A Rapid and Efficient Estimation Procedure for the Negative Binomial Distribution

This paper presents a new noniterative procedure for estimating the parameters of a negative binomial distribution. The procedure uses the first moment equation and an equation based on the weighted sample mean, with weights ω_x∝ α^z. The selection of a value for α is examined. A simulation study has been carried out and also the method has been applied to the 35 data sets analysed by Martin and Katti (1965, Biometrics) in order to compare it with the method of moments and with the method of maximum likelihood (ML). We conclude that the new procedure has greater relative efficiency than the method of moments; it gives estimates which are consistently close to ML and are easy to calculate.     

Wavelet Transform-Based De-Noising for Two-Photon Imaging of Synaptic Ca2+ Transients

Postsynaptic Ca²⁺ transients triggered by neurotransmission at excitatory synapses are a key signaling step for the induction of synaptic plasticity and are typically recorded in tissue slices using two-photon fluorescence imaging with Ca²⁺-sensitive dyes. The signals generated are small with very low peak signal/noise ratios (pSNRs) that make detailed analysis problematic. Here, we implement a wavelet-based de-noising algorithm (PURE-LET) to enhance signal/noise ratio for Ca²⁺ fluorescence transients evoked by single synaptic events under physiological conditions. Using simulated Ca²⁺ transients with defined noise levels, we analyzed the ability of the PURE-LET algorithm to retrieve the underlying signal. Fitting single Ca²⁺ transients with an exponential rise and decay model revealed a distortion of τ_rise but improved accuracy and reliability of τ_decay and peak amplitude after PURE-LET de-noising compared to raw signals. The PURE-LET de-noising algorithm also provided a ∼30-dB gain in pSNR compared to ∼16-dB pSNR gain after an optimized binomial filter. The higher pSNR provided by PURE-LET de-noising increased discrimination accuracy between successes and failures of synaptic transmission as measured by the occurrence of synaptic Ca²⁺ transients by ∼20% relative to an optimized binomial filter. Furthermore, in comparison to binomial filter, no optimization of PURE-LET de-noising was required for reducing arbitrary bias. In conclusion, the de-noising of fluorescent Ca²⁺ transients using PURE-LET enhances detection and characterization of Ca²⁺ responses at central excitatory synapses.     

Statistical models of synaptic transmission evaluated using the expectation-maximization algorithm.

Amplitude fluctuations of evoked synaptic responses can be used to extract information on the probabilities of release at the active sites, and on the amplitudes of the synaptic responses generated by transmission at each active site. The parameters that describe this process must be obtained from an incomplete data set represented by the probability density of the evoked synaptic response. In this paper, the equations required to calculate these parameters using the Expectation-Maximization algorithm and the maximum likelihood criterion have been derived for a variety of statistical models of synaptic transmission. These models are ones where the probabilities associated with the different discrete amplitudes in the evoked responses are a) unconstrained, b) binomial, and c) compound binomial. The discrete amplitudes may be separated by equal (quantal) or unequal amounts, with or without quantal variance. Alternative models have been considered where the variance associated with the discrete amplitudes is sufficiently large such that no quantal amplitudes can be detected. These models involve the sum of a normal distribution (to represent failures) and a unimodal distribution (to represent the evoked responses). The implementation of the algorithm is described in each case, and its accuracy and convergence have been demonstrated.     

Modelling dendritic spines with the finite element method,investigating the impact of geometry on electric and calcic responses

Understanding the relationship between shape and function of dendritic spines is an elusive topic. Several modelling approaches have been used to investigate the interplay between spine geometry, calcium diffusion and electric signalling. We here use a second order finite element method to solve the Poisson–Nernst–Planck equations and describe electrodiffusion in dendritic spines. With this, we obtain relationships between dendritic geometry and calcic as well as electric responses to synaptic events. Our findings support the hypothesis that spine geometry plays a role shaping the electrical responses to synaptic events. Our method was also able to reveal the fine scale distribution of calcium in spines with irregular shapes.

     

On a Model for Estimating the Parameter of the Human Reproductive Process

Attempt has been made in this paper to estimate certain parameters (data pertaining to which are either not available or easily reportable) of the human reproductive process as the period of postpartum ammenorrhoea (P.P.A.), number of foetal wastages in between live births etc., using a truncated negative binomial probability model. In view of the hypothesis that the probability of foetal wastages varies from mother to mother, the truncated negative binomial distribution has been compounded by weighing with the best prior Beta distribution of the parameter. Estimation has been made by successive approximation using the method of moments.     

