Estimating a state-space model from point process observations

A widely used signal processing paradigm is the state-space model. The state-space model is defined by two equations: an observation equation that describes how the hidden state or latent process is observed and a state equation that defines the evolution of the process through time. Inspired by neurophysiology experiments in which neural spiking activity is induced by an implicit (latent) stimulus, we develop an algorithm to estimate a state-space model observed through point process measurements. We represent the latent process modulating the neural spiking activity as a gaussian autoregressive model driven by an external stimulus. Given the latent process, neural spiking activity is characterized as a general point process defined by its conditional intensity function. We develop an approximate expectation-maximization (EM) algorithm to estimate the unobservable state-space process, its parameters, and the parameters of the point process. The EM algorithm combines a point process recursive nonlinear filter algorithm, the fixed interval smoothing algorithm, and the state-space covariance algorithm to compute the complete data log likelihood efficiently. We use a Kolmogorov-Smirnov test based on the time-rescaling theorem to evaluate agreement between the model and point process data. We illustrate the model with two simulated data examples: an ensemble of Poisson neurons driven by a common stimulus and a single neuron whose conditional intensity function is approximated as a local Bernoulli process.     

Model-based decoding, information estimation, and change-point detection techniques for multineuron spike trains

One of the central problems in systems neuroscience is to understand how neural spike trains convey sensory information. Decoding methods, which provide an explicit means for reading out the information contained in neural spike responses, offer a powerful set of tools for studying the neural coding problem. Here we develop several decoding methods based on point-process neural encoding models, or forward models that predict spike responses to stimuli. These models have concave log-likelihood functions, which allow efficient maximum-likelihood model fitting and stimulus decoding. We present several applications of the encoding model framework to the problem of decoding stimulus information from population spike responses: (1) a tractable algorithm for computing the maximum a posteriori (MAP) estimate of the stimulus, the most probable stimulus to have generated an observed single- or multiple-neuron spike train response, given some prior distribution over the stimulus; (2) a gaussian approximation to the posterior stimulus distribution that can be used to quantify the fidelity with which various stimulus features are encoded; (3) an efficient method for estimating the mutual information between the stimulus and the spike trains emitted by a neural population; and (4) a framework for the detection of change-point times (the time at which the stimulus undergoes a change in mean or variance) by marginalizing over the posterior stimulus distribution. We provide several examples illustrating the performance of these estimators with simulated and real neural data.     

Encoding and decoding spikes for dynamic stimuli

Naturally occurring sensory stimuli are dynamic. In this letter, we consider how spiking neural populations might transmit information about continuous dynamic stimulus variables. The combination of simple encoders and temporal stimulus correlations leads to a code in which information is not readily available to downstream neurons. Here, we explore a complex encoder that is paired with a simple decoder that allows representation and manipulation of the dynamic information in neural systems. The encoder we present takes the form of a biologically plausible recurrent spiking neural network where the output population recodes its inputs to produce spikes that are independently decodeable. We show that this network can be learned in a supervised manner by a simple local learning rule.     

PSdropout卷积神经网络在危化品巡检车中的应用

危化品仓储环境复杂多变,基于卷积神经网络的视觉巡检车需要快速的训练方法以便适用不同的环境,提高卷积神经网络的训练速度是当前亟待解决的问题。迅速在网络中提取有效的神经元,是提高算法训练速度的关键。传统的算法中,全链接层神经元的去留问题通常采用基于伯努力分布假设的Dropout方法,本文提出一种基于泊松分布的Dropout方法。理论上看,在充分利用神经元历史行为的基础上,基于泊松分布与基于伯努力分布的最大似然函数类似。实验结果表明,在保持正确率的情况下,训练提前收敛,节约了训练时间。     

Mutual information expansion for studying the role of correlations in population codes: how important are autocorrelations?

