Changes in interspike interval during propagation: Quantitative description

The intervals between nerve impulses can change substantially during propagation because of conduction velocity aftereffects of previous impulse activity. Effects of such changes on interval histograms and on statistical parameters of spike trains were evaluated for Poisson spike trains and for trains generated by a clock with added random delays. The distribution of short intervals was significantly changed during propagation for these spike trains. Substantial changes in serial correlation coefficients were found in trains with certain initial interval distributions. The relevance of these effects to neural coding is discussed.     

Neuronal spike trains and stochastic point processes. II. Simultaneous spike trains

The statistical analysis of two simultaneously observed trains of neuronal spikes is described, using as a conceptual framework the theory of stochastic point processes.The first statistical question that arises is whether the observed trains are independent; statistical techniques for testing independence are developed around the notion that, under the null hypothesis, the times of spike occurrence in one train represent random instants in time with respect to the other. If the null hypothesis is rejected—if dependence is attributed to the trains—the problem then becomes that of characterizing the nature and source of the observed dependencies. Statistical signs of various classes of dependencies, including direct interaction and shared input, are discussed and illustrated through computer simulations of interacting neurons. The effects of nonstationarities on the statistical measures for simultaneous spike trains are also discussed. For two-train comparisons of irregularly discharging nerve cells, moderate nonstationarities are shown to have little effect on the detection of interactions.Combining repetitive stimulation and simultaneous recording of spike trains from two (or more) neurons yields additional clues as to possible modes of interaction among the monitored neurons; the theory presented is illustrated by an application to experimentally obtained data from auditory neurons.A companion paper covers the analysis of single spike trains.     

The relationship of post-stimulus time and interval histograms to the timming characteristics of spike trains.

PST (post-stimulus time) and interval histograms computed from recorded spike trains are related to an average timing characteristics of the spike train. The exact nature of this relationship varies with recording parameters, interfering signals, the histogram bin width, and the duration of the measurement interval. This work describes the conditions under which a PST histogram can serve as an unbiased estimate of the ensemble average of a spike train's intensity and an interval histogram can serve as an unbiased estimate of the probability density function of the interspike intervals. Simulation studies are used to confirm the validity of the theoretical results. As an example of an application, these results are used to analyze recordings of singleunit activity in the eight cranial nerve.     

Sequential interval histogram analysis of non-stationary neuronal spike trains

The spike interval histogram, a commonly used tool for the analysis of neuronal spike trains, is evaluated as a statistical estimator of the probability density function (pdf) ofinterspike intervals. Using a mean square error criterion, it is concluded that a Parzen convolution estimate of the pdf is superior to the conventional histogram procedure. The Parzen estimate using a Gaussian weighting function reduces the number of intervals required to achieve a given error by a factor of 5–10. The Parzen estimation procedure has been implemented in the sequential interval histogram (SQIH) procedure for analysis of non-stationary spike trains. Segments of the spike train are defined using a moving window and the pdf for each segment is estimated sequentially. The procedure which we have found most practical is interactive with the user and utlizes the theoretical results of the error analysis as guidelines for the evolution of an estimation strategy. The SQIH procedure appears useful both as a criterion for stationarity and as a means to characterize non-stationary activity.Portions of this work were presented at the Symposium on Computer Technology in Neuroscience Research, West Virginia University Medical Center, Morgantown, West Virginia, USA, April, 1975.     

Analysis of point process signals applied to motor unit firing patterns: I. Superposition of independent spike trains

The aim of the present study was to examine whether statistical methods common for the analysis of point process signals could be applied to the electromyogram, in order to extract information concerning the physiological mechanisms involved. This was carried out on the assumption that the electromyogram can be treated as the superposition result of a number of point process signals, each representing the firing pattern of one motor unit. No correlated activity between the different spike trains was assumed at this stage. A digital model for the superposition of event sequences was constructed, assigning to the individual sequences a Gaussian interval distribution. The effects of varying the number of spike trains participating in the superposition process, and changing the mean rates of firing were explored. The statistical methods used in the analysis were serial correlation, event autocorrelation, and power spectrum studies. It has been found that serial correlograms of the superimposed processes may be helpful in detecting the number of spike trains involved in the superposition, whereas power spectrum studies are useful in determining the mean rates of firing of the individual sequences.     

