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.     

A framework for evaluating pairwise and multiway synchrony among stimulus-driven neurons

Several authors have previously discussed the use of log-linear models, often called maximum entropy models, for analyzing spike train data to detect synchrony. The usual log-linear modeling techniques, however, do not allow time-varying firing rates that typically appear in stimulus-driven (or action-driven) neurons, nor do they incorporate non-Poisson history effects or covariate effects. We generalize the usual approach, combining point-process regression models of individual neuron activity with log-linear models of multiway synchronous interaction. The methods are illustrated with results found in spike trains recorded simultaneously from primary visual cortex. We then assess the amount of data needed to reliably detect multiway spiking.     

Attractor reliability reveals deterministic structure in neuronal spike trains

When periodic current is injected into an integrate-and-fire model neuron, the voltage as a function of time converges from different initial conditions to an attractor that produces reproducible sequences of spikes. The attractor reliability is a measure of the stability of spike trains against intrinsic noise and is quantified here as the inverse of the number of distinct spike trains obtained in response to repeated presentations of the same stimulus. High reliability characterizes neurons that can support a spike-time code, unlike neurons with discharges forming a renewal process (such as a Poisson process). These two classes of responses cannot be distinguished using measures based on the spike-time histogram, but they can be identified by the attractor dynamics of spike trains, as shown here using a new method for calculating the attractor reliability. We applied these methods to spike trains obtained from current injection into cortical neurons recorded in vitro. These spike trains did not form a renewal process and had a higher reliability compared to renewal-like processes with the same spike-time histogram.     

Including long-range dependence in integrate-and-fire models of the high interspike-interval variability of cortical neurons

Many different types of integrate-and-fire models have been designed in order to explain how it is possible for a cortical neuron to integrate over many independent inputs while still producing highly variable spike trains. Within this context, the variability of spike trains has been almost exclusively measured using the coefficient of variation of interspike intervals. However, another important statistical property that has been found in cortical spike trains and is closely associated with their high firing variability is long-range dependence. We investigate the conditions, if any, under which such models produce output spike trains with both interspike-interval variability and long-range dependence similar to those that have previously been measured from actual cortical neurons. We first show analytically that a large class of high-variability integrate-and-fire models is incapable of producing such outputs based on the fact that their output spike trains are always mathematically equivalent to renewal processes. This class of models subsumes a majority of previously published models, including those that use excitation-inhibition balance, correlated inputs, partial reset, or nonlinear leakage to produce outputs with high variability. Next, we study integrate-and-fire models that have (nonPoissonian) renewal point process inputs instead of the Poisson point process inputs used in the preceding class of models. The confluence of our analytical and simulation results implies that the renewal-input model is capable of producing high variability and long-range dependence comparable to that seen in spike trains recorded from cortical neurons, but only if the interspike intervals of the inputs have infinite variance, a physiologically unrealistic condition. Finally, we suggest a new integrate-and-fire model that does not suffer any of the previously mentioned shortcomings. By analyzing simulation results for this model, we show that it is capable of producing output spike trains with interspike-interval variability and long-range dependence that match empirical data from cortical spike trains. This model is similar to the other models in this study, except that its inputs are fractional-gaussian-noise-driven Poisson processes rather than renewal point processes. In addition to this model's success in producing realistic output spike trains, its inputs have long-range dependence similar to that found in most subcortical neurons in sensory pathways, including the inputs to cortex. Analysis of output spike trains from simulations of this model also shows that a tight balance between the amounts of excitation and inhibition at the inputs to cortical neurons is not necessary for high interspike-interval variability at their outputs. Furthermore, in our analysis of this model, we show that the superposition of many fractional-gaussian-noise-driven Poisson processes does not approximate a Poisson process, which challenges the common assumption that the total effect of a large number of inputs on a neuron is well represented by a Poisson process.     

The impact of spike timing variability on the signal-encoding performance of neural spiking models.

It remains unclear whether the variability of neuronal spike trains in vivo arises due to biological noise sources or represents highly precise encoding of temporally varying synaptic input signals. Determining the variability of spike timing can provide fundamental insights into the nature of strategies used in the brain to represent and transmit information in the form of discrete spike trains. In this study, we employ a signal estimation paradigm to determine how variability in spike timing affects encoding of random time-varying signals. We assess this for two types of spiking models: an integrate-and-fire model with random threshold and a more biophysically realistic stochastic ion channel model. Using the coding fraction and mutual information as information-theoretic measures, we quantify the efficacy of optimal linear decoding of random inputs from the model outputs and study the relationship between efficacy and variability in the output spike train. Our findings suggest that variability does not necessarily hinder signal decoding for the biophysically plausible encoders examined and that the functional role of spiking variability depends intimately on the nature of the encoder and the signal processing task; variability can either enhance or impede decoding performance.     