The role of correlations in the activity of neural populations responding to a set of stimuli can be studied within an information theory framework. Regardless of whether one approaches the problem from an encoding or decoding perspective, the main measures used to study the role of correlations can be derived from a common source: the expansion of the mutual information. Two main formalisms of mutual information expansion have been proposed: the series expansion and the exact breakdown. Here we clarify that these two formalisms have a different representation of autocorrelations, so that even when the total information estimated differs by less than 1%, individual terms can diverge. More precisely, the series expansion explicitly evaluates the informational contribution of autocorrelations in the count of spikes, that is, count autocorrelations, whereas the exact breakdown does not. We propose a new formalism of mutual information expansion, the Poisson exact breakdown, which introduces Poisson equivalents in order to explicitly evaluate the informational contribution of count autocorrelations with no approximation involved. Because several widely employed manipulations of spike trains, most notably binning and pooling, alter the structure of count autocorrelations, the new formalism can provide a useful general framework for studying the role of correlations in population codes.     

The scaling of winner-takes-all accuracy with population size

Empirical studies seem to support conflicting hypotheses with regard to the nature of the neural code. While some studies highlight the role of a distributed population code, others emphasize the possibility of a "single-best-cell" readout. One particularly interesting example of single-best-cell readout is provided by the winner-takes-all (WTA) approach. According to the WTA, every cell is characterized by one particular preferred stimulus, to which it responds maximally. The WTA estimate for the stimulus is defined as the preferred stimulus of the cell with the strongest response. From a theoretical point of view, not much is known about the efficiency of single-best-cell readout mechanisms, in contrast to the considerable existing theoretical knowledge on the efficiency of distributed population codes. In this work, we provide a basic theoretical framework for investigating single-best-cell readout mechanisms. We study the accuracy of the WTA readout. In particular, we are interested in how the WTA accuracy scales with the number of cells in the population. Using this framework, we show that for large neuronal populations, the WTA accuracy is dominated by the tail of the single-cell-response distribution. Furthermore, we find that although the WTA accuracy does improve when larger populations are considered, this improvement is extremely weak compared to other types of population codes. More precisely, we show that while the accuracy of a linear readout scales linearly with the population size, the accuracy of the WTA readout scales logarithmically with the number of cells in the population.     

The time-rescaling theorem and its application to neural spike train data analysis.

Measuring agreement between a statistical model and a spike train data series, that is, evaluating goodness of fit, is crucial for establishing the model's validity prior to using it to make inferences about a particular neural system. Assessing goodness-of-fit is a challenging problem for point process neural spike train models, especially for histogram-based models such as perstimulus time histograms (PSTH) and rate functions estimated by spike train smoothing. The time-rescaling theorem is a well-known result in probability theory, which states that any point process with an integrable conditional intensity function may be transformed into a Poisson process with unit rate. We describe how the theorem may be used to develop goodness-of-fit tests for both parametric and histogram-based point process models of neural spike trains. We apply these tests in two examples: a comparison of PSTH, inhomogeneous Poisson, and inhomogeneous Markov interval models of neural spike trains from the supplementary eye field of a macque monkey and a comparison of temporal and spatial smoothers, inhomogeneous Poisson, inhomogeneous gamma, and inhomogeneous inverse gaussian models of rat hippocampal place cell spiking activity. To help make the logic behind the time-rescaling theorem more accessible to researchers in neuroscience, we present a proof using only elementary probability theory arguments. We also show how the theorem may be used to simulate a general point process model of a spike train. Our paradigm makes it possible to compare parametric and histogram-based neural spike train models directly. These results suggest that the time-rescaling theorem can be a valuable tool for neural spike train data analysis.     

Representational accuracy of stochastic neural populations.

Fisher information is used to analyze the accuracy with which a neural population encodes D stimulus features. It turns out that the form of response variability has a major impact on the encoding capacity and therefore plays an important role in the selection of an appropriate neural model. In particular, in the presence of baseline firing, the reconstruction error rapidly increases with D in the case of Poissonian noise but not for additive noise. The existence of limited-range correlations of the type found in cortical tissue yields a saturation of the Fisher information content as a function of the population size only for an additive noise model. We also show that random variability in the correlation coefficient within a neural population, as found empirically, considerably improves the average encoding quality. Finally, the representational accuracy of populations with inhomogeneous tuning properties, either with variability in the tuning widths or fragmented into specialized subpopulations, is superior to the case of identical and radially symmetric tuning curves usually considered in the literature.     