神经元放电活动的统计信号分析——Ⅰ.神经元放电活动的统计性质的描述

本文结合随机点过程的统计理论和数字信号分析原理探讨了几种神经元放电活动的信号分析方法.用放电间隔(ISI)的均值函数描述取值的集中位置随时间变化的情况,用ISI的变异系数函数描述放电过程离散程度的变化,用标准化自协方差函数(NACVF)描述放电过程的自相关性质.给出NACVF了与期望密度的关系式.并用实例将这些方法与传统方法作之对比.     

The temporal organization of the functional connection of the motor cortex neurons in the cat]

Conditioned food-procuring response to time (2 minutes interval) was elaborated in cats, multiunit activity of the motor cortex being recorded. On the basis of single spike trains discriminated from the multiunit activity the cross-correlation histograms were built and the spikes composing their peaks were analysed in real time. This secondary analysis of the histograms allowed to ascertain the dynamics of functional connections between the neurons during the phase of active waiting according to the distribution of coincident impulses. A concentration of coincident impulses of simultaneously recorded cells was observed in different moment of time. In some neuronal pairs the concentration of coincident impulses was revealed to the end of the conditioned interval. The data obtained are considered as a manifestation of the conditioned reaction at the level of neuronal interaction.     

Neuronal spike trains and stochastic point processes. I. The single spike train

In a growing class of neurophysiological experiments, the train of impulses (“spikes”) produced by a nerve cell is subjected to statistical treatment involving the time intervals between spikes. The statistical techniques available for the analysis of single spike trains are described and related to the underlying mathematical theory, that of stochastic point processes, i.e., of stochastic processes whose realizations may be described as series of point events occurring in time, separated by random intervals. For single stationary spike trains, several orders of complexity of statistical treatment are described; the major distinction is that between statistical measures that depend in an essential way on the serial order of interspike intervals and those that are order-independent. The interrelations among the several types of calculations are shown, and an attempt is made to ameliorate the current nomenclatural confusion in this field. Applications, interpretations, and potential difficulties of the statistical techniques are discussed, with special reference to types of spike trains encountered experimentally. Next, the related types of analysis are described for experiments which involve repeated presentations of a brief, isolated stimulus. Finally, the effects of nonstationarity, e.g. long-term changes in firing rate, on the various statistical measures are discussed. Several commonly observed patterns of spike activity are shown to be differentially sensitive to such changes. A companion paper covers the analysis of simultaneously observed spike trains.     

Properties of synaptic noise in tonically active human motoneurons

The objective of these experiments was to determine the amount of synaptic noise on the cell membrane at various intervals after an action potential in a motoneuron firing at a specified frequency. Sources of noise such as variations in the level of voluntary drive were minimized by selecting only segments of the spike train in which the unit was running within prescribed frequency limits. The level of the membrane potential of the motoneuron during these intervals was determined using two test “pulses” (compound Ia excitatory postsynaptic potentials) of known amplitude. This enabled the probability of the membrane potential falling within a voltage “window” of known size at known times after the preceding spike to be determined. The probability density histograms showed that the fluctuations of membrane potential about a target interspike trajectory (i.e., the membrane noise) increased with time after the preceding spike. These fluctuations in the membrane potential can be accounted for by a one-dimensional “random walk” model of membrane noise. This model explains the salient features of the interval histograms, such as positive skewness at low target frequencies. A quantitative test of the model demonstrated its applicability to the motor pools of tibialis and masseter.     

Spike Discharge Patterns of Spontaneous and Continuously Stimulated Activity in the Cochlear Nucleus of Anesthetized Cats

Interspike interval histograms of spontaneous and stimulated activity were computed from spike discharges of single units in the cochlear nucleus. These histograms indicate that a number of different types of spontaneous discharge patterns exist in the nucleus. The type of spontaneous activity of a given unit is related to its activity in response to continuous tones. Correlations were found between the discharge patterns of units and their anatomical locations within the nucleus.     