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.     

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.     

Estimating the temporal interval entropy of neuronal discharge

To better understand the role of timing in the function of the nervous system, we have developed a methodology that allows the entropy of neuronal discharge activity to be estimated from a spike train record when it may be assumed that successive interspike intervals are temporally uncorrelated. The so-called interval entropy obtained by this methodology is based on an implicit enumeration of all possible spike trains that are statistically indistinguishable from a given spike train. The interval entropy is calculated from an analytic distribution whose parameters are obtained by maximum likelihood estimation from the interval probability distribution associated with a given spike train. We show that this approach reveals features of neuronal discharge not seen with two alternative methods of entropy estimation. The methodology allows for validation of the obtained data models by calculation of confidence intervals for the parameters of the analytic distribution and the testing of the significance of the fit between the observed and analytic interval distributions by means of Kolmogorov-Smirnov and Anderson-Darling statistics. The method is demonstrated by analysis of two different data sets: simulated spike trains evoked by either Poissonian or near-synchronous pulsed activation of a model cerebellar Purkinje neuron and spike trains obtained by extracellular recording from spontaneously discharging cultured rat hippocampal neurons.     

脉冲神经元的信息处理

介绍累积放电脉冲神经元的数学描述;讨论脉冲神经元如何将激励信号转化为脉冲序列;讨论脉冲神经元如何将输入脉冲序列转化为输出脉冲序列。实验结果表明脉冲神经元具有很好的信息表示能力、信号鉴别能力和图像信号重构能力。给出利用脉冲神经网络进行图像信号处理的方法。     

Spatiotemporal spike encoding of a continuous external signal

Interspike intervals of spikes emitted from an integrator neuron model of sensory neurons can encode input information represented as a continuous signal from a deterministic system. If a real brain uses spike timing as a means of information processing, other neurons receiving spatiotemporal spikes from such sensory neurons must also be capable of treating information included in deterministic interspike intervals. In this article, we examine functions of neurons modeling cortical neurons receiving spatiotemporal spikes from many sensory neurons. We show that such neuron models can encode stimulus information passed from the sensory model neurons in the form of interspike intervals. Each sensory neuron connected to the cortical neuron contributes equally to the information collection by the cortical neuron. Although the incident spike train to the cortical neuron is a superimposition of spike trains from many sensory neurons, it need not be decomposed into spike trains according to the input neurons. These results are also preserved for generalizations of sensory neurons such as a small amount of leak, noise, inhomogeneity in firing rates, or biases introduced in the phase distributions.     

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.     

Supervised Learning of Logical Operations in Layered Spiking Neural Networks with Spike Train Encoding

Few algorithms for supervised training of spiking neural networks exist that can deal with patterns of multiple spikes, and their computational properties are largely unexplored. We demonstrate in a set of simulations that the ReSuMe learning algorithm can successfully be applied to layered neural networks. Input and output patterns are encoded as spike trains of multiple precisely timed spikes, and the network learns to transform the input trains into target output trains. This is done by combining the ReSuMe learning algorithm with multiplicative scaling of the connections of downstream neurons. We show in particular that layered networks with one hidden layer can learn the basic logical operations, including Exclusive-Or, while networks without hidden layer cannot, mirroring an analogous result for layered networks of rate neurons. While supervised learning in spiking neural networks is not yet fit for technical purposes, exploring computational properties of spiking neural networks advances our understanding of how computations can be done with spike trains.     

A very simple spiking neuron model that allows for modeling of large, complex systems

This letter introduces a biologically inspired very simple spiking neuron model. The model retains only crucial aspects of biological neurons: a network of time-delayed weighted connections to other neurons, a threshold-based generation of action potentials, action potential frequency proportional to stimulus intensity, and interneuron communication that occurs with time-varying potentials that last longer than the associated action potentials. The key difference between this model and existing spiking neuron models is its great simplicity: it is basically a collection of linear and discontinuous functions with no differential equations to solve. The model's ability to operate in a complex network was tested by using it as a basis of a network implementing a hypothetical echolocation system. The system consists of an emitter and two receivers. The outputs of the receivers are connected to a network of spiking neurons (using the proposed model) to form a detection grid that acts as a map of object locations in space. The network uses differences in the arrival times of the signals to determine the azimuthal angle of the source and time of flight to calculate the distance. The activation patterns observed indicate that for a network of spiking neurons, which uses only time delays to determine source locations, the spatial discrimination varies with the number and relative spacing of objects. These results are similar to those observed in animals that use echolocation.     