Implications of neuronal diversity on population coding

In many cortical and subcortical areas, neurons are known to modulate their average firing rate in response to certain external stimulus features. It is widely believed that information about the stimulus features is coded by a weighted average of the neural responses. Recent theoretical studies have shown that the information capacity of such a coding scheme is very limited in the presence of the experimentally observed pairwise correlations. However, central to the analysis of these studies was the assumption of a homogeneous population of neurons. Experimental findings show a considerable measure of heterogeneity in the response properties of different neurons. In this study, we investigate the effect of neuronal heterogeneity on the information capacity of a correlated population of neurons. We show that information capacity of a heterogeneous network is not limited by the correlated noise, but scales linearly with the number of cells in the population. This information cannot be extracted by the population vector readout, whose accuracy is greatly suppressed by the correlated noise. On the other hand, we show that an optimal linear readout that takes into account the neuronal heterogeneity can extract most of this information. We study analytically the nature of the dependence of the optimal linear readout weights on the neuronal diversity. We show that simple online learning can generate readout weights with the appropriate dependence on the neuronal diversity, thereby yielding efficient readout.     

Fast population coding

Uncertainty coming from the noise in its neurons and the ill-posed nature of many tasks plagues neural computations. Maybe surprisingly, many studies show that the brain manipulates these forms of uncertainty in a probabilistically consistent and normative manner, and there is now a rich theoretical literature on the capabilities of populations of neurons to implement computations in the face of uncertainty. However, one major facet of uncertainty has received comparatively little attention: time. In a dynamic, rapidly changing world, data are only temporarily relevant. Here, we analyze the computational consequences of encoding stimulus trajectories in populations of neurons. For the most obvious, simple, instantaneous encoder, the correlations induced by natural, smooth stimuli engender a decoder that requires access to information that is nonlocal both in time and across neurons. This formally amounts to a ruinous representation. We show that there is an alternative encoder that is computationally and representationally powerful in which each spike contributes independent information; it is independently decodable, in other words. We suggest this as an appropriate foundation for understanding time-varying population codes. Furthermore, we show how adaptation to temporal stimulus statistics emerges directly from the demands of simple decoding.     

Applying the multivariate time-rescaling theorem to neural population models

Statistical models of neural activity are integral to modern neuroscience. Recently interest has grown in modeling the spiking activity of populations of simultaneously recorded neurons to study the effects of correlations and functional connectivity on neural information processing. However, any statistical model must be validated by an appropriate goodness-of-fit test. Kolmogorov-Smirnov tests based on the time-rescaling theorem have proven to be useful for evaluating point-process-based statistical models of single-neuron spike trains. Here we discuss the extension of the time-rescaling theorem to the multivariate (neural population) case. We show that even in the presence of strong correlations between spike trains, models that neglect couplings between neurons can be erroneously passed by the univariate time-rescaling test. We present the multivariate version of the time-rescaling theorem and provide a practical step-by-step procedure for applying it to testing the sufficiency of neural population models. Using several simple analytically tractable models and more complex simulated and real data sets, we demonstrate that important features of the population activity can be detected only using the multivariate extension of the test.     

Distortion of neural signals by spike coding

Analog neural signals must be converted into spike trains for transmission over electrically leaky axons. This spike encoding and subsequent decoding leads to distortion. We quantify this distortion by deriving approximate expressions for the mean square error between the inputs and outputs of a spiking link. We use integrate-and-fire and Poisson encoders to convert naturalistic stimuli into spike trains and spike count and inter-spike interval decoders to generate reconstructions of the stimulus. The distortion expressions enable us to compare these spike coding schemes over a large parameter space. We verify that the integrate-and-fire encoder is more effective than the Poisson encoder. The disparity between the two encoders diminishes as the stimulus coefficient of variation (CV) increases, at which point, the variability attributed to the stimulus overwhelms the variability attributed to Poisson statistics. When the stimulus CV is small, the interspike interval decoder is superior, as the distortion resulting from spike count decoding is dominated by a term that is attributed to the discrete nature of the spike count. In this regime, additive noise has a greater impact on the interspike interval decoder than the spike count decoder. When the stimulus CV is large, the average signal excursion is much larger than the quantization step size, and spike count decoding is superior.     