Rate-coding of spinal motoneurons with high-frequency magnetic stimulation of human motor cortex

Rate-coding in spinal motoneurons was studied using high-frequency magnetic stimulation of the human motor cortex. The subject made a weak contraction to cause rhythmic (i.e., tonic) discharge of a single motor unit in flexor (or extensor) carpi radialis or tibialis anterior, while the motor cortical representation of that muscle was stimulated with brief trains of pulses from a Pyramid stimulator (4 Magstim units connected by 3 BiStim modules). An "m@n" stimulus train consisted of m number of pulses (1-4), with an interpulse interval (IPI) of n ms (1-6). Peristimulus time histograms were constructed for each stimulus condition of a given motor unit, and related to the average rectified surface electromyography (EMG) from that muscle. Surface EMG responses showed markedly more facilitation than single-pulse stimulation, with increasing numbers of pulses in the train; responses also tended to increase in magnitude for the longer IPI values (4 and 6 ms) tested. Motor-unit response probability increased in a manner comparable to that of surface EMG. In particular, motoneurons frequently responded twice to a given stimulus train. In addition to recruitment of new motor units, the increased surface EMG responses were, in part, a direct consequence of short-term rate-coding within the tonically discharging motoneuron. Our results suggest that human corticomotoneurons are capable of reliably following high-frequency magnetic stimulation rates, and that this activity pattern is carried over to the spinal motoneuron, enabling it to discharge at extremely high rates for brief periods of time, a pattern known to be optimal for force generation at the onset of a muscle contraction.     

The correlation grid: analysis of synchronous spiking in multi-dimensional spike train data and identification of feasible connection architectures

This paper presents a visualization technique specifically designed to support the analysis of synchronous firings in multiple, simultaneously recorded, spike trains. This technique, called the correlation grid, enables investigators to identify groups of spike trains, where each pair of spike trains has a high probability of generating spikes approximately simultaneously or within a constant time shift. Moreover, the correlation grid was developed to help solve the following reverse problem: identification of the connection architecture between spike train generating units, which may produce a spike train dataset similar to the one under analysis. To demonstrate the efficacy of this approach, results are presented from a study of three simulated, noisy, spike train datasets. The parameters of the simulated neurons were chosen to reflect the typical characteristics of cortical pyramidal neurons. The schemes of neuronal connections were not known to the analysts. Nevertheless, the correlation grid enabled the analysts to find the correct connection architecture for each of these three data sets.     

Statistical Analysis of Membrane Potential Fluctuations: Relation with Presynaptic Spike Train

In a study of integration at the single neuron level, the relationships between the postsynaptic membrane potential and the presynaptic spike train were analyzed. Fluctuations in membrane potential of neurons in the visceral ganglion of Aplysia were measured and described by histograms. The histogram estimates the probability density function of the membrane potential. Comparisons were made among histograms when there was no synaptic input, and when there was a single input in which variations were made in the PSP (postsynaptic potential) sign, i.e. excitatory or inhibitory, and arrival statistics, e.g. slow or fast, regular, Poisson-like, or patterned. This was examined in cells where the membrane potential was constant and in cells in which there was spontaneous pacemaker activity. The form of the histogram depended on whether the neuron was spontaneously quiescent or a pacemaker, or whether it received presynaptic input and, if it did, on the sign and temporal characteristics of such input. From such histograms the mean firing rate of output spike trains can be predicted; additional information of a temporal nature is required, however, to predict features of the interval structure of the output train. Suggestions are made concerning the way the nervous system might utilize the information summarized in the membrane potential histogram.     

A Pseudo-Markov Model for Series of Neuronal Spike Events

Spike trains of spontaneous neuronal activity in the rabbit brain are submitted to statistical analyses based on the following pseudo-Markov model. The nerve cell is supposed to alternate between a bursting and a resting state. The numbers of consecutive spikes within each state are assumed to be independent integer-valued random variables with discrete probability distributions. Given the state, the interspike intervals are independent real-valued random variables. The two state semi-Markov model is obtained as a special case when the discrete distributions are geometrical. Statistical second-order properties of recorded spike trains are compared with those predicted by the model on the basis of known first-order properties. For that purpose, serial correlation coefficients and intensity functions for spike trains produced by the model are computed. A comparison between observed and predicted results for the spontaneous activity of 17 brain cells yields a good fit in eight cells and discloses some salient features of the statistical structure in the activity of six other cells. By making it feasible to compute theoretical correlograms, the model may advance the understanding of empirical correlograms. The possibilities for integrating this statistical model of spike trains with a model of the mechanism of spike train production are discussed.     

