than that caused by extrinsic environmental factors. This study indicates that both density-dependent and density-independent processes are important to wild turkey population growth, and we make initial suggestions on incorporating both into harvest management strategies.     

Learning Multisensory Integration and Coordinate Transformation via Density Estimation

Sensory processing in the brain includes three key operations: multisensory integration—the task of combining cues into a single estimate of a common underlying stimulus; coordinate transformations—the change of reference frame for a stimulus (e.g., retinotopic to body-centered) effected through knowledge about an intervening variable (e.g., gaze position); and the incorporation of prior information. Statistically optimal sensory processing requires that each of these operations maintains the correct posterior distribution over the stimulus. Elements of this optimality have been demonstrated in many behavioral contexts in humans and other animals, suggesting that the neural computations are indeed optimal. That the relationships between sensory modalities are complex and plastic further suggests that these computations are learned—but how? We provide a principled answer, by treating the acquisition of these mappings as a case of density estimation, a well-studied problem in machine learning and statistics, in which the distribution of observed data is modeled in terms of a set of fixed parameters and a set of latent variables. In our case, the observed data are unisensory-population activities, the fixed parameters are synaptic connections, and the latent variables are multisensory-population activities. In particular, we train a restricted Boltzmann machine with the biologically plausible contrastive-divergence rule to learn a range of neural computations not previously demonstrated under a single approach: optimal integration; encoding of priors; hierarchical integration of cues; learning when not to integrate; and coordinate transformation. The model makes testable predictions about the nature of multisensory representations.     

Interspike interval dependency from arterial chemoreceptors

The carotid body impulse generator has been previously characterized as a Poisson-type random process. We examined the validity of this characterization by analyzing sinus nerve spike trains for interspike interval dependency. Fifteen single chemoreceptive afferents were recorded in vivo under hypoxic-hypercapnic conditions, and approximately 1,000 consecutive interspike intervals for each fiber were timed and analyzed for serial dependence. The same set of intervals placed in shuffled order served as a control series without serial dependence. The original spike interval trains showed significantly negative first-order serial correlation coefficients and less variability in joint interval distributions than did the shuffled interval trains. These results suggest that the chemoreceptor afferent train is not random and may reflect a negative feedback system operating within the carotid body that limits variation about a mean frequency.     

随机森林算法基本思想及其在生态学中的应用——以云南松分布模拟为例

通常来讲,生态学者对于解释生态关系、描述格局和过程、进行空间或时间预测比较感兴趣。这些工作可以通过模拟输出值(响应)与一些特征值(即解释变量)的关系来实现。然而,生态数据模拟遇到了挑战,这是因为响应变量和预测变量可能是连续变量或离散变量。需要解释的生态关系通常是非线性的,并且解释变量之间具有复杂的相互作用关系。响应变量和解释变量存在缺失值并不是不常有的现象,奇异值也经常出现在生态数据中。此外,生态学者通常希望生态模型即要易于建立又易要于解释。通常是利用多种统计方法来分析处理各种各样情景中出现的独特的生态问题,这些模型包括(多元)逻辑回归、线性模型、生存模型、方差分析等等。随机森林是一个可以处理所有这些问题的有效方法。随机森林可以用来做分类、聚类、回归和生存分析、评估变量的重要性、检测数据中的奇异值、对缺失数据进行插补等。鉴于随机森林本身在算法上的优势,将就随机森林在生态学中的应用进行总结,对建模过程进行概述,并以云南松分布模拟研究为例,对其主要功能特点进行案例展示。通过对随机森林的一般术语、概念和建模思想进行介绍,有利于读者掌握本方法的应用本质,可以预见随机森林在生态学研究中将得到更多的应用和发展。     

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.     