Studies on the Analysis of Fluorescence Decay Data by the Method of Moments

The method of moments, as presented by Isenberg and Dyson (1969; Biophys. J. 9:1337) has been shown to be a reliable way of obtaining the amplitudes and time constants of several simultaneously emitting species, even in the presence of an overlapping excitation. Recent improvements in the method include (a) a component incrementation test for determining the number of relaxations, (b) a procedure, which we call exponential depression, for dramatically improving convergence, and (c) a new algorithm for implementing the method of moments on a digital computer with a high degree of flexibility and efficiency. These improvements, as well as new general theory, are described and tested using both synthetic and real experimental data. Component incrementation consists of examining models with increasing numbers of exponential terms. Given adequate precision, we find that an analysis for N + 1 components, of data that are actually represented by N components, provides the correct amplitudes and time constants plus an N + 1 term with an insignificant amplitude. Exponential depression is a transformation in which the original excitation and fluorescence, E(t) and F(t), are multiplied by exp (-λt), where λ is an arbitrary parameter. While the convolution is invariant to this transformation, the proper choice of λ greatly reduces the number of iterations necessary to obtain the amplitudes and time constants and may even improve their accuracy. In addition, an appendix by John P. Mullooly presents a statistical analysis of the effect of counting error on the method of moments estimates of fluorescence decay parameters, applicable when data are obtained by the monophoton technique. Formulas are derived that give the approximate precision of the decay parameters for the general case of N exponential components, with calculational details for one and two component systems.     

Analysis and implications of equivalent uniform approximations of nonuniform unitary synaptic systems

Real synaptic systems consist of a nonuniform population of synapses with a broad spectrum of probability and response distributions varying between synapses, and broad amplitude distributions of postsynaptic unitary responses within a given synapse. A common approach to such systems has been to assume identical synapses and recover apparent quantal parameters by deconvolution procedures from measured evoked (ePSC) and unitary evoked postsynaptic current (uePSC) distributions. Here we explicitly consider nonuniform synaptic systems with both intra (type I) and intersynaptic (type II) response variability and formally define an equivalent system of uniform synapses in which both uePSC and ePSC amplitude distributions best approximate those of the actual nonuniform synaptic system. This equivalent system has the advantage of being fully defined by just four quantal parameters: ?, the number of equivalent synapses;p, the mean probability of quantal release; mu, mean; and sigma(2), variance of the uePSC distribution. We show that these equivalent parameters are weighted averages of intrinsic parameters and can be approximated by apparent quantal parameters, therefore establishing a useful analytical link between the apparent and intrinsic parameters. The present study extends previous work on compound binomial analysis of synaptic transmission by highlighting the importance of the product of p and mu, and the variance of that product. Conditions for a unique deconvolution of apparent uniform synaptic parameters have been derived and justified. Our approach does not require independence of synaptic parameters, such as p and mu from each other, therefore the approach will hold even if feedback (i.e., via retrograde transmission) exists between pre and postsynaptic signals. Using numerical simulations we demonstrate how equivalent parameters are meaningful even when there is considerable variation in intrinsic parameters, including systems where subpopulations of high- and low-release probability synapses are present, therefore even under such conditions the apparent parameters estimated from experiments would be informative.     

Use of the beta-binomial distribution in dominant-lethal testing for "weak mutagenic activity".

When the results of biological experiments are tested statistically for a possible difference between a treatment and its control, the statistical inferences are valid only if the statistical procedure is derived from a model that fits the experimental results satisfactorily. In this paper it is shown that a beta-binomial distribution provided a better fit than a binomial distribution when the data used were based on a large number of counts of dead and total implants on dominant-lethal tests on mice. This suggests that the probability P that an implant will die is not constant over the experimental units. Tests derived from the beta-biomial distribution have been used and their results compared with those of the tests based on the binomial or constant P assumption. The tests based on the binomial model are erroneously too severe when P is not constant in a group. The problem of knowing whether the males or the females should be considered as the experimental units is considered. In this paper, calculations are carried out for the two situations. This problem will be further studied by computer simulation and the result will be presented in a next paper.It is also shown that a negative binomial distribution could be fitted to the dead implant counts. No test based on this model was used because it ignores the total implants. No familiar distribution could be fitted to the total implant counts.     