Faithful representation of stimuli with a population of integrate-and-fire neurons

We consider a formal model of stimulus encoding with a circuit consisting of a bank of filters and an ensemble of integrate-and-fire neurons. Such models arise in olfactory systems, vision, and hearing. We demonstrate that bandlimited stimuli can be faithfully represented with spike trains generated by the ensemble of neurons. We provide a stimulus reconstruction scheme based on the spike times of the ensemble of neurons and derive conditions for perfect recovery. The key result calls for the spike density of the neural population to be above the Nyquist rate. We also show that recovery is perfect if the number of neurons in the population is larger than a threshold value. Increasing the number of neurons to achieve a faithful representation of the sensory world is consistent with basic neurobiological thought. Finally we demonstrate that in general, the problem of faithful recovery of stimuli from the spike train of single neurons is ill posed. The stimulus can be recovered, however, from the information contained in the spike train of a population of neurons.     

Spike-frequency adapting neural ensembles: beyond mean adaptation and renewal theories

We propose a Markov process model for spike-frequency adapting neural ensembles that synthesizes existing mean-adaptation approaches, population density methods, and inhomogeneous renewal theory, resulting in a unified and tractable framework that goes beyond renewal and mean-adaptation theories by accounting for correlations between subsequent interspike intervals. A method for efficiently generating inhomogeneous realizations of the proposed Markov process is given, numerical methods for solving the population equation are presented, and an expression for the first-order interspike interval correlation is derived. Further, we show that the full five-dimensional master equation for a conductance-based integrate-and-fire neuron with spike-frequency adaptation and a relative refractory mechanism driven by Poisson spike trains can be reduced to a two-dimensional generalization of the proposed Markov process by an adiabatic elimination of fast variables. For static and dynamic stimulation, negative serial interspike interval correlations and transient population responses, respectively, of Monte Carlo simulations of the full five-dimensional system can be accurately described by the proposed two-dimensional Markov process.     

Optimal tuning widths in population coding of periodic variables

We study the relationship between the accuracy of a large neuronal population in encoding periodic sensory stimuli and the width of the tuning curves of individual neurons in the population. By using general simple models of population activity, we show that when considering one or two periodic stimulus features, a narrow tuning width provides better population encoding accuracy. When encoding more than two periodic stimulus features, the information conveyed by the population is instead maximal for finite values of the tuning width. These optimal values are only weakly dependent on model parameters and are similar to the width of tuning to orientation or motion direction of real visual cortical neurons. A very large tuning width leads to poor encoding accuracy, whatever the number of stimulus features encoded. Thus, optimal coding of periodic stimuli is different from that of nonperiodic stimuli, which, as shown in previous studies, would require infinitely large tuning widths when coding more than two stimulus features.     

Implications of noise and neural heterogeneity for vestibulo-ocular reflex fidelity

The vestibulo-ocular reflex (VOR) is characterized by a short-latency, high-fidelity eye movement response to head rotations at frequencies up to 20 Hz. Electrophysiological studies of medial vestibular nucleus (MVN) neurons, however, show that their response to sinusoidal currents above 10 to 12 Hz is highly nonlinear and distorted by aliasing for all but very small current amplitudes. How can this system function in vivo when single cell response cannot explain its operation? Here we show that the necessary wide VOR frequency response may be achieved not by firing rate encoding of head velocity in single neurons, but in the integrated population response of asynchronously firing, intrinsically active neurons. Diffusive synaptic noise and the pacemaker-driven, intrinsic firing of MVN cells synergistically maintain asynchronous, spontaneous spiking in a population of model MVN neurons over a wide range of input signal amplitudes and frequencies. Response fidelity is further improved by a reciprocal inhibitory link between two MVN populations, mimicking the vestibular commissural system in vivo, but only if asynchrony is maintained by noise and pacemaker inputs. These results provide a previously missing explanation for the full range of VOR function and a novel account of the role of the intrinsic pacemaker conductances in MVN cells. The values of diffusive noise and pacemaker currents that give optimal response fidelity yield firing statistics similar to those in vivo, suggesting that the in vivo network is tuned to optimal performance. While theoretical studies have argued that noise and population heterogeneity can improve coding, to our knowledge this is the first evidence indicating that these parameters are indeed tuned to optimize coding fidelity in a neural control system in vivo.     