Retinal responses to white-noise modulated current stimuli

Electric current was injected into a rabbit's eye with white-noise modulations of the current amplitude. A variable D.C. bias was added to the whitenoise stimulus to study the effects of stimulus bias. For each bias level, the ERG response to the electrical stimulus was cross-correlated with the random stimulus to estimate first-and second-order Wiener kernels. The kernels indicated both linear and nonlinear characteristics of the Electrical ERG. The results for the zero biased stimulus are particularly relevant for clinical testing because the root mean square (RMS) level of the stimulus was less than 0.2 mA.This work was supported by NIH Grant Number EY03022     

Searching for Optimal Sensory Signals: Iterative Stimulus Reconstruction in Closed-Loop Experiments

Shaped by evolutionary processes, sensory systems often represent behaviorally relevant stimuli with higher fidelity than other stimuli. The stimulus dependence of neural reliability could therefore provide an important clue in a search for relevant sensory signals. We explore this relation and introduce a novel iterative algorithm that allows one to find stimuli that are reliably represented by the sensory system under study. To assess the quality of a neural representation, we use stimulus reconstruction methods. The algorithm starts with the presentation of an initial stimulus (e.g. white noise). The evoked spike train is recorded and used to reconstruct the stimulus online. Within a closed-loop setup, this reconstruction is then played back to the sensory system. Iterating this procedure, the newly generated stimuli can be better and better reconstructed. We demonstrate the feasibility of this method by applying it to auditory receptor neurons in locusts. Our data show that the optimal stimuli often exhibit pronounced sub-threshold periods that are interrupted by short, yet intense pulses. Similar results are obtained for simple model neurons and suggest that these stimuli are encoded with high reliability by a large class of neurons.     

Ca2+ Dynamics during Membrane Excitation of Green Alga Chara: Model Simulations and Experimental Data

Kinetic investigations of stimulus response coupling in the green alga Chara have revealed that an intermediate second messenger is formed in the process of membrane excitation. This second messenger links electrical stimulation to the mobilization of Ca2+ from internal stores. In the present work, the experimentally based kinetic model, which describes the stimulus-dependent production of the second messenger and Ca2+ mobilization, is combined with a model for inositol 1,4,5-trisphosphate (IP3)-and Ca2+-sensitive gating of a Ca2+-release channel in endomembranes of animal cells. The combination of models allows a good simulation of experimental data, including the all-or-none-type dependence of the Ca2+ response on stimulus duration and complex phase locking phenomena for the dependence of the Ca2+ response on stimulation frequency. The model offers a molecular explanation for the refractory phenomenon in Chara, assigning it to the life time of an inactive state of the Ca2+-release channel. The model furthermore explains the steep dependence of excitation on strength/duration of electrical stimulation as a consequence of an interplay of the dynamical variables in the model.     

Dynamics of freely oscillating and coupled hair cell bundles under mechanical deflection

In vitro, attachment to the overlying membrane was found to affect the resting position of the hair cell bundles of the bullfrog sacculus. To assess the effects of such a deflection on mechanically decoupled hair bundles, comparable offsets were imposed on decoupled spontaneously oscillating bundles. Strong modulation was observed in their dynamic state under deflection, with qualitative changes in the oscillation profile, amplitude, and characteristic frequency of oscillation seen in response to stimulus. Large offsets were found to arrest spontaneous oscillation, with subsequent recovery upon reversal of the stimulus. The dynamic state of the hair bundle displayed hysteresis and a dependence on the direction of the imposed offset. The coupled system of hair bundles, with the overlying membrane left on top of the preparation, also exhibited a dependence on offset position, with an increase in the linear response function observed under deflections in the inhibitory direction.     

Random Wiring,Ganglion Cell Mosaics,and the Functional Architecture of the Visual Cortex