Generalized Binomial Regression Model

The generalized negative binomial distribution has been found useful in fitting over-dispersed as well as under-dispersed count data. We define and study the generalized binomial regression model which is used to predict a count response variable affected by one or more explanatory variables. The methods of maximum likelihood and moments are given for estimating the model parameters. Approximate tests for the adequacy of the model are considered. The generalized binomial regression model has been applied to two observed data sets to which binomial regression model was applied earlier.     

Multinomial N-mixture models for removal sampling

Multinomial $N$-mixture models are commonly used to fit data from a removal sampling protocol. If the mixing distribution is negative binomial, the distribution of the counts does not appear to have been identified, and practitioners approximate the requisite likelihood by placing an upper bound on the embedded infinite sum. In this paper, the distribution which underpins the multinomial $N$-mixture model with a negative binomial mixing distribution is shown to belong to the broad class of multivariate negative binomial distributions. Specifically, the likelihood can be expressed in closed form as the product of conditional and marginal likelihoods and the information matrix shown to be block diagonal. As a consequence, the nature of the maximum likelihood estimates of the unknown parameters and their attendant standard errors can be examined and tests of the hypothesis of the Poisson against the negative binomial mixing distribution formulated. In addition, appropriate multinomial $N$-mixture models for data sets which include zero site totals can also be constructed. Two illustrative examples are provided.     

Behavioral evidence of a latency code for stimulus intensity in mormyrid electric fish

Mormryid electric fish (Gnathonemus petersii) respond to novel stimuli with an increase in the rate of the electric organ discharge (EOD). These novelty responses were used to measure the fish's ability to detect small changes in the amplitude and latency of an electrosensory stimulus. Responses were evoked in curarized fish in which the EOD was blocked but in which the EOD motor command continued to be emitted. An artificial EOD was provided to the fish at latencies of 2.4 to 14.4 ms following the EOD motor command.Novelty responses were evoked in response to transient changes in artificial EOD amplitude as small as 1% of baseline amplitude, and in latency as small as 0.1 ms. Changes in latency were effective only at baseline delays of less than 12.4 ms.The sensitivity to small changes in latency supports the hypothesis that latency is used as a code for stimulus intensity in the active electrolocation system of mormyrid fish. The results also indicate that a corollary discharge signal associated with the EOD motor command is used to measure latency.Abbreviations EOD electric organ discharge - ELL electrosensory lateral line lobe - epsp excitatory post synaptic potential     

Two New Discrete Distributions

CONSUL and JAIN (1973a) introduced a generalized Poisson distribution, which has applications in reliability theory and many biometric studies, and described some of its properties. Here we obtain two new distributions treating two of the parameters of the above distribution as random variables having gamma and absolute value distributions. One of the new distributions is related to the negative binomial distribution. Their moments also have been obtained.     

Network Adaptation Improves Temporal Representation of Naturalistic Stimuli in Drosophila Eye: II Mechanisms

Retinal networks must adapt constantly to best present the ever changing visual world to the brain. Here we test the hypothesis that adaptation is a result of different mechanisms at several synaptic connections within the network. In a companion paper (Part I), we showed that adaptation in the photoreceptors (R1–R6) and large monopolar cells (LMC) of the Drosophila eye improves sensitivity to under-represented signals in seconds by enhancing both the amplitude and frequency distribution of LMCs'' voltage responses to repeated naturalistic contrast series. In this paper, we show that such adaptation needs both the light-mediated conductance and feedback-mediated synaptic conductance. A faulty feedforward pathway in histamine receptor mutant flies speeds up the LMC output, mimicking extreme light adaptation. A faulty feedback pathway from L2 LMCs to photoreceptors slows down the LMC output, mimicking dark adaptation. These results underline the importance of network adaptation for efficient coding, and as a mechanism for selectively regulating the size and speed of signals in neurons. We suggest that concert action of many different mechanisms and neural connections are responsible for adaptation to visual stimuli. Further, our results demonstrate the need for detailed circuit reconstructions like that of the Drosophila lamina, to understand how networks process information.     

The Hyper-Negative Binomial Distribution

A generalized family of the negative binomial distribution is introduced in a paper by Srivastava, Yousry and Ahmed (1986). It is a solution of the difference equation and is called the hyper-negative binomial distribution. Certain properties including the moments of the distribution are presented. Moment estimators of the parameters are obtained and goodness of fit is illustrated for chromatid aberration data.     