Finding the event structure of neuronal spike trains

Neurons in sensory systems convey information about physical stimuli in their spike trains. In vitro, single neurons respond precisely and reliably to the repeated injection of the same fluctuating current, producing regions of elevated firing rate, termed events. Analysis of these spike trains reveals that multiple distinct spike patterns can be identified as trial-to-trial correlations between spike times (Fellous, Tiesinga, Thomas, & Sejnowski, 2004 ). Finding events in data with realistic spiking statistics is challenging because events belonging to different spike patterns may overlap. We propose a method for finding spiking events that uses contextual information to disambiguate which pattern a trial belongs to. The procedure can be applied to spike trains of the same neuron across multiple trials to detect and separate responses obtained during different brain states. The procedure can also be applied to spike trains from multiple simultaneously recorded neurons in order to identify volleys of near-synchronous activity or to distinguish between excitatory and inhibitory neurons. The procedure was tested using artificial data as well as recordings in vitro in response to fluctuating current waveforms.     

Analysis and modelling of variability and covariability of population spike trains across multiple time scales

As multi-electrode and imaging technology begin to provide us with simultaneous recordings of large neuronal populations, new methods for modelling such data must also be developed. We present a model of responses to repeated trials of a sensory stimulus based on thresholded Gaussian processes that allows for analysis and modelling of variability and covariability of population spike trains across multiple time scales. The model framework can be used to specify the values of many different variability measures including spike timing precision across trials, coefficient of variation of the interspike interval distribution, and Fano factor of spike counts for individual neurons, as well as signal and noise correlations and correlations of spike counts across multiple neurons. Using both simulated data and data from different stages of the mammalian auditory pathway, we demonstrate the range of possible independent manipulations of different variability measures, and explore how this range depends on the sensory stimulus. The model provides a powerful framework for the study of experimental and surrogate data and for analyzing dependencies between different statistical properties of neuronal populations.     

On the relation between encoding and decoding of neuronal spikes

Neural coding is a field of study that concerns how sensory information is represented in the brain by networks of neurons. The link between external stimulus and neural response can be studied from two parallel points of view. The first, neural encoding, refers to the mapping from stimulus to response. It focuses primarily on understanding how neurons respond to a wide variety of stimuli and constructing models that accurately describe the stimulus-response relationship. Neural decoding refers to the reverse mapping, from response to stimulus, where the challenge is to reconstruct a stimulus from the spikes it evokes. Since neuronal response is stochastic, a one-to-one mapping of stimuli into neural responses does not exist, causing a mismatch between the two viewpoints of neural coding. Here we use these two perspectives to investigate the question of what rate coding is, in the simple setting of a single stationary stimulus parameter and a single stationary spike train represented by a renewal process. We show that when rate codes are defined in terms of encoding, that is, the stimulus parameter is mapped onto the mean firing rate, the rate decoder given by spike counts or the sample mean does not always efficiently decode the rate codes, but it can improve efficiency in reading certain rate codes when correlations within a spike train are taken into account.     

What can a neuron learn with spike-timing-dependent plasticity?

Spiking neurons are very flexible computational modules, which can implement with different values of their adjustable synaptic parameters an enormous variety of different transformations F from input spike trains to output spike trains. We examine in this letter the question to what extent a spiking neuron with biologically realistic models for dynamic synapses can be taught via spike-timing-dependent plasticity (STDP) to implement a given transformation F. We consider a supervised learning paradigm where during training, the output of the neuron is clamped to the target signal (teacher forcing). The well-known perceptron convergence theorem asserts the convergence of a simple supervised learning algorithm for drastically simplified neuron models (McCulloch-Pitts neurons). We show that in contrast to the perceptron convergence theorem, no theoretical guarantee can be given for the convergence of STDP with teacher forcing that holds for arbitrary input spike patterns. On the other hand, we prove that average case versions of the perceptron convergence theorem hold for STDP in the case of uncorrelated and correlated Poisson input spike trains and simple models for spiking neurons. For a wide class of cross-correlation functions of the input spike trains, the resulting necessary and sufficient condition can be formulated in terms of linear separability, analogously as the well-known condition of learnability by perceptrons. However, the linear separability criterion has to be applied here to the columns of the correlation matrix of the Poisson input. We demonstrate through extensive computer simulations that the theoretically predicted convergence of STDP with teacher forcing also holds for more realistic models for neurons, dynamic synapses, and more general input distributions. In addition, we show through computer simulations that these positive learning results hold not only for the common interpretation of STDP, where STDP changes the weights of synapses, but also for a more realistic interpretation suggested by experimental data where STDP modulates the initial release probability of dynamic synapses.     