The architecture of iso-orientation domains in the primary visual cortex (V1) of placental carnivores and primates apparently follows species invariant quantitative laws. Dynamical optimization models assuming that neurons coordinate their stimulus preferences throughout cortical circuits linking millions of cells specifically predict these invariants. This might indicate that V1’s intrinsic connectome and its functional architecture adhere to a single optimization principle with high precision and robustness. To validate this hypothesis, it is critical to closely examine the quantitative predictions of alternative candidate theories. Random feedforward wiring within the retino-cortical pathway represents a conceptually appealing alternative to dynamical circuit optimization because random dimension-expanding projections are believed to generically exhibit computationally favorable properties for stimulus representations. Here, we ask whether the quantitative invariants of V1 architecture can be explained as a generic emergent property of random wiring. We generalize and examine the stochastic wiring model proposed by Ringach and coworkers, in which iso-orientation domains in the visual cortex arise through random feedforward connections between semi-regular mosaics of retinal ganglion cells (RGCs) and visual cortical neurons. We derive closed-form expressions for cortical receptive fields and domain layouts predicted by the model for perfectly hexagonal RGC mosaics. Including spatial disorder in the RGC positions considerably changes the domain layout properties as a function of disorder parameters such as position scatter and its correlations across the retina. However, independent of parameter choice, we find that the model predictions substantially deviate from the layout laws of iso-orientation domains observed experimentally. Considering random wiring with the currently most realistic model of RGC mosaic layouts, a pairwise interacting point process, the predicted layouts remain distinct from experimental observations and resemble Gaussian random fields. We conclude that V1 layout invariants are specific quantitative signatures of visual cortical optimization, which cannot be explained by generic random feedforward-wiring models.     

Generalized additive distributed lag models: quantifying mortality displacement

There are a number of applied settings where a response is measured repeatedly over time, and the impact of a stimulus at one time is distributed over several subsequent response measures. In the motivating application the stimulus is an air pollutant such as airborne particulate matter and the response is mortality. However, several other variables (e.g. daily temperature) impact the response in a possibly non-linear fashion. To quantify the effect of the stimulus in the presence of covariate data we combine two established regression techniques: generalized additive models and distributed lag models. Generalized additive models extend multiple linear regression by allowing for continuous covariates to be modeled as smooth, but otherwise unspecified, functions. Distributed lag models aim to relate the outcome variable to lagged values of a time-dependent predictor in a parsimonious fashion. The resultant, which we call generalized additive distributed lag models, are seen to effectively quantify the so-called 'mortality displacement effect' in environmental epidemiology, as illustrated through air pollution/mortality data from Milan, Italy.     

Facilitating stable representations: serial dependence in vision

We tested whether the intervening time between multiple glances influences the independence of the resulting visual percepts. Observers estimated how many dots were present in brief displays that repeated one, two, three, four, or a random number of trials later. Estimates made farther apart in time were more independent, and thus carried more information about the stimulus when combined. In addition, estimates from different visual field locations were more independent than estimates from the same location. Our results reveal a retinotopic serial dependence in visual numerosity estimates, which may be a mechanism for maintaining the continuity of visual perception in a noisy environment.     

Unraveling proteins: a molecular mechanics study

An internal coordinate molecular mechanics study of unfolding peptide chains by external stretching has been carried out to predict the type of force spectra that may be expected from single-molecule manipulation experiments currently being prepared. Rather than modeling the stretching of a given protein, we have looked at the behavior of simple secondary structure elements (alpha-helix, beta-ribbon, and interacting alpha-helices) to estimate the magnitude of the forces involved in their unfolding or separation and the dependence of these forces on the way pulling is carried out as well as on the length of the structural elements. The results point to a hierarchy of forces covering a surprisingly large range and to important orientational effects in the response to external stress.     

Nonlinear kernels of the human ERG

In this paper we examine the use of a symmetric binary random stimulus for eliciting the ERG, and for calculating the first-order and second-order kernels of a nonlinear functional expansion of the response. We show that if the stimulus is represented in a non-dimensional form, then the units in which all kernels are measured are the same as the units used to measure the response, microvolts in the case of the ERG: further, contributions from all kernels to the response can be added without scale factors. We present the first-order and second-order kernels measured for a population of 15 normal subjects in a clinical setting. The measurements were made at various levels of adaptation ranging from photopic to scotopic conditions. The second-order kernels illustrate the processes of rapid adaptation (<100 ms) in the retina.This research was supported in part by Grants No. EY01526, EY01774, EY01775, and RR07003 from the National Institutes of Health