Statistical characteristics of unit activity in spinal locomotor centers

A statistical analysis of unit activity in spinal locomotor centers was undertaken on immobilized thalamic cats at rest and during generation of efferent discharges. Activation of the spinal locomotor generator was accompanied by shortening of interspike intervals in the spike sequences of neurons and a decrease in their fluctuations. Histograms of interspike intervals became more symmetrical under these circumstances and there was a considerable increase in the number of neurons whose activity showed regular fluctuations on autocorrelation histograms. Spike trains at rest were characterized by dependence of successive intervals, which increased during efferent discharge generation. The possible mechanisms of modification of the time structure of unit activity in spinal locomotor centers during their activation are discussed.A. A. Bogomolets Institute of Physiology, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Neirofiziologiya, Vol. 12, No. 2, pp. 192–198, March–April, 1980.     

Use of `relative-phase' analysis to assess correlation between neuronal spike trains

 We propose a new method of studying the correlation between neuronal spike trains. This technique is based on the analysis of relative phase between two point processes. Relative phase here is defined as the relative timing difference between two spike trains normalized by the associated interspike interval of one cell. This phase measurement is intended to reveal the relative timing relationship between spike trains atdifferent firing rates. We apply this method to a numerical example and an example from two cerebellar neuronal spike trains of a behaving rat. The results are compared with classical cross-correlation analysis. We show that the technique can avoid some of the limitations of cross-correlation methods, reveal certain statistical dependencies that cannot be shown by cross correlation, and provide information as to the direction of influence between two spike trains. Received: 8 November 2001 / Accepted: 30 September 2002 / Published online: 24 January 2003 Correspondence to: Y. Chen (e-mail: chen@nsi.edu, Fax: + 1-858-626-2099) Acknowledgements. Research for this paper was supported by the Alafi Family Foundation and the Neurosciences Research Foundation.     

Analysis of extracellular potentials using an IBM PC

1. A system has been developed for using IBM PC-compatible computers in combination with a Grafitek Data Logging Interface to record spike trains on magentic discs for later analysis. 2. The times and amplitudes of spikes detected on two input channels are recorded, together with a third channel containing information on computer-generated stimuli and keyboard-activated event markers. In excess of 50,000 spikes can be recorded with a computer having 640 k of Random Access Memory. 3. The recorded spike trains can be reconstructed on the computer monitor and keyboard-controlled window discriminators can be used to select the spikes for analysis by amplitude. 4. The same recorded data can be analysed to produce displays of spike count against time, amplitude histograms, inter-spike interval histograms, peri-stimulus time histograms(PSTH), raster displays and auto- and cross-correlations between activity on the two channels. Each spike is identified by number, allowing easy location of the start and finish of the section of data to be analysed, and the PSTH, raster and correlation analyses allow pretriggering to investigate event occurring before stimulation. 5. The axes of the displays histograms can be adjusted to produce optimum displays, and hard copy can be produced on dot matrix printers or digital plotters. 6. Quantitative analysis enables comparison between different recordings and treatments.     

On the Classification of Experimental Data Modeled Via a Stochastic Leaky Integrate and Fire Model Through Boundary Values

We present a computational algorithm aimed to classify single unit spike trains on the basis of observed interspikes intervals (ISI). The neuronal activity is modeled with a stochastic leaky integrate and fire model and the inverse first passage time method is extended to the Ornstein-Uhlenbeck (OU) process. Differences between spike trains are detected in terms of the boundary shape. The proposed classification method is applied to the analysis of multiple single units recorded simultaneously in the thalamus and in the cerebral cortex of unanesthetized rats during spontaneous activity. We show the existence of at least three different firing patterns that could not be classified using the usual statistical indices. PACS: 87.19.La MSC: 60K30, 60J60, 65C40, 62P10     

Quantification of Clustering in Joint Interspike Interval Scattergrams of Spike Trains

Joint interval scattergrams are usually employed in determining serial correlations between events of spike trains. However, any inherent structures in such scattergrams that are often seen in experimental records are not quantifiable by serial correlation coefficients. Here, we develop a method to quantify clustered structures in any two-dimensional scattergram of pairs of interspike intervals. The method gives a cluster coefficient as well as clustering density function that could be used to quantify clustering in scattergrams obtained from first- or higher-order interval return maps of single spike trains, or interspike interval pairs drawn from simultaneously recorded spike trains. The method is illustrated using numerical spike trains as well as in vitro pairwise recordings of rat striatal tonically active neurons.