Routing and scheduling in heterogeneous systems: a sample pathapproach

Consider the problem of routing customers to a set of K parallel servers that have different rates. Each server has a buffer with infinite capacity. The arrival process is general and the service times are assumed to be i.i.d. exponential random variables. Using sample path arguments, we show that, given any Bernoulli policy π, there exists another policy ρ which outperforms π by partially using a randomized version of a round-robin policy. Moreover, ρ is easily specified and implemented. We present extensions of our results to systems with finite capacities and service times that have an increasing hazard rate. Finally, a similar result is shown to hold in the context of scheduling customers from a set of K parallel queues     

Nonlinear population codes

Theoretical and experimental studies of distributed neuronal representations of sensory and behavioral variables usually assume that the tuning of the mean firing rates is the main source of information. However, recent theoretical studies have investigated the effect of cross-correlations in the trial-to-trial fluctuations of the neuronal responses on the accuracy of the representation. Assuming that only the first-order statistics of the neuronal responses are tuned to the stimulus, these studies have shown that in the presence of correlations, similar to those observed experimentally in cortical ensembles of neurons, the amount of information in the population is limited, yielding nonzero error levels even in the limit of infinitely large populations of neurons. In this letter, we study correlated neuronal populations whose higher-order statistics, and in particular response variances, are also modulated by the stimulus. Weask two questions: Does the correlated noise limit the accuracy of the neuronal representation of the stimulus? and, How can a biological mechanism extract most of the information embedded in the higher-order statistics of the neuronal responses? Specifically, we address these questions in the context of a population of neurons coding an angular variable. We show that the information embedded in the variances grows linearly with the population size despite the presence of strong correlated noise. This information cannot be extracted by linear readout schemes, including the linear population vector. Instead, we propose a bilinear readout scheme that involves spatial decorrelation, quadratic nonlinearity, and population vector summation. We show that this nonlinear population vector scheme yields accurate estimates of stimulus parameters, with an efficiency that grows linearly with the population size. This code can be implemented using biologically plausible neurons.     

Compound binomial processes in neural integration

Explores some of the properties of stochastic digital signal processing in which the input signals are represented as sequences of Bernoulli events. The event statistics of the resulting stochastic process may be governed by compound binomial processes, depending upon how the individual input variables to a neural network are stochastically multiplexed. Similar doubly stochastic statistics can also result from datasets which are Bernoulli mixtures, depending upon the temporal persistence of the mixture components at the input terminals to the network. The principal contribution of these results is in determining the required integration period of the stochastic signals for a given precision in pulsed digital neural networks.     

Finite population estimation under generalized linear model assistance

Finite population estimation is the overall goal of sample surveys. When information regarding auxiliary variables are available, one may take advantage of general regression estimators (GREG) to improve sample estimates precision. GREG estimators may be derived when the relationship between interest and auxiliary variables is represented by a normal linear model. However, in some cases, such as when estimating class frequencies or counting processes means, Bernoulli or Poisson models are more suitable than linear normal ones. This paper focuses on building regression type estimators under a model-assisted approach, for the general case in which the relationship between interest and auxiliary variables may be suitably described by a generalized linear model. The finite population distribution of the variable of interest is viewed as if generated by a member of the exponential family, which includes Bernoulli, Poisson, gamma and inverse Gaussian distributions, among others. The resulting estimator is a generalized linear model regression estimator (GEREG). Its general form and basic statistical properties are presented and studied analytically and empirically, using Monte Carlo simulation experiments. Three applications are presented in which the GEREG estimator shows better performance than the GREG one.