A simple model of long-term spike train regularization

A simple model of spike generation is described that gives rise to negative correlations in the interspike interval (ISI) sequence and leads to long-term spike train regularization. This regularization can be seen by examining the variance of the kth-order interval distribution for large k (the times between spike i and spike i + k). The variance is much smaller than would be expected if successive ISIs were uncorrelated. Such regularizing effects have been observed in the spike trains of electrosensory afferent nerve fibers and can lead to dramatic improvement in the detectability of weak signals encoded in the spike train data (Ratnam & Nelson, 2000). Here, we present a simple neural model in which negative ISI correlations and long-term spike train regularization arise from refractory effects associated with a dynamic spike threshold. Our model is derived from a more detailed model of electrosensory afferent dynamics developed recently by other investigators (Chacron, Longtin, St.-Hilaire, & Maler, 2000;Chacron, Longtin, & Maler, 2001). The core of this model is a dynamic spike threshold that is transiently elevated following a spike and subsequently decays until the next spike is generated. Here, we present a simplified version-the linear adaptive threshold model-that contains a single state variable and three free parameters that control the mean and coefficient of variation of the spontaneous ISI distribution and the frequency characteristics of the driven response. We show that refractory effects associated with the dynamic threshold lead to regularization of the spike train on long timescales. Furthermore, we show that this regularization enhances the detectability of weak signals encoded by the linear adaptive threshold model. Although inspired by properties of electrosensory afferent nerve fibers, such regularizing effects may play an important role in other neural systems where weak signals must be reliably detected in noisy spike trains. When modeling a neuronal system that exhibits this type of ISI correlation structure, the linear adaptive threshold model may provide a more appropriate starting point than conventional renewal process models that lack long-term regularizing effects.     

基于脉冲神经网络的红外目标提取

模拟生物信息处理机制,设计了一种用于红外目标提取的脉冲神经网络(SNN)。首先,利用输入层脉冲神经元将激励图像转化为脉冲序列;其次,采用中间层脉冲神经元输出脉冲的密度编码红外图像目标的轮廓像素和非目标轮廓像素;最后,根据输出层神经元输出脉冲的密度是否超过阈值提取红外目标。实验结果表明,设计的脉冲神经网络具有较好的红外目标提取性能,并且符合生物视觉信息处理机制。     

面向图像识别的多层脉冲神经网络学习算法综述

相较于第1代和第2代神经网络,第3代神经网络的脉冲神经网络是一种更加接近于生物神经网络的模型,因此更具有生物可解释性和低功耗性。基于脉冲神经元模型,脉冲神经网络可以通过脉冲信号的形式模拟生物信号在神经网络中的传播,通过脉冲神经元的膜电位变化来发放脉冲序列,脉冲序列通过时空联合表达不仅传递了空间信息还传递了时间信息。当前面向模式识别任务的脉冲神经网络模型性能还不及深度学习,其中一个重要原因在于脉冲神经网络的学习方法不成熟,深度学习中神经网络的人工神经元是基于实数形式的输出,这使得其可以使用全局性的反向传播算法对深度神经网络的参数进行训练,脉冲序列是二值性的离散输出,这直接导致对脉冲神经网络的训练存在一定困难,如何对脉冲神经网络进行高效训练是一个具有挑战的研究问题。本文首先总结了脉冲神经网络研究领域中的相关学习算法,然后对其中主要的方法:直接监督学习、无监督学习的算法以及ANN2SNN的转换算法进行分析介绍,并对其中代表性的工作进行对比分析,最后基于对当前主流方法的总结,对未来更高效、更仿生的脉冲神经网络参数学习方法进行展望。