On Testing Equality of Means of Correlated Variates with Incomplete Data

A statistic is proposed for testing the hypothesis of equality of the means of a bivariate normal distribution with unknown common variance and correlation coefficient when observations are missing on one of the variates. Expressions for the second and fourth central moments of the statistic are obtained. These moments are used to approximate the distribution of the statistic by a Student's t distribution under the null hypothesis. The powers of the test are computed and compared with those of the conventional paired t and the other known statistics.     

Distribution of Polymorphus minutus among its intermediate hosts

Hirsch R. P. 1979. Distribution of Polymorphus minutus among its intermediate hosts. International journal for Parasitology10: 243–248. In 1971, Crofton investigated patterns of distribution of Polymorphus minutus in the intermediate host, Gammarus pulex. Among his conclusions were: (1) P. minutus populations occur in patterns similar to negative binomial distributions, and (2) parasite-induced host mortality results in patterns similar to truncated (high end) negative binomial distributions. Those conclusions, however, were not tested by statistical analyses. To test Crofton's observations, Chi-square goodness of fit tests were applied to data used by Crofton and an additional two stations sampled by Hynes & Nicholas in 1963. Analyses were expanded to include five theoretical distributions, four patterns of host mortality and various rates of host mortality. Truncated forms of negative binomial, positive binomial and Poisson distributions were also investigated where nontruncated distributions failed to fit observed distributions. It was found that negative binomial distributions most frequently describe patterns of P. minutus distribution with the exception of one population described by Poisson and another by positive binomial distributions. Crofton's assumption that truncated distributions result from parasite-induced host mortality seems unlikely in light of those analyses.     

M-channels modulate the intrinsic excitability and synaptic responses of layer 2/3 pyramidal neurons in auditory cortex

Neurons in the auditory cortex are believed to utilize temporal patterns of neural activity to accurately process auditory information but the intrinsic neuronal mechanism underlying the control of auditory neural activity is not known. The slowly activating, persistent K⁺ channel, also called M-channel that belongs to the Kv7 family, is already known to be important in regulating subthreshold neural excitability and synaptic summation in neocortical and hippocampal pyramidal neurons. However, its functional role in the primary auditory cortex (A1) has never been characterized. In this study, we investigated the roles of M-channels on neuronal excitability, short-term plasticity, and synaptic summation of A1 layer 2/3 regular spiking pyramidal neurons with whole-cell current-clamp recordings in vitro. We found that blocking M-channels with a selective M-channel blocker, XE991, significantly increased neural excitability of A1 layer 2/3 pyramidal neurons. Furthermore, M-channels controled synaptic responses of intralaminar-evoked excitatory postsynaptic potentials (EPSPs); XE991 significantly increased EPSP amplitude, decreased the rate of short-term depression, and increased the synaptic summation. These results suggest that M-channels are involved in controlling spike output patterns and synaptic responses of A1 layer 2/3 pyramidal neurons, which would have important implications in auditory information processing.     

Synaptic contacts between embryonic Xenopus neurons and myotubes formed from a rat skeletal muscle cell line

Acetylcholine receptors (AChRs) accumulate at the junctional region during early development. In an attempt to characterize this process of AChR accumulation, we combined embryonic Xenopus neurons with myotubes formed from a rat skeletal muscle cell line. Xenopus neurons in culture are known to induce AChR accumulation in Xenopus muscles [Anderson, M. J., Cohen, M. W., and Zorychta, E. (1977). J. (London), 268, 731–756]. Rat myotubes, however, do not exhibit AChR accumulation in culture even when they are functionally innervated by the fetal rat spinal cord explant [Kidokoro, Y. (1980) Develop. Biol., 78, 231–241]. Establishment of synaptic transmission was examined electrophysiologically by recording synaptic potentials, while the distribution of AChR clusters was visualized using fluorescent α-bungarotoxin. Our results indicate that embryonic Xenopus neurons formed functional synaptic contacts but did not cause AChR accumulation in L6-myotubes. It seems that the ability of a nerve to cause AChR accumulation is separate from that to form the functional synapse. We also found that the mean amplitude of synaptic potentials in L6-myotubes interacted with Xenopus neurons was about half of that in L6-myotubes innervated by fetal rat spinal cord explants. Possible explanations for this finding are discussed.